Universal Polynomials in Lambda Rings and the K-Theory of the Infinite Loop Space Tmf

Universal Polynomials in Lambda rings and the K-theory of the infinite loop space tmf

John R. Hopkinson

MMath. Mathematics University of Warwick, 2001

Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology

June 2006

@2006 John R. Hopkinson. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created.

Signature of Author -J \ RL Department of Mathematics / // May 1, 2006 (/ / Certified by ( / /Vt Michael J. Hopkins Pr or of Mathematics, Harvard University

_ A Thesis Supervisor

r i8 ,,,~~ Accepted by avel I. Etingof Professor of Mathematics Chairman, Department Committee on Graduate Students

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MASUCHUSE!TS iSUE OF TECHNOLOGY 1 J UN 0 5 2006 52LIBRARIES LIBRARIES Universal Polynomials in Lambda rings and the K-theory of the infinite loop space tmf by John R. Hopkinson

Submitted to Department of Mathematics on 4th May, 2006 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

ABSTRACT The algebraic structure of the K-theory of a topological space is described by the more general notion of a lambda ring. We show how computations in a lambda ring are facilitated by the use of Adams operations, which are ring homomorphisms, and apply this principle to understand the algebraic structure. In a torsion free ring the Adams operations completely determine the lambda ring. This principle can be used to determine the K-theory of an infinite loop space functorially in terms of the K-theory of the corresponding spectrum. In particular we obtain a description of the K-theory of the infinite loop space tmf in terms of Katz's ring of divided congruences of modular forms. At primes greater than 3 we can also relate this to a Hecke algebra.

Thesis Supervisor: Michael Hopkins Title: Professor of Mathematics, Harvard University

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2 Acknowledgements

When I thought about the people who I wanted to give my thanks to in these acknowledgements, they fell into two groups: those who made it possible to write this thesis and those who made it enjoyable. In no real order, they were all the following. My advisor Mike, for his many suggestions, but mostly his wit and wisdom. Mark and Haynes, for agreeing to sit on my committee, and also for many corrections and stylistic suggestions, and advice through my work at MIT. The Kennedy Memorial Trust and the department of maths for all their support. Josh, for various productive conversations. The UMO staff, who were always welcoming. Victor and Sigurdur, for making my qualifiers a pleasure, and for always being around to sign my registration forms. Harriet, who provided me with family in the US. My friends, who were willing to talk any time I was recharging my brain, but are too numerous to list. Finally, I want to thank the Bubblesss, for her helpful suggestions on the content; she'll appreciate that I put them here.

3 Contents

1 Introduction 5

2 A-rings 7 2.1 The polynomials Pm,nand P ...... 8 2.2 Adams operations ...... 10 2.3 The operation P ...... 12 2.4 The universal polynomials revisited ...... 15 2.5 K-Theory examples ...... 17

3 Bousfield's Theorem 20 3.1 Z/2-graded p-adic A-rings and Z/2-graded p-adic p-modules. 21 3.2 The free OP-ring functors ...... 23

3.3 K*(QoE; Zp) ...... 25 3.4 Examples ...... 28

4 The K-homology of tmf 32 4.1 K-cohomology and K-homology ...... 32 4.2 The ring of divided congruences ...... 33 4.3 Formulae for the generators of D ...... 36 4.4 D as Kotmf ...... 41 4.5 Diamond operators ...... 45

5 The K-cohomology of tmf and QO°tmf 46 5.1 Hecke Algebras and duality ...... 46 5.2 K*(Q°tmf; Zp) for p> 2 ...... 50 5.3 K*(Qf tmf; Z2) ...... 52

A Combinatorial proof of the formula for P3,, 55

B The canonical differential 62

C Tables of Pn and Pm,n 67

4 1 Introduction

The main goal of this thesis is to describe the K-cohomology of the infinite loop space Qftmf with p-adic coefficients. The algebraic structure of the K- theory of a space is what is called a A-ring. This structure can be concretely described if we consider the example of the K-theory of a finite CW-complex X. In this case the group K°X is obtained by considering the monoid of vector bundles on X under the operation of direct sum, and introducing additive inverses. Specifically we take the free abelian group generated by the monoid and quotient by terms of the form E + F- E F. A basic theorem, the splitting principle, says that any element of K°X can be represented by (the pullback of) a sum of line bundles. There are various operations on K°X corresponding to the various constructions that can be performed on vector bundles. For example it becomes a ring under the operation of tensor product. There is also the operation of taking the nth exterior power. On a sum of line bundles E = 1 Ei we have An E = E E E(n) n!oESm rcsm which, upon identifying tensor products in different orders, is the nth elementary symmetric function in the Ei (provided n < ra). The operations An are not ring homomorphisms (at least for n > 1). The additive behaviour is relatively simple to describe but the multiplicative behaviour is more complicated. However the theory of symmetric functions allows us to describe the action of An on a product. Specifically, it is a classical theorem that any symmetric polynomial can be written as a polynomial in the elementary symmetricfunctions. If we considera productE 0 F = (Z E/) 0 (Z Fj) then the nth exterior power will be symmetric under permutations of the Ei and under permutations of the Fj. It follows that An(E 0 F) will be a polynomial in the Ak(E) and At(F) for k, < n. This polynomial is independent of the number of summands, so long as there at least k, and is denoted Pn. Similarly we can consider what happens if we take Am (AnE). The result will be the mth elementary symmetric function in the sumnmands of the nth elementary symmetric function. This will still be symmetric in the original summands and so is a polynomial in AkE for k < mn. Again this is independent of the number of summands provided that there are at least mn, and the resulting polynomial is denoted Pm,n. In section 2 we will calculate some of these polynomials Pn and Pm,n, giving formulae when it is reasonable to write them down, and giving an effective method to compute them in general. The key idea is that to understand these operations that are not ring homomorphisms,we need to relate them to

5 operations which are. These operations are called the Adams operations, which can be defined topologically, but are more easily described algebraically. In our example of K°X, the Adams operation 4 k can be defined as the operation which sends a sum of line bundles to the sum of their kth (tensorial) power. In the theory of symmetric functions, the Adams operations are the power sums. The power sums form a rational basis for the ring of symmetric functions. This gives us a way to compute the polynomials P, and Pm,n, and is also the starting point for the theory that we will use to describe K*(fQ°tmf; Zp). The formula for the elementary symmetric functions in terms of power sums gives us the statement that in a torsion free ring, the Adams operations determine the A operations. The torsion free condition reflects the need to cancel coefficients. Bousfield generalized this theorem to describe Z/2-graded A-rings, which are an algebraic model of K*X. The torsion free condition can be re- moved if we introduce p-local, or p-adic coefficients, at the expense of replacing the Adams operation 4P (which is a ring homomorphism) with a non-additive operation OP. The point is that once we introduce the coefficients, we only need to worry about p-torsion. The operation 0P is related to 4P by the formula P = x +po(x).

When we try to write AP in terms of the Adams operations, we have a formula OP = pAp + -. and so OP is exactly what we want to get by cancelling the p. If we have OPin advance then we do not need to worry about p-torsion. Removing the torsion free requirement allows us to describe the p-adic K- theory of an infinite loop space in terms of the p-adic K-theory of the corresponding spectrum. The idea is that with p-adic coefficients, we have stable Adams operations Obk for k prime to p. Therefore if we introduce the operation OP freely we should obtain a A-ring isomorphic to the K-theory of the infinite loop space. In section 3 we give a detailed explanation of this theory which we will apply in later sections. We will apply this theory to the infinite loop space of the spectrum tmf. The property of tmf that we will use is that for an even periodic spectrum R, R A tmf can be identified with the stack of elliptic curves E together with an isomorphism of formal groups E GR. In section 4 we use this property to describe the K-homology of tmf in terms of Katz's ring of divided congruences. We can relate K-cohomology to K-homology using the Pontryagin duality functor and so this gives us a description of the K-cohomology of tmf. We expand upon this description in section 5 relating the K-cohomologyto the Hecke algebra corresponding to the ring of divided congruences. We then use Bousfield's theorem to obtain a description of the K-theory of QOOtmf.

6 2 A-rings

We begin with the definition of an abstract A-ring.

Definition 2.1 A A-structure on a commutative unital ring R is a sequence of maps )An for n > 0 which satisfy the following conditions:

1. A°(r) = 1 for all r R

2. A is the identity map S. A(1) = for n > 1 4. A"n(r+ s) = E=O Ak(r)An k(s) for all r, s E R 5. An(rs) = Pn(A'(r),...,An(r);A'(s),..., An(s)) for all r,s R 6. Am (An(r))= Pm,n(A'(r),... ,Am n(r)) for all r E R where Pn and Pm,nare certain universal polynomials with integer coefficients.

We can think of any A-ring as a power series ring as follows. We make 1 + R[[t]] into a A-ring by defining addition to be multiplication of power series, multiplication by the formula

00 oo 00 (1 + E rntn)( + E Sntn) = 1 + E Pn(rl, ..., rn; 81, , Sn)tn n=l n=l n=l and the A operations by the formula

00 00 An(l+ E rmtm)= 1+ E P,n(rl,..., rmn)tm). m=1 m=1 The fact that this does define a A-ring structure on 1 + R[[t]]is a consequence of certain identities among the universal polynomials (see [Tal69]). The conditions in the definition combine to show that the total A-operation At(r) = nl An(r)tn defines an injective A-homomorphism from R into 1 + R[[t]]. That is to say it is an identity preserving ring monomorphism which com- mutes with the A-operations. In particular the injectivity of the map follows from condition 2.

7 2.1 The polynomials Pm,n and Pn

The universal polynomials Pm,n and Pn that occur in the definition of a A-ring are somewhat mysterious. The polynomial Pm,n(el, ..., emn) can be described ([Tal69]) as the coefficient of tm in the product rIl

Lemma 2.1 For all m and p we have

(e;l1)-=P ) Z_(_1)k (p m_k)'

Proof We prove this by induction on p, noting that for p = 0 the formula reads 1 = 1. Now for p > 0 the binomial recurrence formula and induction give us

(m(r-i - 1 = r)(nP m m1 h))

=l (7)- ( (_1)k+lm ))

k=O _k

8 Proposition 2.1 With the convention that e = 1 we have for all n that n P2,n(el., e2n)= (-1)k+len-ken+k k=l1 Proof We can represent a symmetric polynomial by looking at single monomial of each type in the polynomial. For example x2x2x3 and x lx2x4 have the same type. In the product en-ken+k and P2,n each monomial that occurs has the 2 same type as x . x xi+l.. x2n-i for some i. To prove the proposition we will compare the coefficient of this expression that occurs on each side of 2 - the equation. The coefficient of x ... xi2xi+l... x2n-i in en-ken+k is zero if n- k < i and (n(-k-i) otherwise. To see this we simply note that the terms in en-k which can (and do) give this monomial must contain the indeterminates xl through xi but then the remaining n- k- i indeterminates can be any of the 2(n- i) indeterminates xi+l through x2n-i. The polynomial P2,n is obtained by taking the sum of all possible products of 2 distinct terms xs = Xil ... xi, and Xj = Xj*--xjn. Each index set I such that {1,2,...,i C I has a complementary set J such that xxj = X2... 2Xi+l ... X2ni . There are (2(n-_)) ways to choose a suitable index set I. However this would give us the term xIxj twice (once when we choose I and again when we choose J) so the 2 coefficient of x ... xxi+l... x 2ni in P2n is (2(n-i)) = (2(n-it)- 1). Therefore to prove the proposition it is sufficientto show that 2(n -i) 18 _ (1k 2(n -i) A_ d k 2(n-i) 12(n~-) = E(_1)k+1 =_ y (_1)k V,n-i-1 J nk=l ,n-kn-k-i J-i k=O--- ' This followsfrom the lemma with m = 2(n- i) and p = n- i- 1. 0

An alternative way to express the final calculation above is that

2 (-1)k+i -1I, n-k-i2(nk-i) = (1- 1)2(n -) + n-i i) /

The point here is that extending the sum to include k from -(n- i) to -1 gives the same terms repeated, then adding k = 0 gives exactly the binomial expansion of (1 - 1)2(n-i). The case m = 3 is also sufficiently tractable for us to compute.

Proposition 2.2 Let w =-2 Ppio.Lecok-+ i be the+ cube root of unity. Then for all n P3,n(e, te3n)= ~' c?^1,k_eejek lQ

9 wherec,s = 2R(wr) unless r = 0, t, 3t/2. If t > 0 then for r = 0, 3t/2 the value of ct,, is whilefor r = t it is 2R(wr)+ (-l)t+l. If t = r = 0 the coefficient is zero.

An alternative description of the coefficient of eleiek is that it is 1 if two of l - 1,j, k are equal and 2R (w ) otherwise, except when j = n when we must add (-)n-l+1 if < n, and have zero for e. We defer the proof of this proposition for the moment as a different approach will turn out to be much simpler, and indeed give us a more useable formula in general. This formula was discovered using similar reasoning to the m = 2 case and then a Maple program to compute the coefficients. A key point in the calculation is that we can obtain the nt h polynomial from the (n- )th by replacing every monomial elejek by

el+lej+lek+l and adding 0 c, ,3 nkeke3n-k. Indeed this description is easily seen to be equivalent to the one given above and the missing terms in P3,1 give the exceptions in the general case. The calculation reduces to a triangular linear system for the last 3n/2 coefficients which is actually reasonable to solve. In the appendix we give details of this calculation which involves many relations between binomial coefficients.

2.2 Adams operations

The difficulty in calculating the universal polynomials is in large part due to the fact that the A-operations are not ring homomorphisms. Indeed they are are not even group homomorphisms. If we think of our indeterminates as line bundles and elements of R as formal sums of line bundles then instead of taking the elementary symmetric functions to get the A operations we can take the power sums. This gives us the Adams operations 0p. To explain this let us recall some facts about symmetric functions from [Mac79]. With indeterminates x1,x 2,... we call the expression en= '' Xil, .. Xin il <"'

10 follows. For a partition p let mi = mi(/u) be the number of times that i occurs in the partition. Let l(p) be the length of the partition (that is the number of non-zero parts) and [I/ the sum of the parts. Define z = IJibl i"imi! and

ui1 = )!. Then en = E ( )n-lO')--1 zlP IAI=n

Pn= n =( ) use = (el,*--,en)- ]W=n If we think of the indeterminates xi as universal line bundles and the exterior powers as A operations then en corresponds to An and 'bn(xi) = xn. More precisely we have the following theorem. ([Mac79])

Theorem 2.1 If R is a A-ring then for every x R there is a unique A- homomorphism q :A R which maps el to x. This homomorphism maps en to An(x) and pn to bn(x).

Proof Once we define qx(el) = x then requiring ., to be a A-homomorphism forces 0$(en) = q$~(An(el))= An(x). This and the ring homomorphism requirement defines ox uniquely on A. The latter statement is the definition of the Adams operation. n

Another way to express this result is that A is the free A-ring in the single variable el. This theorem means that any formula in A is valid in any A-ring, with appropriate translation. We can apply this to perform calculations for the Adams operations. Firstly we find that the Adams operation can be expressed as the Newton polynomial Vn(A1, ..., An). This is the polynomial above which expresses the power sums in terms of the elementary symmetric functions. This polynomial can be calculated recursively from the relation

n = Alln-1 - A2 4/n-2 +... + (-1)nAn-1 4l + (-1)n+'nAn.

To prove this we note that in terms of the indeterminates xi this reads

En AdXi Ad Xi -d Xil XA 2 + * - zI =-zI l- I +±... In the right hand side the jth term has monomials of lengths j and j + 1, for j < n. The longer terms from the jth term exactly cancel the shorter ones from the (j + l)th. For j = n we only have terms of length n which are cancelled by

11 the terms from the (n- 1)th term. After all the cancellation all that remains on the right hand side is the terms of length 1 which is exactly the left hand side. Next we have the equations

,in X + E yj)= ZX + E yjn= on (Z Xi)+ Vpn(Zy ) and n(E Xi Yj) (Xiyj)n =,On (Zxi)+bn (Zyj) which show that on is a ring homomorphism. Similarly (on(EXi) = Om(EX') = ZXn= (EX) shows that bmebn= Vhmn.Also if p is a prime then the congruence ( xi)P E x mod p shows that V(x) xP mod p. This last property allows us to define a new operation OPon R by the formula OP(x)= - ( P One can write down formulae for OP(x+ y) and OP(xy)just using the fact that ?,p is a ring homomorphism. It also follows that P(1) = 0 and pk0P = pvk for all k. Conversely if we have an operation P which satisfies these formulae then the operation bPdefined by the usual equation is a ring homomorphism.

2.3 The operation OP

The Adams operations make most calculations in a A-ring much simpler. In this section we apply this principle to prove that if we have operations Ak for k prime to p then by introducing the operation OPwith its usual formal properties we obtain the Ak with k a multiple of p. To compute P(An ) we first compute OP(An ) using the basis of power sums for the ring of symmetric polynomials. The advantage of using the power sums instead of the elementary symmetric functions is that the Adams operations are described by the formula V(pn) = pp, which follows easily from the fact that '/P is a ring homomorphism. Combining this with the two inversion formulae gives us the formula that we want. The formula for OP(An ) will then allow us to prove Proposition 2.2.

Lemma 2.2 Modulo decomposables we have

op(An)- (1)(P-,)n,\pn

12 Proof Working mod decomposables we have pa = (-1)n- l An and so applying 'P we obtain 1(-1)nOP(An) = Ppn= (-1) 1 pnAn Therefore (-1)n-lpnOP(A n) = (-l1)Pn-lpnAP" which gives the result in a torsion free ring. However this is a universal formula so it must hold in general. []

For p = 3 we can be more specific.

2 ri/3 Proposition 2.3 Let w = e and define cr,8 to be 2R(wr) if r 7 0, s and 1 whenr =O, s. Then

03 (An) = Cj-i,k-iAiAAk. i+j+k=3n, iQ.'k

Proof To compute 03 we first compute the Adams operation 03. We have two different formulae that we can use. The first comes from the Newton formula:

)3(An) = (n) 3 + 3(A 3 (An ) - Al(An),2(An))

The second formula comes from the formulae for elementary symmetric functions and power sums: 3 ( n) = E (-1)n-I(A)z.-lp3 IjA=n /() \4(A)

= Z (-1)n-()zo II( (-1)3i-z()_3i UL) ) I[l=n i>0 }v=3i The first formula allows us to simplify the calculation of the second, based on our knowledge of the universal polynomials Pm,n. Firstly it tells us that OP(An ) has degree 3. Secondly from the remarks in the appendix we see that all the monomials of degree 3 are of the form Cl,j,kAl+l Aj+lAk+l where clj,kAlXjAkis a monomial in ?p3(An-1). We should notice that the formula for the coefficients in the theorem is consistent with this. The conclusion that we draw from this is that in the second formula we only need to consider the terms which give us monomials of degree 2 or less. The higher degree monomials must either cancel out or be known. This means that in the second formula we can restrict our partitions to have length no more than 2. To perform this calculation let us first assume that n is odd. The partitions A are of the form (k, n- k) or simply (n). From the definition we calculate that

13 Z(k,n-k) = k(n - k) and Z(n) = n. Therefore from the formula for elementary symmetric functions in terms of power sums we have

n o3 (xn)= (-) -- n + E (T_ kP3(n-k) P+ , k=1 k(n - k)' 3kp3(n-k) ignoring terms of higher degree. Similarly from the formula for power sums in terms of elementary symmetric functions we find that in Pm the terms of degree 2 are m-l m (-1)mkAm-kL L~2.j~~~ + MA~m 2/2)2' k=l where the last term is zero for m odd. We also have the degree 1 term m(-1)m-lAm. Putting these expression together we find that in 'b 3(An ) the terms of degree less than 3 are

3n L2.J - (3n(-).-,n. + 3n(l)3. Z AA.-k)+ '/1, k=l

n E k~n---k)°3k-1)3-kk=1~~ ~ ~~~~~~= ~-k

2 L 3a 3 3 3 = 3) - 3 y A;)a -k + 9 SA k (n-k) k=1 k=l If we divide by 3 we see that the coefficient of AkA3n- k in 03 (An) is 1 for k = 0, -1 for k > 0 but not a multiple of 3, and -1 + 3 = 2 for k > 0 a multiple of 3. In other words it is k,3n-k. This completes the proof for n odd.

For n even the only difference is that the term P32n/2 comes with a coefficient 2 2 1/2 which means that the coefficient C3n/2,3n/2 of (3n/ ) is not 2 but 1 as claimed. O

As a corollary to this theorem we can prove proposition 2.2. Proof of proposition 2.2 From the Newton formula and the calculation of A2 (An) we have

3 (n) 03 (n) = A'(An)A2 (An) n = An(](-1)k+lAn-kAn+k) k=l = (-1)n+1An(A2n)+ ...

14 where we ignore the terms of degree 3. If we compare the formulae for the 3 coefficients in P3,,, and (An) we see that the difference is indeed just the (-1)n+lA(A 2 n) that we have here. ° It is interesting to compare this proof to the combinatorial calculation given in the appendix. The description of the coefficients as real parts of powers of a cube root of unity, which is so crucial to the latter calculation, bears no obvious relation to how they arise here. Indeed in the above we only need to deal with real numbers.

2.4 The universal polynomials revisited

The Adams operations give us a much more effective way to compute the universal polynomials P,n and Pn. Let us first consider the polynomial Pn which gives An(rs) in terms of Ai(r) and Ai(s) for i < n. To compute this we look at %bn(rs). On the one hand this is v(Al(rs), ..., An(rs)) while on the other it is bn (r),O'(s) = vn(A(r),..., An(r))vn(Al(s),..., An(s)). Setting these two expressions equal and rearranging allows us to write An(rs) in terms of Ai(r), Ai(s), and Ai(rs) for j < n. This gives a recursive algorithm to compute Pn for n as large as we please. In the appendix we tabulate values of Pn for n < 10. A general formula would be difficult to write down but from the first few cases we might conjecture that the sum of the coefficients is always zero, for n > 1. This indeed is the case as we now show.

Lemma 2.3 The sum of the coefficientsin the nth Newton Polynomial v is (_1 )n+l.

Proof We can calculate the sum of the coefficients of vn(ei, -...,en) by evaluating Ei= xTnat values of the indeterminates xi such that the ei all take the value 2 1. This means that -xi should be a root of 1 + x + x + . + xn for each i. These roots, which we denote wi, are the (n + 1)th roots of unity, except for 1. Then Exi = (l) n Elin = (-l) n EW 1= (-l)n Ewi = (-)n+l

since 1 + E wi = 0.

Theorem 2.2 The sum of the coefficients in Pn is zero for n > 1 and 1 for n = 1.

15 Proof The case n = 1 holds since P1 = elfi. Now suppose that the result holds for all k < n for some n > 2. The equality vn(\(rs), ..., An(rs))= n(Al(r),. n(r))zn(Al() An()) and the fact that the sum of the coefficients of a product of polynomials is the product of the sums shows that the sum of the coefficients of the polynomial vn(Al(rs),..., An(rs)) is ((-1)n+l) 2 = 1 for each n. However we also have vn(Al(rs),..., A(rs)) = Al(rs)nl(Al(rs) A l(rs)) -A2(rs)Vn2(A(rs),,An-2(rs)) + -- (-1)l nAn(rs). By induction when we sum coefficients this reads 1 = 1 * 1-0*1+0*1 +-.-+(-)n+ln*x where x is the unknown sum of coefficients in An(rs)= p (Al(r) An(r);A(s) A(s)) Solving this gives x = 0 as claimed. [

We now turn to the polynomials Pmn. In calculating P3,n we used the best method available to us. That was firstly to calculate O'm(An) using the inversion formulae and then rearrange the Newton formula to calculate Am(An ) recursively. We actually fixed m but it is easier to fix n instead as this means you are calculating what you will need for the next value of m. Again we have tabulated some values in the appendix and can calculate the sum of the coefficients. In the calculations below we abuse notation and make no distinction between an expression and the sum of the coefficients in that expression.

Lemma 2.4 The sum of the coefficientsin Opm(An) is 1 if m is odd and (-1) n if m is even. Proof We have n= E (-1)nl()z-lp IAI=n so Om (en) = E (1)nI((J)z lpmt - Il=n n+ l By the previous lemma Pn = (-) so if m is even Pm = (-1) (1). Therefore for m even m(en) = (-1)n E Z- 1. IIL=n

16 If m is odd then Pm., = (-1)l( )- k where k is the number of odd parts of ,u. Note that n- k is even, since mod 2, it is the sum of the even parts. Therefore 6(en)= z(_j)n-21(M)+kz-l = Z-1 =~~~~~~ = A 11,1=n jpl=n

since n- 21() + k is even. To prove the lemma it remains to show that ZI1I=n z 1 = 1. This is a classical result that goes back to Cauchy, though perhaps a more accessible proof can be found in [Dwy38] [

Theorem 2.3 The sum of the coefficients in Pm,nis 1 if n is odd or m = 1 and 0 for n even and m > 1.

Proof We prove this by induction on m, the case m = 1 holding since A(A n ) = An . To prove the induction step we use the newton formula again:

¥ m (An) = Al(An)?,m-l(An) - A2(An)Obm-2(An) +... (1)m+lmAm(An).

In n is even this reads

1 = Al(An)- A2(AXn) + -.. + (- 1)m+lmAm(An)

by the lemma so by induction we find that Am (An) = 0 for m > 1. If n is odd it reads n 2 m n ) 1 = -A (A ) - A (A) -. M A (A if m is odd and -1 = A (An) + A2(AXn) + ... mAm (An)

if m is even. In either case the equation gives Am(An)= 1 as required. a]

2.5 K-Theory examples

The definition of a A-ring is set up so that for a CW-complex X, the ring K°X has a A-ring structure with Ak induced by the operation of taking the kth exterior power of a vector bundle. In this section we give explicit computations in the cases X = S2, and X = BU to illustrate the operations Akand P.

KoS2n Let us recall from [HatOl] the ring structure on K0S2n. As an Abelian group it is freely generated by 1, the trivial line bundle, and the external product

17 a = (H - 1) * * (H -1) of n copiesof the generatorH -1 of K0S2. Here H is the canonical line bundle over S2 = CIP1. The multiplication is trivial since (H - 1)2 = 0. If m denotes a trivial vector bundle of rank m then the kth exterior power of m is a trivial vector bundle of rank (k) = m(m-1) ...(m-k+l) Therefore on the first summand we have Ak(i) = (k) wherek~~~~k the given formula makes sense for all values of m. To calculate the Adams operations we note that on a sum of line bundles L1 + .. + Lm, pk acts by taking it to the sum Lk +. + Lk . A trivial line bundle raised to a power is still a trivial line bundle = m-p so pk is the identity on trivial bundles. Finally this means that OP(m) p To compute the A-operations on oa we first have to figure out the Adams operations. The Adams operations are additive so once we know them on vector bundles we can extend them to virtual bundles too. Once we know tpk for all k we can define Ak on virtual bundles using the Newton formula. To determine opk(O)we first compute that on S2 we have k(H- 1) = Hk _ 1 = ((H-1) + )k - 1 = k(H-1) where the last equality uses (H- 1)2 = 0. Then the equation

ok(x * y) = k(x) * pk(y) shows that o~k acts by multiplication by k on oa. Now we can show that Ak(a) = (1)k+1kn-1a. Indeed this is true for k = 1 and if it is true for k then the Newton formula shows that kna = ()k+lkAk(oa) mod terms of degree 2. The latter all vanish by the induction hypothesis since a2 = 0 and so dividing by k gives the result. The formula for 4 P(a) also gives that OP(a)= pn-1a. This calculation illustrates two important points. The first is that if we want to determine a A-ring structure then we may well have to find the Adams operations first. The second is that if we can do this, then so long as the ring is torsion free (to allow us to cancel the k) we will have determined the A-operations. This latter statement is the content of the following theorem of Wilkerson.

Theorem 2.4 If R is a torsion free unital commutative ring equippedwith ring endomorphisms ?/k : R -+ R for k > 1 such that

1. ?P1 = Id

2 Vpkm = km 3. P(r) _ rP mod pR for p prime

then R has a unique A-ring structure with the given pk as Adams operations.

18 KOBU The space BU is a topological direct limit of spaces BU(n). The space BU(n) is the classifying space for complex vector bundles of rank n. In can be constructed as the complex Grassmannian Gn(C ° °) of all n-planes in C'. Atiyah and Hirzebruch proved that as a ring K°BU(n) can be computed as the completed representation ring R(U(n)) which itself is the ring of invariants in R(T(n)) under the action of the Weyl group, where T(n) is the maximal torus of diagonal matrices. Since R(T(n)) = Z[[zl, ..., zn]] and the Weylgroup is the full symmetric group permuting the variables we find that R(U(n)) is also a power series ring in n variables, which are the elementary symmetric polynomials in the zi. Taking a limit we find that K ° BU is the power series ring in infinitely many variables. The proof of the theorem gives a useful description of the variables in K°BU(n). The cohomology of Gn(Co) is shown in [HatO1] to be a polynomial ring in the Chern classes cl,...,cn of the universal bundles over Gn(Coo). The operation of taking the ith Chern class of a vector bundle extends to operations Ci: Ko°(X) -- H 2i(X) which behave additively much the same as A-operations. As before we consider the ring homomorphism that results by looking at power sums instead of elementary symmetric functions and we obtain the Chern character ch = Zk chk = Ek Vk(C1, ..., ck)/k!. Formally we can say that if L is a line bundle then ch(L) = e(L). The Chern character gives a map to the direct product of the individual cohomology groups rather than the direct sum, which in the case of Gn(Co) means power series in the ci rather than polynomials. The key theorem is that if X is a CW-complex with torsion free homology then ch: k°(X) - H*(X) Q is injective. In the case of X = BU we have one element i E K°(BU) given by the identity map of BU and using the exterior power operations we obtain a family of elements A(i).

Theorem 2.5 The elements ch()An(i))E H*(BU)0Q form a set of polynomial generators and so K°(BU) = Z[[A(i), A2(i)...]].

Proof We give a sketch and refer to [Hir61] for details. The idea is to introduce the total y-operation defined by yt(x) = At/-t(x). The -operations ?i, defined by yt(x) = E Yi(x)ti,give another set of generators for the polynomials in the Ai , so it is sufficient to prove that the Chern characters of the 'ii are a set of generators. If we have At(xi) = (1 + xit) then ch(At(xi)) = (1 + exit) and so if we introduce new indeterminates zi = xi - 1 we have ch(yt(zi)) = (1 + (exi - 1)t).

19 Now ei -1 = xi + decomposables so any polynomial in the xi can be written as a polynomial in ex - 1 plus higher terms. To prove the theorem we start with a maximal torus T(n) in U(n) and consider the product B2k of n copies of CPk. The cohomology of this product is a polynomial ring in n variables xi modulo terms of degree k + 1 and using the Chern character we find that ° K B 2k is the same truncated polynomial ring in the zi. Taking an inverse limnit over k we obtain that K°BT(n) is a power series ring in n variables. Next a naturality argument gives us K°BU(n) and another inverse limit gets us to K*BU ° The power series ring K°BU is a completed version of the free A-ring in one variable. In fact it is the inverse limit over n of the quotient by the ideal generated by terms of degree at least n. This means that the actions of Ak and 9P are given by the universal formulae described above. The y-operations introduced in the proof of theorem 2.5 give rise to the so 2 called -y-filtrationI = '(I) D (I) D) _... of a A-ideal I. Here Ln(I) is the ideal generated by products yil(xl)... Yik(xk) where il + ..-- + ik > n and xi E I. I is called y-nilpotent if it is nilpotent and yi(x)= 0 for x E I and i sufficiently large. We remark that in the above calculation the augmentation ideal of K*B2k is y-nilpotent, the operations Yibeing zero for i > n, and any polynomial with zero constant term raised to the (k + 1)th power being zero.

3 Bousfield's Theorem

In the previous section we saw how in a torsion free unital commutative ring, a A-ring structure is equivalent to a set of Adams operations. We also saw how in K-theory the operation of taking the kth power of a line bundle gives rise to Adams operations and hence a )A-ringstructure on the K-theory of a space. A natural question to ask is whether the K-theory of a spectrum E is also a A- ring. The answer is generally no because the Adams operation pk only defines a stable operation if k is a unit in the coefficient ring. If we were to work over the p-adics for p a prime then we would have all the Adams operations save for those which are a multiple of p, plus we wouldn't have to worry about the ring being torsion free. In this case by formally introducing P, or indeed as we saw previously even P, we would expect to obtain a A-ring. We would also expect this A-ring to be closely related to K°(Qf'E; Zp). This is the content of Bousfield's theorem. The object of this section is to understand the theorem in detail so that we may apply it to the spectrum tmf and give explicit details. There are two main issues in stating the correct theorem. The first is determining the formal algebraic properties that characterise K*(X; Zp) and K*(E; Zp) for a space X and spectrum E. This involves the notions (in the language of

20 Bousfield) of Z/2-graded p-adic A-rings and Z/2-graded p-adic 0p-modules. The second issue is establishing which properties will be required so that the introduction of the P operation to K*(E; Zp) gives a unique (p-adic) A-ring structure. The first of these conditions is that K*(E; Zp) should be torsion free, due to this being the class of spectra that can be constructed starting from E = K. The second of these conditions is more subtle and relates to the -filtration of the augmentation ideal (K°,K'). The operations Ok must satisfy the relations k(y) = ky mod r2( K0(E; Zp)) for y E K°(E; Zp) and pk(x) = x mod F2(Kl(E; p)) for x K(E; Zp). If K*(E; p)/F 2 (K*(E; p)) = 0 then these conditions are automatic and the operation pP, or equivalently OP, can be introduced freely to give K*(Q°OE;Zp). Otherwise these conditions must be included as part of the data when we introduce P. The space K*(E; Zp)/r2 ( K*(E; Zp)) can be identified as the first and second stable cohomology groups of E which gives a more useable criterion for K*((QE; Zp) to be a free object.

3.1 Z/2-graded p-adic A-rings and Z/2-graded p-adic - modules.

If X is a connected CW-complex then by considering the finite subcomplexes of X and regarding Zp as a (topological inverse) limit of finite p-torsion groups we can write K*(X; Zp) as a limit of rings R with the following properties:

1. R is an augmented, Z/2-graded commutative ring.

2. R° is a A-ring and R 1 has endomorphisms O'~for n > 1 such that Al = Id, opJopk = pjk and Opk(xy) = k(x)Opk(y) for x R and y R. FurthermoreAk(xy) = (-l1)fn-Jiii (x)i (y)... Vi (x)qii (y) for x,y E R 1 where the sum is over all strictly increasing partitions (il, ... , ij) of k.

3. The augmentation ideal R = (0, R') is finite p-torsion and 'y-nilpotent.

An inverse limit of such rings R is called a Z/2-graded p-adic A-ring. A ring R satisfying properties 1,2 and 3 also has the following properties:

4. Fr2 (R) = {x E Rl(OP)nx= 0 for some n > 0}.

5. OP(y)= y mod rP2 (R ) for y ER0 .

6. ky = ky mod 2 (R0 ) for y E AO.

2 l 7. OpkX x mod P (R ) for x G R1.

21 8. The Adams operations k: R satisfy = ikpk+p" for k > 0 and somer. 9. pk(xy) = kpk(x)bk(y) for x, y E R1.

There are several remarks to be made here. Firstly R' gives rise to a A-ring Z E R1 with trivial multiplication, where (-1)k+lpk corresponds to Ak. Then BPis identified with 9Pso that F2(R1) = {x R (P)nx = 0 for some n > 0} and property 7 is consistent with property 5. We note that to obtain the A-ring corresponding to the group R we apply the functor which is the left adjoint to the forgetful functor from the category of A-rings to the category of abelian groups with endomorphisms V)n. The maps in the latter category are homomorphisms that commute with the opn. Secondly the operations ibk on R can be thought of as pk/k as in [Bou96] p17. In this case the formula in property 9 becomes pk(xy) = k(x),pk(y) for x,y R1 which is consistent with [GoeO4]. We note that this formula comes from the fact that the Adams operations commute with the Bott periodicity isomorphism only up to multiplying by k. Finally we remark that the formula for An(xy) with x, y E R1 is seen to be equivalent to property 9 when we consider the formula en = El'l=n(-1)n-l()z;lp, and remark that since squares are zero in R', we only need to look at the partitions I with no multiplicities greater than 1. If we ignore the operations /)k for k prime to p then we arrive at the definition of a Z/2-graded p-adic OP-ring. Specifically it is an inverse limit of rings R P: ° 1 1 p over Zp with operations R -R and Opb:R -+ R , such that 9 defined by the usual formula is a ring homomorphism on R° , and VP on R 1 is a group homomorphism. With the convention 4p = 0P on R1 in 4 and 5 as remarked above, R must satisfy properties 1, 3 and 5. We don't have the -- operations so property 4 is used here as a definition and -y-nilpotent reduces to just nilpotent. Furthermore the operations must be consistent with properties 2 and 9, which means we have the formulae ,P(xy) = 'P(x) 9 (y) for x E R° and y E R1 and OP(xy) = pP(x)P(y) for x, y E R1 . Bousfield proves that if we take a Z/2-graded p-adic 0P-ring and endow it with Adams operations Ok for k prime to p that satisfy properties 7,8 and 9 above, together with the usual properties described in section 2.2, then the ring has a unique structure as a Z/2-graded p-adic A-ring with the given Adams operations. We remark that property 2 is straightforward so the difficulty here is in proving that /i vanishes on the augmentation ideal for sufficiently large i, so that R is 7y-nilpotent. This theorem can be viewed the other way around too. If we consider K*(E; Zp) for E a spectrum, then additively we have a limit of Z/2-graded finite abelian p-groups which are equipped with Adams operations V)kfor k prime to p. This means that ?bksatisfy property 8 plus the conditions b1 = Id and VPJVbk= ,)jk .

22 A space which satisfies these properties is called a Z/2-graded p-adic Vp-module. These properties mean that the action of Z+ \ pZ extends to a continuous action of the p-adic units Zp. We can make such a space into a Z/2-graded p-adic P-ring by simply applying a functor which forces conditions 1,3 and 5 to be satisfied and conditions 7 and 9 will naturally be satisfied too. Bous- field's theorem is that when we apply this functor to K*(E; Zp), at least when it is torsion free, then we will get K*(QE; Zp), where the subscript 0 denotes the baseloop component of the loopspace.

3.2 The free SP-ring functors

Bousfield constructs (Z/2-graded p-adic) P-rings out of p-profinite abelian groups using the adjoint functor method for various forgetful functors. To motivate the definitions let us consider the case when our abelian group has Adams operations, so that when we make it into a OP-ring it becomes a A-ring. In a A-ring R the ideal P2((R) is a A-idealso we can consider a decomposition R= z + r(R) +R/p 2(R) which is preserved by the A-operations. The summands are also closed under addition and multiplication but the sum is not necessarily direct. In the p-adic case this decomposition is a limit over ae of decompositions R= z + F2(Ra)+ Ra/P2(Ra)

P where p2(,) = {x E =-l(oP)nx 0 for some n > 0} and 0 acts as the identity on R/F 2(R ). This means that these decompositions can be described without reference to the Adams operations Ok for k prime to p. If we forget the operational structure but remember the decomposition then we should be able to recover the original A-ring from this. Encoding the decomposition as the map - R/r 2 (R) gives us the functor W. This can be built up out of functors T and J that are left adjoint to functors which remember R and R/p 2(R) respectively. T is the simplest way to introduce 0P freely but it for- gets the decomposition of R so it doesn't give us the A-ring that we want unless p2 (R) = R. Restricting (the right adjoint of) T to rings which satisfy this condition gives us the functor T, which is the special case W(M -- 0). Similarly if it happened that Pr2(R) = 0 then the functor J would give us what we want and so J is the special case W(0 c H). We recall the definitions of T, J, W and T from [Bou96] T is left adjoint to the functor sending a Z/2-graded p-adic 0-ring R to the Z/2-graded p-profinite abelian group R.

23 J is left adjoint to the functor sending a Z/2-graded p-adic OP-ring R to the Z/2-graded p-profinite abelian group R/p 2 R. Here p2R is defined as the limit of p2Ra where R = lim Ro. W is left adjoint to the functor sending a Z/2-graded p-adic 0P-ring R to the map of Z/2-graded p-profinite abelian groups R -+ R/r 2R. T is left adjoint to the functor sending a connective Z/2-graded p-adic Or-ring R to the Z/2-graded p-profinite abelian group R. Here connectivemeans that /r 2R= 0. To evaluate these functors on specific groups we can use the fact that left adjoints preserve cokernels to reduce ourselves to dealing with free groups. The functors J and T are simple to describe. In the case of T we note that a connective Z/2-graded p-adic Or-ring R can be written as R = Zp ff r2R and the adjunction relation is Homep(TH,Zp · r 2R) = HomAb(H,r2 R).

In the case where H consists of Zp on an even generator x and an odd generator y this formula tells us that

T(Zp(x), zp(y)) = zp[[x, Opx, (9p) 2 x, ...]]6Ap[[y, y, (0p) 2 y,...]].

To find J we use the identifications ([Bou96]) Ro/ 2R0 - {u e RI°loPu= O} and R1/r 2R' - {u G R1 kpu = u} and obtain J(H) = Zp[[H]]6Ap[[H1]] where 0p acts by zero on H ° and OP is the identity on H 1 . Note that we can view this functor as starting with T then introducing by the relations that exist in R/F2 R. Next if we write down what the definitions entail we find that W(M H) = T(M)6J(M)J(H) and so we can compute W from T and J. Specifically, the action of OPon a tensor product is given by the formula OP(r0 s) = O(r) 0 sP + rP0 O(s) +poP(r) 0 OP(s) which extends the action on the two factors. In this case the tensor product is over J(M) which means that we introduce relations given by equalizing the actions on the two factors. The action of J(M) on J(H) is given by the natural map J(M -- H). The action of J(M) on T(M) is also given by a map J(M) -- T(M). M is naturally a subset of T(M) and this map extends the inclusion to J(M), whose elements are formal 4_combinations of elements of M. Another way to describe this map is as the adjoint of the map obtained

24 by composing the adjoint of the identity of T(M) (which is the inclusion of M in T(M)) with the projection to T(M)/F 2 T(M). This just leaves us needing to describe T. We can describe T(Zp, 0) as follows. Let Tp(Z/pm ) be the cokernel of the OP-endomorphismof Zp[x, 0Px, (P) 2x, ...] which sends x to pmx. Since OP(pmx) = pmOPx mod decomposables, this is a polynomial ring with coefficients in Z/pm , though with a more complicated OP

T operation. Let I C Tp(Z/p m ) denote the OP-ideal generated by (Tp(Z/pm))n together with (OP)n(OPx- X) for x G Tp(Z/pm). Then Tp(Z/pm )/I, is a p-adic OP-ring and we have T(Zp, O)= limmlimnT(Z/pm )/In.

As a ring this is a power series ring in infinitely many generators as with T, but the 9P operation is more complicated. The functor T is not the easiest to to understand so a reasonable question to ask, given the decomposition of R, is whether we can build W out of J and the functor To which is left adjoint the functor sending R to r2R. In fact since 2R = {x E Rl(OP)nx = 0 for some n > O} in the finite case, we see that OP-maps preserve these spaces and so To(M) is simply the connective P ring T(M) viewed as a non-connective ring. Intuitively the functor sending 2 2 2 R R/p R to T0 (r R)®J(R/rPR) will be W provided that we know how to reconstruct R from P2 R?and R/rP2 R. Specificallywe have the following proposition.

Proposition 3.1 Let C be the category of Z/2-graded p-adic OP-rings R together with a splitting R = P2(R) R/P2 (R) that is preserved by the morphisms. If Wo is the left adjoint to the functor sending R E C to the split surjection R - R/rF2(R) then Wo(M E H - H) = To(M)J(H).

Proof This is just an exercise in definition chasing. l

This proposition is a generalisation of the case H = 0 which gives the functor T. However the naturality condition seems unlikely to be satisfied which limits its usefulness.

3.3 K*(Qo E; Zp)

The free OP-ring functors above are defined on the category of Z/2-graded p- profinite groups. When we come to describe K*(Qo E; Zp) we will apply them to Z/2-graded p-adic 0h-modules to obtain Z/2-graded p-adic A-rings. More

25 precisely let A-ring, -mod, P-ring and Ab denote respectively the categories of Z/2-graded p-adic A-rings, /2-graded p-adic O-modules, Z/2-graded p- adic P-rings and Z/2-graded p-profinite abelian groups. Then we have a commutative diagram of forgetful functors

A-ring , +-mod I I OP-ring - Ab

where the horizontal arrows are the various functors defined above and the vertical arrows keep the underlying space but forget the operational structure. These diagrams correspond to a commutative diagram of adjoints

A-ring - +-mod TI I OP-ring Ab

where the functor I introduces the operations 4 k for k prime to p, and is in fact the same functor in both vertical arrows. The functors referred to below are the top arrows in this diagram, but in fact they are the same functors as the bottom arrows defined in the previous section. This is because the bk P P commute with O and so introducing 0 after the Ok is the same operation as introducing P before the Ok. We are now in a position to state Bousfield's theorem. Let E be a spectrum with O-connected section E(O) and consider the Atiyah-Hirzebruch spectral sequence for K*(E(0); Zp). The picture for E2 is shown below.

t . H2 2 0 H 1 H 2 H 3 1 0 0 0 H 1 H 2 H3

-1 0

0 1 s

Here every group to the left of the picture is zero since we take the zero connected section of E. This means that there are no differentials that hit the groups Hi(E(O); Zp) and H 2 (E(0); p) and so there is an edge map K*(E; Zp)H K*(E; Zp) - K*(E(0); Zp) -+ {H 2(E(0); Zp), H'(E(0); Zp)}.

26 We remark that this is a quotient map and so is surjective.

Theorem 3.1 If E is a spectrum with K*(E; Zp) torsion free then

K*(QoE; Zp) - W(K*(E; Zp)H)

as Z/2-graded p-adic AP-rings. In particular if E is also O-connectedwith H2(E; Zp) = H'(E; Zp) = 0 then K*(Q°°E; Zp) T(K*(E;Zp)).

Both statements in the theorem are consequencesof the followingproposition. (The former requiring a bar spectral sequence argument while the latter being trivial.)

Proposition 3.2 If E is a spectrum with K*(E; Zp) torsion free then

K*(Q°°E(2); Zp) - T(K*(E; Zp)).

Proof The proof of this statement is to check that it works for E = K and E = EK and then build up the class of spectra with torsion free K-theory from these. Let us consider the case of E = K. According to [Tor77]K' (K; Zp) = 0 and K°(K; Zp) is the completed group ring Zp[[Zp]]. Here Zp is the group of units in Zp and a E Zp corresponds to the stable Adams operation ?/i. This means that as a Z/2-graded p-adic p-module K*(K; Zp) is a free module on the single even generator x corresponding to the identity map. When we introduce OPby applying T we therefore obtain the free Z/2-graded p-adic A-rings on an even generator, which is Zp[[x, A2x, A3X, ...]]. The 2-connected cover of K has zero space BSU, which is the 2-connected cover of BU. The other spaces in K(2) are given by the homotopy equivalences QBSU _ SU, QSU _ BU, QBU U and QU Z x BU. The calculation of K*BSU is similar to that of K*BU is section 2.5. The cohomology of BSU is again a polynomial ring in infinitely many chern classes, with c omitted. The conclusion is that K° (BSU; Zp) Z[[y, A2y, A3y, .. ]], where y corresponds to the covering BSU -+ BU. Therefore we see that in this case K*(BSU; Zp) - T(K*(K; Zp)) which is what we want. For E = EK the indices are shifted so now K*(EK; Zp) is a free module on the single odd generator x. Now we get that T(K*(EK; Zp)) is the free Z/2-graded p-adic A-rings on an odd generator, which is the completed exterior algebra Ap[[x, A2X, A3 x, ... ]]. Shifting indices in the 2-connected cover we have the zero space SU. The calculation of K*U can be found in [Ati67] and with suitable modifications for SU, we have that K'(SU; Zp) is also a completed exterior algebra on infinitely many generators.

27 3.4 Examples

In the previous section we saw how Bousfield's theorem applies to the the spectra K(2) and EK(2). Now we look at some other examples to help us to understand the theorem further. Given a spectrum E we will want to compute the stable cohomology groups H2 (E(0); Zp) and Hi(E(0); Zp) of the 0-connected cover of E. If we know the homotopy groups r E and ir2 E then we can use the following theorem.

Theorem 3.2 If E is a CW-spectrum and G is any coefficientgroup then HI(E (0)) --irE

H2(E(0)) - coker(x7: rlE --* r2E) and so H'(E(O); G) ~ Hom(7rE, G) H2(E(0); G) - Hom(H2(E(0)), G) (3 Ext(irlE, G). Proof The formula for H1 (E(0)) follows from the Hurewicz isomorphism. To compute H2(E(0)) consider the cofibration E(1)

E(O) - Hr1,E where HirlE is the Eilenberg-Maclane spectrum. The stable homology groups Hn+l(K(II, n)) and Hn+2(K(II, n)) are computed in [Mac54] as 0 and II 0 Z/2 respectively. Therefore part of the long exact homology sequence for the fibration is

7rE 0 Z/2 7 r2 E H2(E(0)) HI(HirlE) = 0 where the second group is given by the Hurewicz isomorphism for E(1) and 7 is the non-zero element of 7r. This sequence gives the formula for H 2(E(O)). The formulae for the cohomology groups follow from the universal coefficient theorem. 0

We remark that in the preceding proof we could equally well have used the long exact sequence in cohomology. The stable cohomology groups we would need ae Hn(K(II, n); G) = Hom(II, G), Hn+ (K(l, n); G) = Ext(II, G) and Hn+2 (K(II, n); G) = Hom(II 0 Z/2, G). The cohomology long exact sequence then gives us isomorphisms Hi(E(0); G) - Hom(7rE, G) and H2 (E(0); G)

28 (Xr))* ker (Hom(-7r2E, G) (* Hom(7r1E 0 Z/2, C)) ®Ext(7rlE, C) which amounts to the same as above. We can use this theorem to look at some examples. Let us begin with the spectrum K. This has homotopy groups 7r2 = Z and 7r2n+1= 0. This means that H 1(K(O);Zp) = 0 and H 2 (K();p) = Zp. The space POOKis x BU so the connected component of the base loop is just BU. We saw that K°BU is the free Z/2-graded p-adic A-ring on an even generator, while K°(K; Z) Zr[[Zp]. Therefore we have the formula

W(Zp[[, ]] __+ZP) ZP[[X, A2(X),...]].

A second example is the sphere spectrum S ° . The K-cohomology of this spectrum is simply the coefficient ring Zp in degree zero. The groups r So and ° 7r2S are well known to be Z/2 generated by q and 772 respectively. From the formulae in theorem 3.2 this gives us HlS°(O) = Z/2 and H2S°(0) = 0 and so we have H'(S-(0);p) = 0 and H2(S0(O);Zp)= Ext(Z/2,Zp). This last group is zero for p > 2 and Z/2 for p = 2. The space Q'S ° is what is usually called QSg. The integral cohomology of this space was calculated by [Pri70] as the cohomology of the classifying space of the infinite symmetric group, BSoo. This isomorphism is induced by a map so we can apply the Atiyah- Hirzebruch spectral sequence to deduce that the K-cohomology of QSo is also isomorphic to the K-cohomology of BSo. This latter ring can be calculated using Atiyah's theorem that for a finite group G, K*BG is isomorphic to the completed representation ring R(G). Atiyah proves that the dual of the representation ring of Sn is the ring of symmetric functions of degree n in n variables. Taking a direct sum over n gives the polynomial ring Z[cl, c2, ...] in the elementary symmetric functions. Dualising and completing we have the free Z/2-graded p-adic OP-ringon an even generator. Therefore in this example the theorem gives us the formulae

W(Zp - 0) = T(Zp) OPx...]] OZp[[x, for p > 2 and W(Z 2 -+ Z/2) Z 2[[x, 9x, ...]]. Note that these rings, as in the example above, are power series rings in infinitely many generators. The difference is in the grading of the generators: the operation A increases the grading by i while 0P increases it by p. More generally we could consider the Moore spectrum MA for an Abelian group A. The cohomology of this spectrum (by definition) consists of A in dimension zero only. Therefore the Atiyah-Hirzebruch spectral sequence is trivial and gives us that K°MA = A also. This means that to apply Bousfield's theorem we have to assume that A 0 ,p is a torsion free group, or equivalently

29 that A has no p-torsion. The homotopy groups of MA for torsion free A can be found using the following lemma.

Lemma 3.1 If A is a torsion free abelian group then rnMA - A 0 lrnS°.

Proof Let O k F2

be a free resolution of A, which occurs as the homology of a cofibration sequence Vs° --+ Vs° - MA. In fact we would construct MA as the cofibre of a map between a wedge of spheres like this. The long exact sequence of homotopy groups has terms

° ° -*F 1 0 irnS l F2 0 irS - rnMA

and so can be broken up into short exact sequences

0 --+ coker(k ® 1) - xrMA ker(k 1) 0.

To identify the terms in this sequence we can tensor our free resolution of A with 7rkS° and obtain an exact sequence

F1 7rkS° F2 rkS° -- A rkS° -- 0.

This tells us that the cokernel in the short exact sequence above is A 0 rnS ° and the kernel (by definition of Tor) is Tor(A, r, _lS°). Since we are assuming that A is torsion free this Tor term vanishes and the short exact sequences become isomorphisms rnMA _--A 0 7rnS°. [

From this lemma and the previous proposition we deduce the isomorphisms H'(MA(O);Z) Hom(A 0 /2,Zp) = 0 and H2(MA(O);Zp) _ Ext(A Z/2, Zp) for torsion free A. In fact these results still hold under the weaker assumption that A has no p-torsion, or indeed with no assumptions on A.

Lemma 3.2 If A is an Abelian group then H1(MA(O);Zp) - Hom(A0 Z/2, Zp)= 0

and H2 (MA(O); Zp) Ext(A 0 Z/2, Zp) - Hom(A, Z/2 0 Zp).

30 Proof The result for H 1 is clear since the tor term vanishes when n = 1. For n = 2 we - find that H2MA (0) - coker(xr) r2MA/(A 0 Z/2) - Tor(Z/2, A) 2(A) so that the Hom term in H 2MA vanishes here too. The final isomorphism can be obtained using the following result from [CE99]: If C is torsion free and B is a torsion module then the sequence O-C - Q C - Q/z C--O is exact and so Hom(B,Q/Z 0 C) - Ext(B, C). In our case B is the torsion module A 0 Z/2 and C is the p-adics. There- fore we have isomorphisms Ext(A 0 Z/2, Zp) - Hom(A ® Z/2, Q/Z 0 Zp) - - Hom(A, Hom(Z/2, Q/Z Zp)) Hom(A, 2(Q/Z) 0 p) -- Hom(A,Z/2 ®ZZp).

We now state our conclusion in the following proposition. Proposition 3.3 Let p be a prime and let A be a abelian group that has no p-torsion. If p > 2 or p = 2 and A = 2A then K°(Q'°MA; Zp) is the free p-adic OP-ringon A Zp. If A/2A $ 0 then K°(QO°MA;Z 2) - W(A 0 Z2 Hom(A/2A, Z/2)). Proof The proposition follows from Bousfield's theorem and the isomorphisms

Hom(A, Z/2 0 Z2) - Hom(A, Z/2) - Hom(A, Hom(Z/2, Z/2)) -Hom(A 0 Z/2, Z/2) for p = 2. (For p > 2 the first term is of course zero.) O In the subsequent sections we will apply Bousfield's theorem to the example E = tmf to obtain a description of K*(Qo tmf; Zp). For the moment let us record the computation of H' (tmf (0); Zp) and H 2 (tmf (0); Zp). Lemma 3.3 For all primes p H'(tmf (0); Zp) = 0. For p > 2, H 2(tmf (0); Zp) = 0 while 2 H (tMf (0); Z2 )- Z/2. Proof The homotopy groups of tmf agree with those of the sphere spectrum in small ° dimensions so the result is the same as for S . O This result means that at odd primes K*(Qotmf; Zp) is a free OP-ring.

31 4 The K-homology of tmf

4.1 K-cohomology and K-homology

In the previous section we described Bousfield's theorem which gives the Z/2- graded p-adic A-ring K*(Qo E; Zp) functorially in terms of K*(E; Zp) when the latter is torsion free. Our next objective is to apply the theorem to the case E = tmf. To describe the K-cohomology of the spectrum tmf it will be con- venient first to consider K-homology. To this end we now describe the relation between K-homology and K-cohomology with various coefficients. Let Zpoo denote the p-torsion subgroup of Q/Z. Also for a locally compact Hausdorff abelian group A let A# be the Pontryagin dual of continuous characters of A into S 1. (Note that if A is finite then such characters must map into Q/Z.) The relevant result is the following (corollary 2.3 of [Bou99])

Proposition 4.1 For a space or spectrum X there are natural isomorphisms Kn(X; ZP) Kn(X; Zpoo)# Kn(X; Z/p) K. (X; Z/pi)# fori, n E Z with i > O.

Proof To prove this statement we simply need to check that the expressions on the right hand side define a cohomology theory in X and that the coefficients agree (naturally). The first statement follows from the fact that the Pontryagin duality functor is exact and transforms direct sums into direct products. To see exactness it suffices to show that S1 = R/Z is injective. To prove the injectivity of R/Z we use Baer's criterion, which says that for a ring R, an R module M is injective if and only if every R module homomorphism from an ideal of R into M, extends to all of R. In our case R = so an ideal of R is of the form nZ for some n. If f: nZ -+ R/Z is a homomorphism then we can find some x R/Z such that nx = f(n). We can extend f to all of by defining f(1) = x. To see the preservation of limits we simply note that the Pontryagin duality functor is representable and hence continuous. It remains to check that the coefficients agree. In the first case this means we want an isomorphism Zp - Hom(Zpoo,R/Z) - Hom(poo, Zpoo). The map sending a p- adic number to the map multiplication by a gives us such an isomorphism. In fact since this map is induced by the multiplication map of the spectra KZp and KZpoo this gives us the naturality that we want. The argument in the second case is the same. [

32 As a corollary to this proposition we find that if Kn(X; Zp) is torsion free then Kn (X; Z/p) - (Kn (X; Zp)/pKn(X; Zp))#. In addition to the functorial description of K*(Q E; Zp), Bousfield proves that is torsion free when K*(E; Zp) and so there is the following statement in terms of mod p K-homology.

Theorem 4.1 If E is a spectrum with K*(E; Zp) torsion free then there is a Z/2 graded Hopf algebraisomorphism K*(QoE;Z/p) - (W(K*(E; Zp)H)/pW(K*(E;Zp)i)) #

The K-homology groups of an infinite loopspace, with mod p coefficients, are an attractive object to study, as they are a commutative cocommutative Hopf algebra. These objects can be understood quite well. For example in the case where E is 2-connected we find that K*(Q'E; Z/p) is a tensor product of Witt and exterior Hopf algebras.

4.2 The ring of divided congruences

According to proposition 4.1 we can recover the K-cohomology of a spectrum with Zp coefficients from the K-homology with coefficients in Zp. We will see below that we can describe the K-homology of tmf in terms of Katz ring of divided congruences. We describe this ring now. We recall some definitions from [Gou88] Let N be an integer prime to p and A be a p-adic ring (meaning that A is a Zp-algebra which is complete and Hausdorff in the p-adic topology). If E is an elliptic curve over A then a level N structure is an inclusion of the cyclic group scheme of order N over A, N, into E. Let M(A, k, N) denote the space of classical holomorphic modular forms of weight k and level N over a p-adic ring A. This A-module can be viewed as containing functions f of isomorphism classes of triples (E, w, L), consisting of an elliptic curve E over A, a nowhere vanishing differential w, and a level N structure . These functions are natural with respect to A and are homogeneous of degree -k in w, meaning that for any A E A, f(E,Aw,t) = A-kf(E,w,t). The q- expansion of a modular form f is defined to be the value of f on the triple (Tate(q), Wcan,Ltcan). Tate(q) is the elliptic curve with canonical Weierstrass equation 3 y2+xy=x +h 4x +h 6 5S where h =-5S and h = -5s+7S for Sk = n ~~nk 4 3 6 12 5 =h cnonical-q The canonical differential is dx/(2y + x) and Lcan() = "mod q". We refer to [Kat76] for full details. If p 5 then the Tate curve can be given by the equation

2 3 y = 4X E4 + E6 := 4X392X + 93 12 216

33 with canonical differential dx/y. In this case M(A, k, N) consists of the homogeneous polynomials of degree k in the Eisenstein series E4 and E 6 (which have degrees 4 and 6 respectively) with coefficients in A (or equivalently polynomials in 92 and g93).The Eisenstein series have q-expansions

E4 (q) = 1 + 240 E.o 3(n)qn n=1 and oo E6(q) = 1- 504 o5Eas(n)qn n=1 where ak(n) is the sum of the kth powers of the divisors of n, which gives an explicit way to compute q-expansions in general. Now let us suppose that B is a p-adically complete discrete valuation ring, and that K is its field of fractions. We remark that this means that B is the topological inverse limit of the discrete rings B/pnB, and that the localization, Bp, of B at its maximal ideal can be viewed as the ring of K-endomorphisms of K/Bp. More pertinently though is that this condition assures us that any elliptic curve over B, with ordinary fibres, admits a trivialization. The module of divided congruences of weight less than or equal to k is defined by k Dk(B,N) = {f e M(K,j,N)If(q) E B[[q]]}. j=0 The ring D(B, N) of divided congruences of modular forms defined over B is the direct limit over k of the modules Dk(B, N). The point of this is that the individual summands in f need not have q-expansions with coefficients in B, but that the denominators are cancelled in the whole sum. Katz showed that the ring of divided congruences (or rather its p-adic completion) is, in some sense, the coordinate ring of the moduli problem of 'elliptic curves over p-adic rings together with isomorphisms of their formal groups with the multiplicative group Gi'. More precisely, for a p-adic ring A we consider the set of isomorphism classes of triples (E, A, ) where E is an elliptic curve over A, 0b: E Gm is an isomorphism of formal groups over A, and L is a level N structure which composes with 0 to give the canonical inclusion. This set has the structure of an algebraic stack M. The functor which assigns this set to a p-adic ring A is representable by a p-adic ring W = W(Zp, N) (which should be thought of as the coordinate ring of the stack). This means that for any p-adic ring A, the set of (p-adically) continuous homomorphisms from W to A (or equivalently the stack morphisms from SpecA to M) can be naturally identified with M. The elements of W are called generalized p- adic modular functions. Restricting attention to B-algebras, for a p-adic ring

34 B, we obtain in the same way the representing space W(B, N) = W6B of generalized p-adic modular functions over B. Given a triple (E, b, ) and a generalized p-adic modular function f we can apply the homomorphism corresponding to (E, 0, t) to f to obtain a value f(E, q, L) E A. (Note that we regard ourseves as evaluating f on the triple.) The q-expansion f(q) of f is defined in this setting by evaluating f on the triple consisting of the Tate curve over B((q)) with its canonical trivialisation and level N structure. The key facts are that the q-expansion map W(B, N) B((q)) is injective with flat (over B) cokernel, and that the smallest ring that f is defined over is that generated by the coefficients of f (q). These facts are called the q-expansion principle. The function f is called holomorphic if its q-expansion lies in B[[q]]. The ring of holomorphic generalized p-adic modular functions over B is denoted V = V(B, N). Clearly the q-expansion principle remains true if we replace W by V. We can regard classical modular forms f as generalized p-adic modular functions by identifying f with the function whose value on the triple (E, 0, ) is given by f(E, O* ( d't) , l), where dt/(1 + t) is the canonical differential on Gm. With this identification, Katz proves the following theorem:

Theorem 4.2 If B is a p-adically complete discrete valuation ring then there is an injection D(B, N) V(B, N). Moreover the image is dense.

Proof The map is constructed using the q-expansion principle. Given I E D(B, N) the fact that B is a DVR implies that (q) is torsion in B((q))/V and so by flatness, there is some f E V with the same q-expansion as f. The proof that the image is dense consists of explicitly constructing certain divided congruences d which form a sequence of Artin-Schreier generators for V. We refer to [Gou88] for details. El The second statement in this theorem refers to the p-adic topology. It follows that the p-adic completion lim D(B, N)/pnD(B, N) of D(B, N) is equal to V. It remains to explain in what sense V is the coordinate ring of a moduli problem. V = V(Zp, N) can be constructed as a limit of rings Vn,m, which in each case are the coordinate ring of an affine scheme. For M > 0 this scheme is M(Np m ) ®Z/pn, where M(Np m ) is obtained by adding cusps to the moduli space of elliptic curves with an arithmetic level Npm structure. For m = 0 we have to delete the supersingular points in M(N) to obtain the affine scheme of ordinary points, and Vn,0 is the coordinate ring of M°rd(N) 0 Z/pP. Then V = lim lim Vn,m. n m

35 As above we take the completed tensor product over Zp with B to obtain V(B, N). An alternative approach, which we shall use below, is that described in [Kat75] The ring VB is defined to be the representing object for the functor MtriV(A) of isomorphism classes of trivialized elliptic curves over a B-algebra A. Mt iv is constructed from the stack of trivialized elliptic curves by passing to isomorphism classes. The q-expansion map is defined by evaluation on the Tate curve with its canonical trivialization and using this to define D = DB, the q-expansion principle and theorem 4.2 still hold. This approach is preferable because in the above we need N > 3 for M°rd(N) to be representable by a scheme. To recover VB from V(B, N) we need to take the ring of invariants under the action of GL(2, Z/N).

4.3 Formulae for the generators of D

The generators dn for the ring of divided congruences constructed by Katz in his proof of theorem 4.2 are not uniquely determined. For p > 5 Katz gives a formula for a particular choice of dn in terms of the canonical differential. Specifically if we set t = x/y and expand w = dx/y in terms of t we obtain w = E ant'- 1dt n>1l

t h for some an E Z[g2 , 93].The dn can be described by saying that an is the n Witt polynomial in the d/. This means that n apn= i=O Given ap. for all n we can construct dn inductively from this formula by rearranging and dividing by pn. In appendix B we carry out the computation of an for n < 125, expanding and correcting the formulae given by Katz. This gives us the following explicit formulae for certain dn. p = 5 : The coefficients of t4, t24 and t' 2 4 are 16g2,1081344(240gg2 - 7g6 - 60g) and

995569969321779190762968186882(14726735444678400g38g2 8 +2908841831540736Og 2gg2 - 10570523441402496093gOg95 ° _ 218 3 +43119208g3 - 609616768696320093 g2 4 +888298720100352009g69g - 288697084032614400g 3 g92 +37563035811840g2 0 + 14430842538840g4g24 -780963243278400g36g2 1 - 72682814985gg9227)

36 respectively. Therefore we have

do = al = -2

a5 - d 169g2+ 32 5 5

a 25 - 5d5 -do d2 25

1081344(240g3g2 - 7g926- 60934) (1692+32)5 + 1048576 25

7569408 6 51904512 3 2 12976128 4 ' _5 92+ 9293- 5 93 - 25 g5 g3 1048576 5 2097152 4 8388608 3 15625 92 - 3125 92 3125 92 16777216 2 16777216 20937965568 3125 92- 3125 92 + 15625 and d3 can be written down from this, but is too cumbersome to print. Note that in Katz original paper a = -2 was erroneously replaced by 1 in the formula for d. Given these formulae and the q-expansions for 92 and 9g we can check that q-expansions of these forms do indeed lie in [1/6][q]. For example 1692+32 5 has q-expansion

1 16( + 00 §(T (1 + 240E u3(n)q ) + 32) n=1

1 400 20 oo =5(- + 320 a3(n)qn) - + 64E 3(n)qn. n=1- 3 n=1 p = 7 : From our calculations in appendix B we have

a 7 = 9693

6 a49 = 997103501312(6336g8 + 236544g23g - 221760g4g26+ 18480g2g9 - 91g12) and so for p = 7 we find do -2 96 128 dly 93+-7 6317647784312832 8 33694121516335104 2 49 93 + 7 9

37 31588238921564160 4 6 26323532434636802_ 12962345517056 12 9392 + 7 Z929 7 '

75144747810816 7 100192997081088 6 400771988324352 5 5764801 93 823543 3 823543 93 890604418498560 4 1187472557998080 3 949978046398464 2 823543 3 823543 3 823543 ' As before we can check that these forms are defined over Z[1/6]. For example the q-expansion of d is

1 -96 (1 - 504 o5 (n)qn + 128) n=1

-1 ( 11489 +224 E Us(n)qn) n=1 164 = + 32 a(n)qn. n=1 p-ll: We give one more calculation, for p = 11. Again from appendix B we have

all = -25609293

° a 121 = -272699558638918420400406398026907648g92 +721965202905751390557818724689134498086912000g3 4g92 +38443459611963324456100955940478173315072009g3g 1 -159592308010745044228570454931282362735001600g3692 -64449329349467926294051602606095761145856000g8 8

+400690687612692021759589392202469646438236160g 3Og25 +6418996802845451753788655804174252900352000g3 8 g23

-78349442143946992777379665639561494528000gg 42

-92077910537260035303339986591981661176463360093g2 2 -10135258109755976453350509164485662474240g32 0 +428578145060906627158316119737772277760g3g22 7 . Therefore do= -2 2560 2048 di -11 9392 +

38 d -=a92349485996472393858691826872186165211955200 21 2 11 583545163895041068526241436743113900032000 18 3 + 11 93g2

65633200264159217323438065880830408916992000 14 9 + 11 93 92

920779105372600353033399865919816611764633600 12 12

121 9 3 92 921387100886906950304591742225969315840 20 11 93 71226765585406357070345150581419540480004 24 11 g392 364264261466083656145081265638608769489305601o 15 + 13 9211 5859029940860720572186509327826887376896000 8 11 g32 38961649550991511559846919976161116160 2 27 + 11 9392 14508391637340458566233677721025669339545600 16 6 11 9392 24790868967174401854582399820627968 30 11 92 68953501833760746541571357357938184604037939200 + ~3138428376721 30948500982134506872478105600000000000 11

+ 3138428376721 3 2 24758800785707605497982484480000000000 10 10 285311670611 93 92 99035203142830421991929937920000000000 9 9 ±+ 285311670611 9392 237684487542793012780631851008000000000 8 8 285311670611 9392 3802951800684688204490109616128000000007 7 + 285311670611 9392 425930601676685078902892277006336000000 6 6 285311670611 32 340744481341348063122313821605068800000 5 5 + 285311670611 g9392 1947111321950560360698936123457536000004 4 285311670611 9392

39 77884452878022414427957444938301440000 3 3 + 285311670611 9293

20769187434139310514121985316880384000 2 2

285311670611 g3 g2 3323069989462289682259517650700861440 + 285311670611 9392-

Let us try to show that the q-expansion of-29°g39 2 + 2048 has coefficientsin Z[1/6]. If we multiply by 11 then the constant term is 502 + 2048 = 6__ . Since 165968 = 11 x 15088 this coefficient is okay. For n > 1 the coefficient of qn is so (240,3 (n) -504,5 (n) -240 504 o-1 qis (240 3(n)-504 5 (n) - 504 l a(k)o5(n-k)) so it is necessary to show that the term inside the parentheses is divisible by 11. Reducing mod 11, then dividing by 2 this becomes 5(n) - o3 (n) -2 Ek=l 3(k) 5 (n- k) which should be divisible by 11. A direct proof of this congruence has proved to be elusive, which demonstrates how remarkable it is that every coefficient in these q-expansions lies in Z[1/6]. In the course of searching for a proof we did however uncover the following fact.

Proposition 4.2 Half the number of integer solutions to the equation x2 +xy +3y 2 = n is equal to o.5(n) mod 11.

Proof The form x2 + xy + 3y2 has discriminant 1 - 4 3 =-11. The number of equivalence classes of quadratic forms of discriminant -11 is equal to the number of triples (a, b, c) of integers satisfying b2 - 4ac = -11 and either -a < b a < c or 0 b a = c. (See [Ken82] p323) In this case we see that b2 - 4ac < b2 - 4b2 = -3b 2 so we must have b = ±1. In either case the inequalities force a = 1 and c = 3. However for b = -1 we contradict b 0 and b > -1 so we must have b = 1. It follows that x2 + xy + 3y2 represents the unique class of forms of discriminant -11. A theorem due to Dirichlet ([Ken82] p307) states in this case that for n > 0 prime to -11, the number s(n) of solutions to x2 + xy + 3y2 = n is given by

s(n) = 2E ( n din

where (L) is Kronecker's symbol. Specifically (ll) is zero if 11 divides n, 1 if n is a quadratic residue mod 11 and -1 if n is a quadratic non-residue mod 11. Both this expression and a5 (n) are multiplicative so we can assume that n 5 8 is a prime power. If n = pr where p is not 11 then cr5(n)= ~ = 0 p g. If p is a

40 quadratic residue mod 11 then p58 is a tenth power so is 1 mod 11. Otherwise 5 p 8 = -1 mod 11. It follows that o5(n) mod 11 = s(n)/2 in this case. If n = 11r we claim that there are exactly 2 solutions to x2 + xy + 3y2 = n and so the formula holds in this case too. For n = 11 it is easy to see that the only solutions (x, y) are (-1,2) and (1,-2) so this is okay. For higher powers of 11 we will show that both x and y are divisible by 11 and so by dividing by 121 and using induction on r we will be done. (We should note that the only solutions with n = 1 are (1,0)and (-1,0).) To prove that x and y are both divisible by 11 we use the standard algebraic reduction that says for a solution (x, y) to x2 + xy + 3y2 = n we obtain a solution (X, Y) to X 2 + 1 1Y2 = 44n where X = -11ily and Y = 2x + y. If 11 divides n then 121 divides 11Y2 and 2 so 11 divides Y. Now for r > 1 we have 113 divides X and so 11 divides y. This and the fact that 11 divides 2x + y allow us to conclude that 11 divides x. [5

The same proof will show that 03(n) mod 7 is half the number of integer 2 2 solutions to x + xy + 2y - n. Indeed whenever we have a prime p of the form 4k + 3 (k > 0) for which x2 + xy + (k + 1)y2 represents the unique class of forms of discriminant -p, we will find that 2k+1 (n) mod p is half the number of solutions to x2 + xy + (k + 1)y2 = n. Note that p = 3 doesn't work because the automorphism group of x2 + xy + y 2 has 6 elements rather than 2. The next primes that work are p = 19, p = 43 and p = 67. A necessary condition for x2 +xy+ (k + 1)y2 to represent the unique class of forms of discriminant -p for larger primes is that both k + 1 and k + 3 must be prime, for if either were composite, the factors would give different values for a and c that represent a different class of forms of discriminant -p. The only part of the proof which is not clearly the same is that (-1,2) and (1,-2) are the only solutions for n = p. To see this we change the equation to (2x + y)2 + py 2 = 4p. It follows that IyIis at most 2. The values y = 1, 0,-1 are easily ruled out if p > 3 since neither 3p nor 4p is a square. This just leaves us with the two stated solutions. Higher Primes The formulae for ap in appendix B for larger primes p will give many more congruences mod p between 3 (m) and o5(n) for various m and n. These formulae will be quite large for any primes bigger than 11, and so are perhaps most useful simply stated as congruences for the q-expansions of various polynomials in g92and 93.

4.4 D as Kotmf.

In section 4.2 we defined the ring of divided congruences and explained how its p-adic completion can be viewed as the coordinate ring of a certain mod-

41 uli problem. Next we show that this ring is precisely Kotmf. We give two related but philosophically different approaches to this calculation, based on two different characterisations of tmf. The first constructs tmf directly as a homotopy inverse limit and then uses various homotopy theory results to calculate Kotmf as an inverse limit. The second approach is to describe K Atmf directly as a stack of trivialized elliptic curves, as considered above, and then to prove that this stack is in fact a scheme. This approach is more direct, and in fact will be used to complete the calculation from the first approach, but the first approach has the advantage of being described using only homotopy theory. The following lemma is the key algebraic step.

Lemma 4.1 If J is an elliptic curve and T + O(T5 ) is a coordinate near infinity. Then there exists a unique pair (x', y') of Weierstrass coordinatesfor J such that x'/y' T mod T5.

Proof If we choose an arbitrary pair (x, y) of Weierstrass coordinates then any such coordinates (x', y') are related to (x, y) by a transformation of the form x' = A-2x + r, y' = A-3y + A- 2sx + t. We can expand x and y as Laurent series in 2 2 3 2 T to get x = T- (Uo+ ulT + u2T +" ) and y = T- (vo + vT + v2T +.. ) where u0 and vo are invertible and ug = v2. Substituting these power series into x' and y' and expanding x'/y' as a power series in T (with the help of Maple) gives 2) x/y = luoT _l(-uiVo + UoV1 + lsuo T2 -- Vo V 2

2 2 - 2 3 A(-uovo2 -2uovoAsu 1 +U2V? +rvO2A Ulv OV1 +uOVl + 2U2V AS + A2S2UO3)V + vVo 3 3 2v 2 -A(2uov~2su2A + UoV2V3 + UoVO2At- u 3 o + UlV2V2 + Ul oAs- 2vouovv 2 2 2 2 3 -4V o UOVlAsu1 - 2V0oAsu2 2 - 3voA0 s uUi + U2V2Vl + rvOA vVl + rv2 A suO 2 2 3 3 4 4 5 -ulvov + uov3 +3UVA + 3U3V1 A s + As u4)T /v + O(T ) We have to solve for A, r, s and t such that this expression is T mod T5. This gives us 4 equations in 4 variables (from the coefficients of T, T 2, T 3, T4 ) with Jacobian matrix

VO 2 0 ~0~~~~~)"%-? * 0 0 VaA 0 0 0

42 The first equation gives A = U0 and so the Jacobian is invertible. The inverse function theorem therefore tells us that there is a unique solution to the equations. In fact with some more algebra we find that this solution is

V0 Uo VI S = - U1 Vo U0 (-U UoVo + V U - VUllUO+ U2Vo)Uo r -' 3 2 1

t =3(-U3UoVo + V3 uO -U 2 VUO + U2 UlVo)Uo

The spectrum tmf is constructed to have the mapping properties of a universal elliptic cohomologytheory. Specifically,we can construct a functorial diagram of Landweber exact elliptic cohomology theories, El, each of which is an Eoo spectrum satisfying further properties from obstruction theory. This means in particular that each Ei is complex orientable, and the formal group associated to this orientation comes with an isomorphism to the formal completion of an elliptic curve E over Ei*. Furthermore on finite CW-complexes X we have Ei*(X) = E* ®Mmu*MU*(X), where the MU* action on E* comes from the complex orientation. For any spectrum R, there is a map from R A tmf to the homotopy limit Holim(R A Ei). The first result from homotopy theory is that this map is a homotopy equivalence. A proof of this statement would involve the nilpotence theorems of [Smi88] but it is certainly reasonable given the relation of Ei to MU. This equivalence means that for any R, there is a spectral sequence

lim rt(R A Ei) ==> rt- 8(R A tmf).

The second result that will allow us to calculate 7r*(K A tmf) = K*tmf is the following.

Proposition 4.3 If R is complex orientable then the derived functors lim2 are zero for s > O. In particular,

7r*(RA tmf) = lim REi.

Proof We may assume that R is even and periodic, so that we can associate to it a formal group GR. This is sufficient because it is true for the periodic

43 version of MU, which we denote MU{u'l}, and we can extend to other R by naturality. In this case R A tmf can be identified with the stack of elliptic curves E together with an isomorphism of E with CR. (For R = K this would be the stack of trivialized elliptic curves.) If J is the universal trivialized elliptic curve over W then this stack is naturally identified with the stack of isomorphisms Iso(J, GR) (since such isomorphisms classify such E). We can think of the R A Ei as forming an etale cover of this stack. Then by

definition, the 5 th cohomology group of the stack is the lim' that we want. A basic fact about the category of stacks is that it contains the category of schemes as a subcategory, and that the sth cohomoloy of a scheme is zero for s > 0. Therefore it remains to show that Iso(J, GR) is a scheme. If we choose a coordinate on GR then pulling back that coordinate defines a map from Iso(J, R) to the stack A of coordinates on JR. A can be identified with MU{u± 1} Atmf. We claim that A is an affine scheme and so this map identifies Iso(J, GR) as a closed subscheme. This follows from the lemma: If we let bn denote the coefficients of T' for n >)5 then

A = SpecZ[al, a2, a3, a4, a6][u1][b 5, b6, ...].

The details of this can be found in [Rez]. Note that an element of r2n is identified with an element of ir0 by multiplying by uE. D

Corollary 4.1 For any p-adic ring B, the ring Ko(tmf; B) can be identified with the p-adic completion of ring of divided congruences DB, that is VB. Proof The previous proposition gives us two different ways to see this statement. The direct approach is to observe that Spec iro(K Atmf) = Iso(Js, Gm) = Spec VB Under this identification, forms of weight 2i must be multiplied by v-1 (v being the Bott class) to put them in 7r0. The alternative method is to calculate KEi directly as in [Lau99] and use the statement of the proposition to obtain Ktmf as a limit. o

Corollary 4.2 There is an isomorphism of Zp-modules K° (tmf ; Zp) - lim hom (Vz/pi,2/p i ) for any prime p. Proof This follows from proposition 4.1 and the previous corollary. One simply needs to note that the continuous character must map pi-torsion to pi-torsion. D In section 5 we will expand upon this description and then apply Bousfield's theorem to obtain K* (Qtmf; Zp).

44 4.5 Diamond operators

The ring of divided congruences has a natural action of Zp by what are called diamond operators. The p-adic K-homology of a spectrum also has an action of Zp given by the stable Adams operations. In this section we define each of these actions and show that they correspond under the identifications of the previous sections. The diamond operators act on generalized p-adic modular functions through varying the trivialization. Specifically, given a generalized p-adic modular function f and a E Zp we define (a)f by the equation (a)f (E, ) = f(E or-1 0).

This action preserves the ring V and so gives rise to an action on D. If (a)f = akf then f is said to have weight k. Under the identification of classical modular forms as generalized p-adic modular functions, a form of weight k becomes a function of weight k. It follows that if f = E i D wherefi has weighti then (a)f = E ai fi. The stable Adams operations are natural transformations of K-homology. For an integer k they can be defined axiomatically as follows. From the work of Miller [Mil89] it follows that natural transformations of a Landweber exact homology theory are determined by their action on coefficients and on line bundles. As we noted in section 3, the operation ok acts by sending the Bott class v to kv, while it acts on line bundles by raising them to the kth power. We remark that Miller phrased this in the language of formal group laws. In this language ok is described as the ring endomorphism of K, = Z[v± 1] determined by sending v to kv, together with the strict isomorphism

[k]: Gm ?ikm of the multiplicative formal group law. To see this we just have to note that the multiplicative formal group law x + y- xy corresponds to the tensor product of the line bundles (1 - x) and (1 - y). Of course this definition requires us to be able to divide by k so over the integers we only obtain two operations. Working with p-adic K-theory we obtain Obk for any k prime to p. We can then extend this to any p-adic unit by continuity. Given this description of the stable Adams operations, we can easily prove the result that we want.

Proposition 4.4 Under the identification of Kotmf with the completion of D, the Adams operation ok A 1 correspondsto the diamond operator (k-i).

45 Proof We showed above that if f = E fi E D where fi has weight i then ()f = E aifi. Thus (k-l)fi = k-fi. Our identification of Kotmf with D requires us to multiply fi by v-i to put it in degree zero. Therefore we have (k A 1)(v-ifi) = (k(v-i)f i = k-=v- fi, and so opk A 1 also acts by multiplying fi by k- . 0,

In terms of the representing spectra, if we think of K A tmf as the stack of trivializations of the universal elliptic curve, then the diamond operator (k-1), which act by multiplying the trivialization by k is represented by a map of the spectrum K. This map is precisely the Adams operation obk.

5 The K-cohomology of tmf and Q tmf

5.1 Hecke Algebras and duality

In section 4 we showed that there is an isomorphism of Zp-modules

K*(tmf; Zp) - lim hom (Vz/pi,Z/p i) for any prime p. There is a duality between the ring of generalized p-adic modular functions and the ring of Hecke operators acting on V. Given the formula above we can expect there to be a relation between the K-cohomology of tmf and this Hecke algebra. We will use this relation below to obtain a description of the K-cohomology of Qfttmf. For the moment let us assume that p > 5 and that is an integer prime to p. We recall the definition of Hecke algebras from [Gou88] The Hecke operator T acts on the ring V of generalized p-adic modular functions as follows. If (E, k)is a trivialized elliptic curve then for every subgroup H of order 1, the projection r: E -->E/H induces an isomorphism on formal groups. Therefore we obtain a trivialized elliptic curve (E/H, T7r'). If f E V then T}f is defined by the formula Tf (E, q) = E f(E/H, 7r-1) where the sum is over all subgroups of order 1. If f has q-expansion f(q) = anq" then Tf (q)= alq + (()f )(ql). For = p it can be shown [Gou88] that there is a linear operator Up on V which on q-expansions sends E anqn to E anpq . If we denote Up by Tp then we obtain T1 for all according to the following formulae:

46 1. T is the identity.

2. If and m are coprime then Tlm = TITm.

3. If is a prime distinct from p and k > 2 then

Tlk=TITk-l - ()Tk-2

4. For any k > 0, Tpk= Tk

It follows from these formulae that any Tl can be written as a polynomial in the Tq for q a prime dividing 1. Furthermore the Tq commute with each other. It is clear from the modular definitions that the Hecke operators commute with the diamond operators as well.

Definition 5.1 If V' C V is a subring that is preserved by the Hecke and diamond operators then the Hecke algebra Tv, of V' is defined to be the completion of the commutative subalgebraof the space of Zp-linear endomorphisms of V' generated by the Tl for 1 prime to p, Up and the diamond operators. The restricted Hecke algebraTv, is obtained by omitting the Up operator.

As two particular cases we have T = Tv and TD', where D' is the ring of divided congruences of forms of positive weight. To see that D' is preserved by the Hecke operators we can look at the effect on q-expansions. The only concern would be the term j, but the effect of the diamond operator (1) on a form of weight i is to multiply by li, so if we restrict to forms of positive weight this will cancel the 1 in the denominator. Now let us consider the ring VB for a p-adic ring B. For any V' as above there is a bilinear form

Tv, V' B (T,f) , ) a(Tf) where a (Tf) is the coefficient of q in the q-expansion of Tf. In particular if T is the Hecke operator T and f has q-expansion f(q) = aqn then (Tl, f) = al. Varying we should be able to determine all of the coefficients (except the constant term) in the q-expansion of f and hence determine f itself up to an additive constant. This means, for example, that if we restrict attention to space VB/arof forms with zero constant term (parabolic forms) then we obtain an injection

Vpar horn (Tvpar B)

47 where the right hand side consists of continuous (with respect to the p-adic topology) homomorphisms of B-modules. There are several results which strengthen this observation; we mention two.

Proposition 5.1 Let B be a p-adic ring. Then the map f -, (, f) induces an isomorphism of Zp-modules

Var- hom (Tvz ar , B)

(continuous Zp-module homomorphisms). Moreoverf correspondsto an algebra homomorphism if and only if it is an eigenformfor the Hecke and diamond operatorsand a (f) = 1.

Proof We refer to [Gou88] for details. The condition al (f) = 1 corresponds to the condition that ring homomorphisms must respect multiplicative units. Notice that the ring B occurs in the right hand side only as the target space, and not as part of the Hecke algebra. o

Proposition 5.2 Let V be a p-adic ring and define VB = (VB+Qp)/Qp. Then the map f -. (., f) induces an isomorphism of Zp-modules

VB - hom (Tvz, B)

(continuous Zp-module homomorphisms). Moreoverf correspondsto an algebra homomorphism if and only if it is an eigenforrnfor the Hecke and diamond operators and al (f) = 1. For B = Zp there is also an isomorphism

Tvz, hom (Vd,, Zp)

induced by the same pairing.

Proof See [Gou88]. We note that VB has a dense subset consisting of sums of forms, whose q-expansion lies in Qp + qB[[q]]. o

We are interested in the inverse limit over i of the spaces hom (Vzp, Z/pi). We remark that these spaces consist of module homomorphisms. If we consider the subspace of algebra homomorphisms then this space would be exactly the set of isomorphism classes of trivialized elliptic curves over Z/p i , by definition of V. Also we note that this inverse limit does make sense, since VB is contravariant as a functor of Zp-algebras B.

48 Now let p be any prime and let Pi: Z - Z/p i denote the reduction mod pi map. The map induced by Pi on spaces of generalized modular functions is denoted pi*. For each i there is a map

hom (Vzp, Zp) (i) hom (Vzp, Z/p i ) (_* hom (Vz/pi, Z/p i ) which give rise to a map p: horn (Vz,, Zp) lim hom (Vz/pt, Z/p'). The final statement in proposition 5.2 also shows that the inclusion of Vz in Vp induces a map q: Tvzp - hom (Vzp,Zp) when p 5.

Proposition 5.3 The map p: hom (Vzp,7Zp) -+ lim._hom (Vz/pi,Z/p i ) is an isomorphism, while q : Tvz - hom (Vzp,Zp) is injective but not surjective. Proof The map 0 is not surjective as any map from Vzp to Zp which does not vanish on the constants does not extend to V1p. It is injective because a map from Vp to Zp must vanish on the constants so is determined by its restriction to Vzp. To prove that p is an isomorphism we first note that Vzp is the direct limit of the Vz1/p, as an algebra, but we can regard the same direct system as a diagram of Zp-modules and obtain the same object as a direct limit of modules. Given this the statement follows by some definition chasing. Specifically an element of the direct limit consists of a family of maps from Vz/pi to Z/p i . These maps factor through each Z/pk for k > i and so factor through Zp. But such a family of maps is precisely a map from the direct limit of the Vz/pi to Zp. El

Corollary 5.1 There is an isomorphism K*(tmf, Zp) -- hom (Vz, Zp) for all primes p and a split inclusion Tvzp * K*(tmf, Zp) for everyprime p 5. Proof This is mostly just a restatement of proposition 5.3. To obtain a splitting of b we note that elements of Tvz can be viewed as elements of hom (Vzp,Zp) which vanish on the constants. Such an element can never be a multiple of a map which doesn't vanish on constants, and so we can construct a right inverse to 0 by extending the identity map to zero on functions which do not vanish on constants. El

49 5.2 K*(Q tmf; Zp) for p > 2

From corollary 5.1, our calculation of the first and second cohomology groups of tmf and Bousfield's theorem, we have the following statement.

Theorem 5.1 For p > 3 there is an isomorphism

K*(D tmf;Zp) 2 T(hom (Vzp, Zp)) whilefor p = 2

K*(Q tmf; Z2 ) - W(hom (V 2 , Z2 ) Z/2).

Let us recall the definitions of T and W to make this statement explicit. T is left adjoint to the functor sending a connective Z/2-graded p-adic OP-ring R to the Z/2-graded p-profinite abelian group R. Therefore for p > 2 and any connective Z/2-graded p-adic OP-ringR, there is an isomorphism

homop (K*(Q tmf ; Zp), R) homAb (hom (Vzp,Zp), f).

Similarly W is left adjoint to the functor sending a Z/2-graded 2-adic 02-ring R to the map A -+ R/ 2 R.k Therefore for any R there is an isomorphism

home2(K*(f2'tmf; Z2), R) - homAb(hom (V 2, Z 2) - Z/2, - R//Fr2).

Let us consider a particular example for p > 2. If Y is a connected CW- complex with Hi(Y; Zp) = 0 for i < 3 then X = K*(Y; Zp) is a connective 2n Z/2-graded p-adic P-ring. In particular Y - S for n > 1 satisfies these hypotheses and according to [HatO1] K* (S2n) = Z generated by (H - 1) *... * (H- 1). It follows that the reduced p-adic K-theory of S2n is a single copy of the p-adics and there is an isomorphism

2 hom0 p (K*(Qo'tmf ; Zp), K*(S n; Zp)) homAb (hom (Vzp, Zp), Zp).

There is a natural map 2 K* .r2nro tmf homep(K*(Qotmf; Zp), K*(S n; Z)) given by applying K-theory to a representing map from S2n to QcOtmf. There is also a map e: Vz - homAb (hom (Vzp, Zp), Zp)

sending v E Vzp to the map which sends a map to its value at v.

Lemma 5.1 The evaluation map e is injective but not surjective.

50 Proof If e(v) = 0 then f(v) = 0 for all f E hom(Vzp,Zp). However if v ~ 0 then since Vz, is torsion free, there is a map sending av to a and the rest of Vzp to zero. Thus if e(v) = 0 then v = 0 and e is injective. It is not surjective because the space of group homomorphisms is much larger than the space of Zp-module homomorphisms. For any copy of Zp in Vzp we obtain a copy of the completed group ring Zp[[Zp]] in the space of group homomorphisms.

We remark that the copies of Zp[[Zp]] correspond to the action of the Adams operations bk for k prime to p. The P operation on the other hand is introduced freely on the left hand side. A natural question to ask is whether any elements of the homotopy groups of fQOtmf correspond to elements of Vzp. For p > 5 we have the following theorem.

Theorem 5.2 With 6 inverted, the homotopy of Qtmf is

ir*Q0tmf - Z[, E4, E6].

Proof For details of this computation and some history see [Bau]. D

Corollary 5.2 For p > 5, the elements pidi E Vzp correspondto elements of the homotopy groups of QOtmf.

Proof From our formula for d/ in terms of ai in section 4.3 we know that pidi is a polynomial in 2 and 93 and so this result follows from the theorem and the equations 1292 = E4 , 21693 = E6. 0

The map 0: Tvzp -+ hom (Vzp, Zp) from corollary 5.1 gives us another way to look at K*(Qotmf; Zp). Applying the functor T we obtain a map

Tq: T(Tvzp) - K*(Qotmf ; Zp).

The Hecke algebra Tvzp is a module over the ring Zp[[Zpx]]under the action of the Diamond operators. It follows that each Hecke operator T1 generates a free A-ring on one generator when we introduce OPfreely with the functor T, and this gives all of T(Tvzp). Let us summarise:

51 Theorem 5.3 For p > 3, K*(Qotmf; Zp) is the free 0P ring generated by the group hom (Vzp,Zp). The A-ring structure is given by the Adams operations, which correspondto the diamond operators on V. If p > 3 then K*(Qotmf; Zp) contains the free 6P ring generated by the Hecke algebra Tvz, as a direct summand.

Proof The only part which requires comment is the last statement. The map : TvzP - hom(Vzp,Zp) is a split inclusion of p-profinite abelian groups. It follows that To is a split injection also. 0

5.3 K*(Qotmf; 2)

For p = 2 we found that there is an isomorphism

K*(Q' tmf; Z2) -W(hom (VZ2,Z2) /2).

We have some understanding of the functor W from our examples in section 2.4. We know that

W(Z 2 - Z/2) - Z2 [[x, 02x, ...1]], and W(zP[[zp ]] - zA zp[[X, 2(X), ...]] for all p. We are interested in Vz2 as a Z2 [[Z ]]-module under the diamond operators and so we would like to understand what we get if we apply W to the composition Z[[z2x]]--z2 - z/2. More generally we have the following construction.

Lemma 5.2 For any compositionM -- H - K of p-profinite abelian groups there is a composition W(M -- H) -- W(M --+K) W(H --+K).

Proof For any R there is an isomorphism between homp W(M -* H); R) and the set of commutative squares M - R

H - R/r2?

52 and similarly for the other maps. Given such a square for M -- K we obtain a square for M - H since the left hand arrow factors through H. Given a square for H K we obtain a square for M - H by mapping M onto the H in the square. This means that we have a composition hom0o (W(H -K); R) homop(W(M -+ K); R) -4homop (W(M H); R) for any R. Now if we take R to be W(H K) we obtain a map W(M H) - W(M -- K) as the image of the identity of W(H K). Similarly we obtain a map W(M K) -- W(H -+ K) as the image of the identity of W(M K). [] We could also view the maps in the lemma as the result of applying W to the squares

~I and

$ 'I which are maps in the category of maps. This allows us to deduce the following result.

Proposition 5.4 The maps

W(Z2[[Z~']]-- Z2) -+ W(Z2[[Z]] Z/2) and

W(Z2[[Z -- Z/2) W(z 2 - Z/2) are surjections. The latter is split.

Proof The functor W is a left adjoint and so is right exact. This means in particular that it takes surjections to surjections. The inclusion of Z2 in Z2[[Z2]] gives a square

Z2 - Z 2 [[Z ]] z/2 ,- Z/2

53 and applying W to this square gives the desired splitting map. []1

We could replace Z2[[Z2]] in the above with horn (Vz 2, Z2 ) and reach a similar conclusion. This allows us to say the following.

Theorem 5.4 The A-ring K*(fQotmf; Z2 ) is a quotient of the free Z/2-graded p-adic A-ring generated by horn(Vz 2, Z2 ). As a P ring is contains the free BP ring on one generator as a direct summand.

Proof We only remark that W(Z 2 -- Z/2) Za [[x, OPx,...]] is the direct summand coming from the split surjection from proposition 5.4. El

54 A Combinatorial proof of the formula for P3,.

In this appendix we give some more details on the calculation of the polynomial P3,n =--P 3 ,n (el, ... , e3n) from the definition of a A-ring. Let us recall the formula for P3,.n-

Proposition A.1 Let w = -2 + i be the cube root of unity. Then for all n

P3,n(el, -e3n)= cjnlkLelejek l

r wherec ,s = 2YR( ) unlessr = O,t, 3t/2. If t > 0 thenfor r = 0,3t/2 the value of cr,8 is 1 whilefor r = t it is 2R(Wr) + ()t+l. If t = r = 0 the coefficient is zero.

3 The definition of P3,n that we use is that it is the coefficient of t in the product lFi1<<...

3 which is a typical monomial in the coefficient of t . In the polynomial P3,n there are two types of monomial that may occur. There are the degree 3 monomials, which are also exactly those of the type el+lej+lek+l for elejek a monomial that may occur in P3,n- 1, and there are monomials of the form eke3n-k. Notice that +' l *' t the coefficient of xr in el+lej+lek+l is the same as the coefficient of Xr,' ,t in elejek. In fact in both cases it is equal to p (p) Z (P) (t-l+r+p) where the sum is taken so that all the numbers are non-negative. Here p is the number of the terms from el that are chosen to lie between r + and r + s. Once these are chosen the terms from ej must include the other s-p such numbers, but then may also have q of the original p as well. We can also see that the coefficient of x r+l' s't is zero in the degree 2 monomials, and that its r s t coefficient in P3 ,n is equal to that of x in P3,n-1. Taking these three facts together we conclude that the coefficient of el+lej+lek+l in P3,n is equal to the coefficient of elejek in P3,n- 1 . This means that we only need to determine the coefficientsof the monomials eke3n._k in P3 ,.. We remark that if we had made this same observation when dealing with P2,n the calculation would have become trivial, as we would only have to determine one coefficient. We have reduced ourselves to dealing with the monomials XOi,3n- 2i. Conve- niently these are equal in number to the monomials eke3n-k so we expect to

55 obtain an invertible linear system for the unknown coefficients. In fact it will turn out to be triangular. The full system has the form

B 0 c b

In this equation c is the vector of known coefficients for the monomials elejek with 1 > 0 and Bc = b is precisely the set of equations that we have in the n- 1 case. The matrices A and L contain the coefficient of xOi,3n- 2i in elejek for > 0 and = 0 respectively. The vector d contains the coefficients of n- xOi,3 2i in P3 ,n and so the system that we have to solve is Lv = d- Ac. We can give explicit formulae for the entries of d, A, and L. In the formulae below we adopt the conventionthat (n) = 0 if m > n or either term is negative. These values correspond to counting impossible choices so in our context they should be zero. This will allow us to have cleaner formulae without affecting our calculation. For example with this convention the formula

(mn) E±Q)(ia ) holds for any positive values of m and n.

Lemma A.1 If the ith row of d correspondsto xOi,3n-2i then it contains the number 1 , ri (3n- 2i 2(n -i + 1)8 6 10 n- , n-i + I if n i and 1 Ax (i) 3n-2i) 2(n-i + ) 3 6 - k, n -i+l 3 ifn=i.

Proof 0 i - 2i The coefficient of x '3n in P3 ,n is the number of ways of partitioning the numbers 1, 12, 2, ..., i, i, i+1, i+2, ..., 3n-i into three distinct sets of n distinct numbers. Let us suppose that we put numbers less than i + 1 in the first set. Choosing the first set involves first picking from i, then n- from the other 3n - 2i so it can be done in (i)I\n-Z] (3n-2i) ways. The second set must contain all the numbers from 1 to i that weren't in the first as we cannot have two equal numbers in the third. Therefore we just have to pick n- i + numbers from the set containing the numbers less than i + 1 that are in the first and the 3n- 2i - (n- 1) numbers bigger than i that are not in the first. Note that by

56 our convention we do not need to worry about whether all this is possible. The productproduct i) ('nl ) ! (2- n-i+l i+)) therefore gives the number of ordered partitions. We don't care about the ordering so we have to divide by 6. The final thing that we have to worry about is that our three sets are all distinct. This will be automatic unless two of the sets do not contain any numbers bigger than i. This can only happen if n = i and either = 0 or = n. If = 0 the second and third sets are both {1,2,..., i} so this should not be counted. If I = n we just have to exclude the 2 cases where either 0 or n of 1 through n are in the second set. This leads us to the expression in the second formula. l

Next we consider the coefficient of xOi,3n - 2i that occurs in eeiek. Lemma A.2 If 1 < j < k then the coefficient of xOi3n- 2i that occursin elejek is >j i 3n-2ia 3n-2i +2a-1 -i+a a=0 5£aJVI-aJV ' Proof The argument is the similar to the previous lemma. This time however the sets do not need to be distinct and the partitions are ordered by 1,j, k so we neither have to divide by 6 nor rule out any cases. a

The previous lemma give us the entries of both A and L from our system of equations. However we should take a closer look at L which has columns that correspond to the monomials eke3n.-k. Proposition A.2 Let the (k + )th column of L correspondto the monomial eke3n-k. Then L is an upper triangular of dimension L2j and in the i row andandjhcolumn jth column itit contains (3n-2i). j-i Furthermore let vo be the vector whose (k + l)th component is 2R(e 2 rki/3). If n is even we modify vo by replacing the 2 in the final component by 1. Then every component of Lvo is (-1)n. Proof The formula for the entries of L comes from the previous lemma. To prove the second claim about L we let w = e2 ri / 3. Consider the ith entry of Lvo. For n odd it is given by 3n 3n i 2Ry.( - 2 k = 2 LR (E3n )k+i k=i k=0 = Wi(1 + )3n-2i = wi(-)3n-2i = (-1)3 n3n-3i = (-l)n

57 Here the key observation for the second equality is that we sum over exactly half the terms in the binomial expansion of (1 + w)3n- 2i, but we multiply by two to account for the other half. The binomial coefficients and the real parts of the powers of w match up perfectly so that the real parts of the two halves are equal. It is not true that the imaginary parts are equal though. For n even the calculation is essentially the same. This time instead of being between consecutiveoccurrences of - in the sequence2, -1, -1,2, - 1,- 1,2, - 1... we would have a 2 in the final position. Our modification to v0 splits this 2 evenly between the two halves and so the second equality still holds. 0

With these results in hand we see that the formula for P3 ,n reduces as in the n = 2 case to some identities among binomial coefficients. The formula for Lvo is our guide as to how to proceed with the calculation. We can generalise it to the case > 0 (where we are looking at monomials elejek).

Lemma A.3 With w as above we have 3n-- L2 (i) (3n-2i) (3n -2i+2a i) )2R(wjl) = (-1) (

j=O a-

3 ( 1 2 In this formula when a n- )/ = 2 we replace it by 1 as before.

Proof First we assume that < i and 3n- 2i > 1. We can sketch the region of summation in the (j, a) plane as shown.

('An-1V/ i \- =/~J -

i-1 i j

From this we can see that we can interchange the order of summation and obtain f ( 3)(3n-2i)2- (3n-2i +2a-l)2 f-) a=O kal-a j=i-a: - - i +a 2,w-1

58 3n-1-2i+2a

3 1 a=0 I (z)nV-i2i ( k=02 (3n-2i+ 2a-+

=E(a)( -a ])(1 a

=Ea=Oa - i\l-(i) (3n-2i) ia ( )(nl1 ()3n-2i+2a-l

a=O I:a a( ) 3n-ia C)32ia- a=O

exactly as above. In this proof we use the identity that

(mIn) =±()(Ia) To see this one just considers how to distribute 1 items among m + n positions by putting a in the first m and 1- a in the remaining n, for various a. From this point of view we can see that if with our convention on binomial coefficients the above equalities will still hold even if we change our limits on a and j to ensure that all the terms make sense for arbitrary values of and i. a

This calculation is very similar to that following proposition 2.1. It is reasonable to expect that in calculating Pm,n by this technique we would be able to use the trick of taking real parts of powers of a primitive mth root of unity in the same way to complete the calculation. The expression on the left hand side in this lemma is close to what we have when we multiply the part of A corresponding to each by a vector like vo, of appropriate dimension. The difference is that the sum in the lemma includes all the terms corresponding to elejek with j < 1 as well. These columns of A are present but are in a block corresponding to the smaller value j. The next lemma is a key observation that allows us to rearrange the triple (1,j, k) without affecting the formula above.

Lemma A.4 Let 1+ j + k = 3n. If p is the difference between 2 of 1,j, k then R(wP) is independent of which 2 we choose.

Proof We simply note that working modulo 3 we have just 0 and ±1 to worry about

59 and R(w) = R(w-1) so the order of the pair that we choose does not matter. Then, for example,we have k = 3n- j - 1 so j - k = 2j + - 3n = I - j modulo 3. 0

Let us now consider how the product Ac + Lv relates to what we obtain by summing the formula in lemma A.3 over 1;. If we sum over I from 0 to 3n and for each for j from 0 to 3`1J we obtain an alternating sum of binomial coefficients which of course is zero. In this sum the sum of the coefficients of terms corresponding to etejek is -3 if I -j is not a multiple of 3. This follows from the previous lemma. If - j is a multiple of 3 then the total coefficient is 6, 3 or 1 according to whether 1,j, k are all distinct, 2 are equal or all are equal respectively. Note that the coefficient 3 when two are equal consists of a 2 from 1,1, 3n- 21 and 1 from 3n- 21,1,1, which is how we modify our coefficients to make the formula work. Now what about Ac+ Lv? Our formula for the coefficient of elejek is that it is 1 if two of 1,j, k are equal, and 2R(wl -j ) otherwise, except when j = n, in which case we have to add (1)n as well. There is also the exception that the coefficient of enenen is zero. Putting this altogether we see that with the coefficients as we claim we have the formula

3(Ac+ Lv)i= 3 (-1)n-- E (l (3n-2i (3 2i + a - )

+2 (zi 3n-2i)a)al-J- (2(na - +J + a)) i a J =0=O In fact by lemma A.4 if we sum of all and rearrange the terms we arrive at 3 times the sum from = 0 to n minus twice the term with = j = k = n. This sum is zero on the one hand and 3(Ac + Lv)i minus 3 times the j = n terms minus twice the n, n, n term on the other. Taking 3(Ac + Lv)i to the other side gives the formula above. If we denote the expression on the right hand side as 3x + 2y then we see that y is 6 times the expression for the ith component of d (unless i = n) so it remains to show the identity 2x = -y. We remark that if n > i each term in y is even so this is certainly reasonable. For n = i we instead want 2x = -y- 1. At the moment the binomial identity that we have to prove is somewhat difficult but it will become clear once we see an alternative expression for the term labelled x. This is supposed to be the sum over of the coefficients of x i 3n - i in elene2.n-iland was arrived at by first considering the terms from el. If we first consider en and so reverse n and in the formula we instead arrive at the expression =0L- 1 i 3n-2i 2(n-i +a)h 1=0 a=O -i+a J

60 L a)(n-a) EZ( 1)n (n - i +aa) a=Oa n a Il=i-a i+a Weobserve that 2 -i a(1) (n-l-(2(n-i+a))= (1 _ 1)2(n-i+a)_ (2(n-i+a)) O Weobserve t at.2 l=i-a - '1) 1-i+a )-'(-I-\n-i+a !s if n > i we have 2x = -y as required. If n = i then the same holds for all a except a = 0 since 0 is not well defined. However if we look at the a = 0 term we have 2x = -2 = -1 - 1 = -y - 1 as required. Note that here only the = n terms contribute. If i > n we should have a similar problem when a = i- n. However in this case we still only get a differencefrom the = n terms and having a = i- n means that all i of the numbers 1 through i must go in the third set. However this set has only n members since = j = n so in this case the problem goes away and we have 2x =-y as required.

This completes the combinatorial proof of the formula for P3,n.

61 B The canonical differential

In this appendix we calculate the modular forms a2i- 1 (in the notation of Katz) that occur as the coefficients of the differential on an elliptic curve, as polynomials in g92and g3. Katz gave formulae up to all, which we will extend to a1ol, and indeed we will note a correction to the formula for all. We recall that in Weierstrass form the curve is given by an equation

y2 4x 3-g 2x - 93 and the canonical differential is w = dx/y. If we define t = x/y then the modular forms a are defined by

w = -a i -l dt.

To calculate this series we introduce s = 1/y and divide the Weierstrass equation by y3 to obtain 3 2 s = 4t - g 2s t g3s3. This formula can be recursively substituted into itself to determine s as a formal power series in t to whatever order of t we require. Indeed if we define 3 s = and = 4t -g2slt - 9g3s3_ 1 for n > 0 then we claim that sn-s n-1 - l mod tn . This is true for n = 1 since s = 4t3 . Now if s- Sn-1 mod tk then s m l mod tk+3m because the smallest term occurring is t3. This means that the definitions of sn+l and sn agree mod tk+4 so they are equal for 4 extra terms. The conclusion is that the sequence sn converges to a unique power series that is s and that sn s mod t4n +3 . To determine the differential in terms of t we first calculate that dx/y = dt + tdy/y from the quotient rule. Also ds = -dy so dy/y = -yds = -ds/s and we obtain w = dx/y = dt - tds/s.

3 2 3 Next we differentiate s = 4t - g2S t - g3S to obtain

2 2 2 ds = (12t - g2s )dt - (2g2st + 3g3s )ds and we substitute this in the above, which gives

2 w = dt- t(12t - 92s) 2 dt. s(1 - 2g2st+ 3g3s ) We can replace s in this formula by the power series calculated above and then formally invert the denominator to obtain the series that gives the formulae

62 for the ati. We note that we have to divide out by t3 before performing the inversion. This means that if we use sn instead of s we will obtain a formula for w that is accurate mod t4n. We carried out this calculation with the aid of Maple and found that the first 50 terms of the series are:

4 6 8 0l° -2 + 169 2t + 96g3t - 192g22t - 2560g39g2t + 2560(g2 - 3g2)tl2 4 16 8 +53760g 39g2t 716892(592 - 48g32)t - 34406493(392 - 2 )t +516096g2(g3 - 20g2)t 2 o - 27033609392(-792g3 + 16g2)t2 2 6 24 +1081344(24092g3g - 7g - 60g4)t -337379328 g2 g3(g3 - 52)t 26 + 18743296g2(6g - 315ga23 + 28Og4)t2 8 +13120307293(489g4 - 400g2g392+ 45g6)t 3° -56229888092(3g 6 + 448g4 - 22423g2)t32 4 2 34 -1853882368g 2g93 (336g - 77092g3g+ 55g26)t +463470592(201609g4g2 - 1344g 6 + 55g9 - 5544gg26)t 36 +158506942464922g93(224g4 - 2242g3g2 + 1lg26)t3 8 4 ° -270952038492(109200gg492 - 26880g6 - 18720g2926+ 143929)t 9 42 -325142446080g 3(4704g493g2- 2548g2g + 91g - 192g6)t +455199424512922(-10560g6 - 2145g2g 6 + 13929+ 18480g4g3)t 44 4 6 -5484069257216939 2(1536g6- 100809392 + 3360gg26 - 91929)t 9 2 48 +9971035--)01312(6336g8 + 23654492g3g6 - 2217609g4926+ 18480g2g - 9g1 )t -23930484031488092g93(35g 9 - 2640g 6 + 73929g4923- 1650g 296)t 5 o

Note that the coefficient of t2k is a2k+1. We calculated as far at t124 which requires determining the 3 1 st iteration of s. For the purposes of section 4 we include these terms below.

1173062942720092(1199212 + 82368g 8 - 81681692g3g6 + 466752g4926- 29172g2g92)t5 2 +18299781906432093(-190080939 6 + 3520g8 + 2827449g4926 2> 9 1)52 54 -44880gg2 + 765g~)t -253381595627520g922(320320g8 - 24752gg29 + 85g 12 - 1345344g2g 6 +510510g 49 )t5 6 - 12031921383014409392 (91520g 8 - 130416092g3g6 + 117374494g 6 -138567g299 + 1938g1 2 )t5 8 -257826886778880(256256g' ° + 440895g; 2 g2 + 42602560926g6 - 11411400g929 4 - 1292g 5 - 19219200g3g8)t 60 +399631674507264009g2293(91291294926- 153753692g3g6+ 256256g 8 2 6 -8299293gg92 + 969g )t -532842232676352092(969g 5 - 61501440g6g 6 - 383040g2912 +12172160g49g + 461260809g892g3- 23429129g°0 )t 6 4 -34101902891286528093(20134409g8923 - 63423369g6g6- 19968g T0

63 +2642640g4g2 - 190190g2g 12 + 1881g15)t66 +1406703494265569280g2(57g 5 + 7468032gag23-6534528g6g6 -2584Og3g~2- 905216g° + 990080gg)t 68 +630143857773772809293(-22364160g ° + 589388800gs92g - 111141888Og3g9 +340372032g4g29- 19835200g]g2-+ 168245g 5)t7 -259471000259788800(-358465536923g ° + 4807g 8 + 112720608g 2 g4 +2715648g12 - 2477376g25g2 - 944706048gg296 + 1545882624g26g38)t72 +19749449848345067520g3g (7920640g] ° - 8712704093g9 + 10735296093gg92 -25049024gg + 1204280gg92 - 8855g25)t74 +43887666329655705692(-14277943680gg2 + 31925836800g6g8- 25741485gg92'5 +13728792Og49 2 - 124156032OOg°0g3 + 361181184g32 + 44275g18)t76 +2501596980790375219293(115115g8 + 292454493g2 - 17892160gg925 +444050880g3g 22 - 494247936gl 0 g2 - 2427110400g36g29 +2831628800g 8)t 78 -34232379737131450368g2(555663360gl2 + 8855g 8 - 7945986048g° 0g3 +1330243200gg268 - 4514764800g g + 356592000gg9392 -5768400g2g2 5)t 8° -9596769704084714291209293(-172931616Og3g9 + 2766905856g6g8 +253955520g34g22 + 49335g 8 - 8691930g2gl 5 + 18522112g 2 -812657664g ° 2 )t8 2 +95967697040847142912(16855121920g3g2 - 79380480g34 -385165872Ogi 2 g 6 - 125149280256g6glO ° - 35790300g 8 g2 +14526255744O29g + 25395552Og 5g4 + 49335g2l)t 8 4 +173317660855769940099072gg2293(-240787456g3 Og° + 30232800g4g92 -889200gg 5 + 532097280gg s - 251536896gg9+ 4485g~8 + 13230080g1 2 )t8 6 +569187720380196847616g2(-8438730300g3g 5 + 104504400gg2918 -712903935744gg899 + 151222046976glg 22 + 844923183104glog6 +3492741120g34 - 192027996160g3g2 - 130065g2l)t88 -5008851939345732259020893(-56408625g32g8 + 2209413024g34g5 -39278453760g °9g + 60008748800g8g9 - 22094130240g93g2 - 15876096g4 +4167475200g32 g3 + 254475g2)t9° 3~~~~~2 1 8 4 +102453789668435432570880Og2(-10015005gg2 + 113109g1 - 2677768192g +60919226368g1 2g3 - 173511010816ga°g6 - 19110478464g6g12 +913368456gg915 + 110416097792ga8g)t92 +79392506545934553841664g2g93(2629575g2 1 + 116931333376093g929g -343434685440g36g1 2 + 28977301584g4g 5 - 1055681003520g °g960 -647927280gg9 8 + 189481205760g 2g3 - 2794192896g3 4)t94 -20711088664156840132608(15107968139264g26g32 + 14669567278080g 2g8 +87121298880g2 8g43 - 2106399404032g 212345 g36- 2967636598784Ogga 0° + 876525g24 -1285328732160g39 4 - 851561568g2lg2 + 401665228893g6)t96 1 + 590155005440 -4639283860771132189704192g22ga(736281g-423860771132189704192 3(736281 2 + 59015500544093 g2

64 -14105353152Og36g 2 - 708186006528g3°g 6 + 10164151536g934g92 +196815962112g223 - 70291415049g14- 200608254gg328 )t98 +270624891878316044399411292(-1 116558OOgg329221 - 35263382889Og3g5 +1273703200094928 + 291834892800g83g 2 + 5221647974400g2g 6 4 +9128755200g6 - 753122304000gg23+ 1051839224 -741909149696093g°929)tl ° 19330349419879717457100800g93(451067904gl6 - 866118176g2g22l -173811499008g34923+ 4013054036992gg22 - 6213761089536g°g 9 -793790908416g6g 5 + 2888487g24 + 49368736032g4g 8 +2509403516928g] 2 926)tm2 -78867825633109247224971264g22(-1661433446400g4 g + 48865689600g]6 +8279070800gg28 - 260362502400g'g25 + 56637g24+ 252923573760093g12 -7919499427840g3°g9 + 7424530713600g29g26- 65447200gg 2l)tO1 4 -8523910880190160209883168768g392(-35150900gg221 + 322191360g16 +2239960800gg98 - 485450085120gOg99+ 27158046720093g2g926 +106981g24 - 31950643200g4g23 -41343276480g6g95 + 2489487616009gg892)t0 °6 +(-7353299750441951439453984452777292595200g34g2 -4353255581424835834815640101383779123209g36g8 +4910953761902303282778196759533406126080g8g5 +22570545067328767612768480056441411993600g]2g -18620699680546233280533996046564164894720g3°g 2 -915443479669088258880047862157148160g38 +4201885571681115108259419687301310054409g36g2 -87932877737813697479160016089907200gg3224 +12274216864919649910194473970067046400g g2 +70145885374894117647193175228416g227)tlo8 +88517536063513202179555983360g 3 g(- 152990329856093g°92g + 1142235494400g3 2g 6 + 168113g24 + 4270858592g4g 8 -89747074560g6g95 + 5119262720g 16 - 60288800gg321 +635708444800gg89 12 - 208865918976g149g)tllO -194933512852581626421444608092(- 189688230051840g34g26 8 -4671141995520g69g + 117536852160g34g2 - 766544688gg24 -274323429918720g °g92 + 419553481052160g]2g + 18441911255040g36g3 5 -156370206720g38 + 565471g27+ 60620579379360g3g2 )tll2 7 4 6 -59543327562243114979641262080g3(407529192g - 18354092630016g g +68827847362560g1 2g9 - 711742512499203g 912 +875393925120g16g9 - 2957446566360936g8 + 124524065952gg 5 8 1 4 +24376529274240g 8g2 - 1589363490gg24 - 16166092809g1 )tl +16161760338323131208759771136g2(- 1567485612g2g24 +1738150135787520g32g - 11641361691960g6gl 8 - 3281716838400g 8 5 +262933070400g34g2 - 919214014118400glog22+ 172352627647200ggl

65 +158999180820480g1692g + 1072445g27 - 1050530301849600g g46)t"4 6 2 +970272699609504473620630470656939 2 (-131398617692160g 3° -1722664944gg2924 - 34819276800ga 8 + 160598310512640g3 2g9 +4075291g + 37640229026400gg39215 + 14857985142092gg -3962129371200g6g8 - 59526786908160g34 g6 +4837703270400g36 g)tll8 +(-272699558638918420400406398026907648g2° +721965202905751390557818724689134498086912000g34929g +3844345961196332445610095594047817331507200g3g2l -159592308010745044228570454931282362735001600g 36g2 -64449329349467926294051602606095761145856000938gg 8 +400690687612692021759589392202469646438236160g3g25 +6418996802845451753788655804174252900352000Og38g23 -783494421439469927773796656395614945280009gg2 1212 -920779105372600353033399865919816611764633600g 3 2 -10135258109755976453350509164485662474240g32 +428578145060906627158316119737772277760gg3222 7)t 2° 2 ~ ~ ~~~10 12 - 122455106226578840463845086986240g3g 2(-30242538992640g3 g2 +7338263137920g8g'5 + 45772491448320g]2g9 - 675892657440gg18 +22707856080gag 1 - 22714168688640gg4 6 + 2877813227520g6 -50487951360ga8- 240179247gg224 + 525844g227)t 22 +99556996932177919076296818688g2(14726735444678400g93g 8 +290884183154073600g3 2gl 2 - 105705234414024960g]°g° 5 + 4311920892g3° -6096167686963200g38g92+ 8882987201003520093g6g6 -288697084032614400g4 g + 37563035811840g93° +14430842538840gg4924 - 780963243278400g6g1 -72682814985gg327 )t 24

66 C Tables of P, and Pm,n

Table of Pn n Pn 1 elfl 2 ef2 + e2 fl2 - 2e2f 2 3 e3 e+ ee 2flf 2 - 3 1e2f3 + e3fl3 - 3e3flf 2 + 3e3 f 3 4 -2ele 3fl + 2e4f22+ 4e4flf 3 - 4e1 2f 4 - 2e2flf3 - 4e4fl2f2 + 4ee 3f4 +e~e2flf 3 + ele3ff 2 - ele3flf 3 + e 4 + 2f 2f 4 ± - 4e4f4 2 5 -5ef 2f 3 + 5e5fx2f3 + 5e5 flf - 5e5f3f 2 - 5e2e3f5 + 5e2e3f 5 + 5e2f 5 5 -5ee 2f 5 - 5 5f1f4 - 5ee4f5 + 5e5f5 + el5 + e5f ±ee+ 2flf 4 -3ele2flf3el 4 --+ ele2f2f 3 - ee 3flf4 - 2eeI 33ff~3+eC2f + 61e3fl 2e3f2f 3 +e2 e3flf22 + 5e2e3flf 4 - e2e3f 2f 3 - 2e2 e3f2f3 + e1e4flf 4 2 +5ele4f 2f 3 - ele4f2f3 - 3ee 4flf + ele4f3f 2 6 eL4e2 fl f 5 + e2e2f 2f 4 - 4e2ef 1 f 5 + e3e3 ff2 4 23e 3 f2f 4 2 + + -e 3e3flf5 - 3eie2e3f ef 6 3f - 2e/f6 ± f + 2 f32 +3e'f 6 + e6f16 - 2e6f23+ 3e6f32- 6e6f6 2e2f2f42e2f ++ 2f l 3ef2f 4 -3ef 2 f4 - 3flf 5 + 3e2e4f32- 2e2 e4 f23- 3e2e4f32+ 21le5fl -3e e f32 + ele e flf f - 3ee e fl2f + 4eIe e f f + 7ele e flf l 5 2 3 2 3 22 3 4 2 3 2 4 2 3 5 -3e2flf 2f3 + e ff 3 _-e2e 4f f4 + 2ele4f2f4 + e2e4ff 5 +e 2 eIf2f22 - 2e2e4 fl3f 3 + 2e2 e4 ff 4 + 2e2 4 2f f4 - 6e2e4 flf 5 +ele 5f 4f2 - 4ele5 f2f2 - ele5 f13f3 + ele5fif 4 - 6ele5f2f4 -ele, 5flf 5 - 12e1 2e3f 6 - 12e6flf 2 f3 - 3e2e4 flf 2f 3 + 4e2 e4flf 2 f3 +7ele 5flf 2f 3 - 64e 2f6 + 9e2e2f6 + 6e1 3 f6 - 62e 4f6 + 6e2e4f6 2 +6ele5f 6 - 6 6ff 2 + 9e6f+ f2 6e6f3f 3 - 6e6f2f 4 + 6e6 f 2f 4 + 6e6flf 5 3 7 7e4e3 7 + 4l7e f27 - 777flf 7f f 4 - 7e 6f7 +7e2e-766f3f 7 - 7ee3f7616f2 7 - 73e 4f 7 f- 7e2e5f 7 + 7le21f7366 + 7e 7f4f1 3 -7e3e4 f 7 + 7e2e5f 7 - 7e5e2f7 + 14e7 fl3f + 7e7f22f3 + 7e7flf32 - 7e7f 3f 4 +e5e2flf6 + eef63 2f 2f5-+-5e3e2flfe3 2 f6f6 6 eeef3f4 3 f4-f-3eefe361f2/f52f5 +5eeflf 6 + e4le3 ff 5 - 2e4 3f2f 5 -4le 3 flf 6 + e2e3flf32 -e2e 3f3 f 4 + 2e e3 f2f 5 + 3e2e3 f2 f5 - 7ee 3 ff 6 + e1e23f2f3 22 2 2 2 3 -2ee2f l f32+ 5e1e2f3f4 + 4ee2ff 5 - 7ee2f 2f5 - 4ele2flf6 7 2 +elf7 + e7f + 7e7f 7 + ee 2e3 flf2f4 -3e 2e3f 3f 4 -4ele 2e3f 5 +6e2e2e3f 2f 5 + 9ee 2e3f1f 6 - 2e2e3f lf 2f 4 - el e2flf 2 f4 +ele.l4 f3f4 + 3e3e4f3f4 - ee 4f2f5 + 2e3e4f2f5 + ele4fl f6 +e3eflf2 - 3e4flf 3 + 5e3e4 flf32 + 3e3e4 f3f 4 - 5e3 e4lf3f4 -7e3e4 f f5 + 7e3e4f 2f 5 + 7e3e4flf 6 + ele5f f3 + 2e1e5f.f 3 +4eje5flf3 -ele 5ff 4 - 7ee 5f3f4 + ee 5ff 5 - 2ee 5f2f5 -e2e5flf 6 + e2 e5 f3f2 - 3e2e5flf23 - 2e2e 5f4f3 + 3e2e5f22f3 -7e 2e5 flfl + 2e2e5 f3f 4 + 7e2e5f3f4 - 2e2e5f2f5 - 3 2 e5f2f5 +7e265flf6 + ee 6f 5f2 - 5ele6f3f22 + 5ele6flf23 - ele6 ff 3

67 2 + f2f -7ele6 f2f-ee2f3 - 4e 6ff + e1e6f3f 4 7ele6f3f 4 - ele6 5 - 21e7ff2f3 +7ele 6 f2 f5 + ele 6flf 6 - 21ee 2 e3f7 + 14ele2e4 f7 f22f +14e7fl f2 f4 - 3e3e4flf2f4 + ele2e4flf 2f3 - 2ele 2e 4 3 2 e f f -ele 2e 4flf3 - 3ele2e4f3f4 + 8elC2e4 flf 2f4 - 2ele2 4 3 4 e f f f +3ele 2e4 ff5 - 4elC2 e4 f 2 f 5 - 8e1e 2e4flf 6 - 3e3 4 2 3 + f2f f -2e3 e4flf 2f4 - 4e2e5f2f2f3 + 3e2e5flf2f4 6e2e5 2 3 7e f f - 7e flf6 -4e 2e5flf2f 4 + 9ele6f2f2f 3 -8ele 6flf 2f4 + 7e7f2f 5 - 7 2 5 7 f2f32 + 8e flf 8 -8e 2e2f8 + 8e8f 2f 6 - 8e61e2f8 - 8e2e6f8 + 12e8 8 7 - flf +2044ef 8 + 2Oe8fl f - 16e8f f - 8ee4f8 + 8ele7f 8 8e8 2 3 - 8e f f 2 +8e8f1f 5 + 8e2e6f 8 - 8esf2f 6 - 8e8f22f4 + 8e8fl5f3 8 2 + 8e3e f -8e fl4f + 8e f f 5 + 12e2e2f 8 - 16e2e32f8 - 8e4e4f8 5 8 8 4 8 3 1 1 3 1~~~ ~~~~~~2 21 f + 2e2e2f42 +8e e f8 + 8e5e3f 8 - 2e4f 3f 5 + 2e4f 2f 6 - 2e42fl 7 3 5 1~~ 2 24 2 21 2ff3 - 4e2e4f42 + 2e4f2f32+ 4ef2f32 - 4ef22f4 - 4efj3f 5 +4e2ef42 2 4 _ 4 3 3 - f24 -4e4f 3f5 +4eff 6 - 4ef 2f6 - 4e4flf7 - 4e e5f42 2e3e5 2 2e e f24 - 4ele f42 -4e 3e5 f4 + 4e2e6f42 + 2e 2e6f24 - 4e2e6f42 - l 7 7 +e64e flf ± e'e2f 2f 6 - 6ee2flf 7 + e2e3f1 3f5 -4e e3f 2f6 2 7 6ele2 2 +9e2e~flf 7 + e5e3f f6 -2e e3 f2 f6 - e5e3 flf 7 + e8lf8 + e4f4 +9lflf+1 3 J6,-- 8 f8+24 4ele2fff f-e flf +e-le2 2e le~flh+ee +2e42f8 + e4f + 6e4f4 + 4e4f8 + e8fl + 8 2 3 2 5 1e3e e f f -3e3e 2e3f 3f5 - 5e3e 2e 3f2f6 + 8e 2e3f 2f + 2 3 1 7 -4ele e3f 2 + e23f22f4 ± 3e2e2f3f5 + 5e2le3f2f6 - 9e2le3f2f6 ef2f -5e2e2flf 7 + e2ef 2f - 2e2ef 22f4 - 7e2ef 3f 5 - 5e2 6 3 -e f2f +2e2e2f 2f6 + 82 e flf 7 + e4e4fl f5 + 3e4e4 f3 5 4 6 +2elef2f6 + ee4 fdlfe~~~~27 + 4ele2e4f4 + eee24f f 2f-2e-e 422f2f3 4 f f -4e2e f f +4e2e 4 f22f4 + 2e2e4 f1f 5 + 8e2e 4f 3f 5 - 2e2e4 6 4 2 6 +8e2e ff - 8ele3e4f42- 4e~flf22f3 + 4e4f1f 2f4 -8e2flf 3f 4 4+8 72e~~flf7 4 43 -3ee f f +8e]fIf2f5 + e3e5fl4f4 + 2e3le5f2f 4 -ele 5fl f 5 5 3 5 +e3e5ff6 - 2e3e5f2f6 - e3le5flf 7 + e1e2e3ff 3f4 -3ele2e3 flf2f5 + 6ele2e3f3f5 + 5elee 3 f2f 6 - 17ee2e3flf7 -2e2Cflf3f4 - e2e23flf2f5 e2ef l f3f4 ± 5e 2 fflf2f5 fif f + e2e e4f2f2f4 - 2e2e2e4f2f4 -ee 2e4flf3f4 -3ee 4 2 5 je22 f gf2 24fA 2if6f 23f flf5f+ I e -4ele 2e4 flf5 + 11ee 2e4 flf 2 f5 - 9ee 2e 4f3 f5 + 4e2le2 4f2f6 flf f -6e2 2e 4f 2f 6 - 1Oele2e4flf 7 - 2e2e4 f f2 f 4 - 4e2e4 2 5 ele e f2 f f +ele 3e4 flf22f3 - 2ele3e4f2 f - ele3e4f2f32- 3 4 2 4 + ele e f f +10ele3 e4f l f3 f4 + 4ele 3e4flf 5 - 10ele3 e4flf 2 f5 3 4 3 5 f2f f -9ele 3 e4 f2f6 16ele3e4 f2f6 + 9ele3e4 flf 7 - 4ee 5 2 4 flf22f +4ele5 flf 3f4 + 3ele5flf2f5 + ele2e5flf2f3 - 3ele2e5 3 e f 2f f -ele 2 e5f2f32 + 5ele2 e5f 2f32- 3ele2e5 f4f 4 + llee 2 5 2 4 -4ele 2e5f2f4 - 10ele2e5flf 3f4 + 8ele2e5f42+ e3e5f2f2 + + f22f - 3e e f3f +3 3ee5fl2f32 - 7e3e5 f2f3 3e 3 e5 f 4f4 8e3e5 4 3 5 5 + e2e f5f +7e3e5f3f5 ± 8e3eff6 8 3e5f 2f 6 - 8e3e5flf 7 6 3 +5ele6f, f -5e e6f2f3 -ele6fl'f4-2ee6f f4 + -e2e6flf5 +8se6f3f5-e 5 -e2e54~fe 6 f 2f6 + 2C2 e6f2f62f± + e426flf7eeff+ + e2e6f4f22 1~~~6f 1 1 1 1 68 -4e 2 e6 fIf3 - 2e2 e6fl5f3 - 9e2e6f2f32f + 2e2e6f2 f32+ 2e2e6fl4f4 -4e 2e6f f4 -2e 2e6fl3f5 - 8e2e6f 3f 5 + 2e2e6f2f6 + 4e2e6f 2f 6 6 2 5 -8e,2 e 6flf 7 + ele7f1 f 2 - 6ee 7 f4f 2 + 9e le7f2f23 - ele7 fl f3 4 -5eie 7f2f + 8ee 7 f2f32 + el e7f f4 + 8e1e7 f2f 4 - ele 7f3f 5 -8e.e 7f 3f 5 + ele7f2f 6 - 8ele7f2f 6 - el e7flf 7 - 32e3e2e3f8 3 +24elee3f8 + 24ee2e4f 8 - 16ee3e4f8 - 16ele2e5f 8- 32e8fl f2f3 +24e 8 fl f22f3 + 24e8 f2f 2f4 - 16e8 flf 3f 4 - 16e8f 1f2f5 + 3ele2e5f3f 5 -8e:e 2e5f l f2 f5 + ele 2e5 f 3f 5 - 3ele2e5f2f 6 + 4ele 2e5 f 2f 6 +9ele 2e5flf 7 - 3e3e5 fl f2 f3 + 6e3e5 flf22f3 - 9e3e5ff 2f4 +e3 e5 flf3 f 4 + e3 e5flf 2f 5 - 5e2e6 fo3f2f3 + 5e2eofl f2f 3 +4eie6ff 2f 4 - 9ele6flf3f4 - 3e+eoff2f5 +8e 2e6flf f 2 2 3 -6e 2e6fj f 2f 4 + 16e2e6f l f 3f 4 + 4e2e6flf 2f 5 + 11el e7f13f 2 f3 -17ele7flf22f3 - 1ele 7f2 f2f4 + 9ele7flf 3f4 + 9ele7flf 2f5 +4efi- 8e8f8 -3e4e 5f33+ 9egf22f5 6f 3f 6 + 6e3e6 f - 9eflf 8 9 e7f -30c ef 9 9e2e7f 9 - 3Oegff -9e3flf + 3ff + 3eff 7 3 -9ele 4f9 + 9egflf24+ 18egflf32- 9e9f 2f7 - 3ef 2f7 + 9e4ef9 +18eef 9 9e3e6f9 + + 93f9e 93ee4f3+ - 9e9f2f3 +9ee 5f9 - 3e3f 4f 5 - 9e9flf 4 - 9e 4e5f + 9e9f 4f5 + 27e5e2fg +3e,'lff2- 3eflf8 + 9e9f2f 7 + 9egf6f 3 - 9eae 6f9 - 9ere4f9 5 -11el fie 3e4f 7 - 9ele8f 9 - 3e2e7 f - 9ef3f 6 - 9ef 4f 5 + 27ef ff22 +3ee 7 f3 9e7e2f4e2f2ee f - 2 e5f 7 -11e2f2e 7f 3f 4 + 9e9 flf42 9 e 2fleff - 3ff -9e,'e 3f9 - 3ee 8f33 9e fe 7f2f 3 ± 9e1f e23f 7 - 3ee 5f 3f6 +3eIe5f33 - 2e4le5f2f7+ e6f2e3f7 -e2f5e7f4 - e3f2e6f7 -e5 j e4f7 + 6e3f2e2f7 + e2f,6e7f3 + e4f]2e5f7+ 6e2f 3e7f32 +e2je2l247+7f7 + efe 4f6 +ef2 3fle 4f4 ee 3f2f3e4f4 22~ 6 e4 fl/ -efall~f6 %f +ef2f7 e¢2fef2 -e2 f12e1 f 3e 4f 4 - e2 e3 2f 3 e4f 4 - 2e 2f2f 3e4f 4 - 2e 2e 3 flf2e 4 f 4 +e2 e3 f2 f32e4 fl + e2e2f2f 3elf 4 + 8e2e3f2f3e4 f4 - 2e2fl2e3e4f 3f 4 -eIJe 3e4f 2f5 + 5efl3e 3e4f6 + elf1 e5f5 - e4f3e 5f6 ±e3Lfe6f4- e 1f66f 5 + elfe6f6 + elf'e7f5 - elfle7f6 -4e f2e5f2f5 -e2 fe 2e5f3f4 -4e2 fe 2e5f5 + 15e2fe 2e5f2f5 +4efle 2e5f6 + 5efle 6f 3f4 + 4e fe 6f 2f 5 - 6e fle 2e3f7 3 +5e f1 2e 4 f7 - 6 fe 7f 2 f3 + 5ef1 e7 f 2f 4 + fle 2e4f 2f 5 -5elfjeC 2 e4 f 6 ef_ e2e5f2 f4 - 5ef1e 6f2 f4 - 4e2f2e 7 f2 f 5 -eClfe2e4 f4 -eifle~f2f 4 - 5elfle4f ef1 e2f8 + efle 6f8 2 5 2 -6elf3esf - elf2e8 f7 - 7e5fle2f8- 7elfl 8f2 + 14elfe 8f23 -elJ'fe 4 57f8 +elf7e 8f2- elf e8f5 +elfle 8f6 - efe 8f3 +eClj+el~ef4e8f4-6efle2f - 66e l9 8 f. 14eefefl2fs e8 feafe]fe1fleJf8 4f4-e4fle e415ef85f8 3 -ef'le 3f8 - 5efl ef 6 + 5elf2e~f 7 - 5e3f le6 f42+ 5e2fle 7f42 +5e2fe 3e 4 f42+ 5elf2e~f 3f4 + 19elflee2 f 8 - lOefle f2 f6 +19elfle2e 4f 8 - 10elfle 2 e6 f 8 + 19elfe 8f22f4 + 19elfle 8 f2f32 -10elflesf3f5-10~f-le 8f 3 f5 --lOelfle10Clfle3¢5f83 e5 fs-3eifle- 3elfle22e 2f42-- - 3elfe2f3ef 4~f2 f32

69 ef42 + 14elfle 2e 6 f42 -6eifie f3 f5 + 14el fl e2f2f 6 -6elfle 3 4 2 f -7elfle42f8 - 5elflef 8 - 7elfle 8f2 - 5elfle 8f + elfle8 8 f f5 e4fle 2e3f2f6- 2e3fl e3f 3f5 -- e3ef2f6 ± 4e fle 2e 3 - e2f7 - 3elfle 4f 2f6 + 2elfle2e2elle4f 3f5 +7elfle2e3f7elfle2 e2f2f6 12elfe- 2 3 7- 1 f3 2 2 f 2~~~~22 f +5elflee4f6 - 12efleie 4f2f6 - 5el ee 4f7 + f 2ef 2 5 f + 3e4fle f 2f6 + eIfe2e 3f 3f 5 - elfle2 e4f 3f5 +4e4fie5f3 5 5 l f f - 13e2f e e4f2f6 +14e3fle2e4f 2f6 - 3e-f2ee 4f2f5 + 5e2fle3e4 3 5 3 - 2efle e f3 2 + 4elfle3 ef 2f 2 -3ef 1e2 e 5f2f4 + e2fle 2 e5f l 3 5 e f f -4e lfl3e 3 e5f6 +2elfle 3 e5 f2f4 + 4el f4e3 e5 f5 + 5elfle 3 5 3 5 le 5f7+ 5efle6f2f4-4efe 6f3f5 +lOelfle3e5f2f6 + 1 5 2 f32- 3efl e e f4 -3elfle 6f 2f6 - elfl e2e6f3 + 7elfle 2e6f2 2 6 f - 3elf3e e f 6 -12efle 2e6f22f4+ 3elf4e2e6f 5 + 10elfle 2e6f 3 5 2 6 f22f +8elfle e f f6 + 3elf2e 2e6f 7 - 12efef 2f - 5e2f1e 7 4 2 6 2 O~~~~3elfle2ef 2f, 2 e4 f, f - 32f 2e3f8 +lOe fle 7f3 fs + 3e2lfl e7 f2f6 + 13e41 f e2e3 8 1 f8 + 13elfl4e f f3 -12elfle 2e4f8 + 11e21fle3e4f8 + 11efle 2 e5 8 2 -3Oelf2e8f22f3- 12efae8f2f4 + lle1fe 8f3f4 + llefe 8f2f5 e ff - elfe 3e5f2 f4 -15el2fle 2e5f 3f5 - 11e2fle 2e5 f2f6 + efe 3 5 3 2 e f f + 14elf e2e6f2f4 +5elf2e 3e5f 3f4 - 15elf e3e5f2f 5 + e1f4e 2 6 2 3 -13elfle e6f3 4 - llefe 2e6f2f5- el f22f5 + eef23f3 2 3 3 3e e f 11ee2f 4f 5 +_le3f22f 5 - 3ele 3e5f3 - 3e2e3e4f 3 - 1 2 6 f + 5e e4f2f7 -9ele2f 3 f6- 9ee2f 2f7 + e1e4f4f 5 - 3ele42f3 6 l e f f f + 8ele 3 e 5f22f5 -2ele 2 e4 f22f5 - 6ele 2e5f22f5 + 5ele 2 5 2 3 4 + e ff f -2ele e f f f4- 2ele3e5f2f 3 6ele2e4 f2f 5 - 2e2 e3 4 3 5 3 5 2 3 efff3 -6e2e3e4f22f5+ 5e2e3e4ff2f5 + 4e2e3e4flf2f6 - 4ele2 6 f4 + 2ele e f23f3- 4eee 3flf2f6 +4ele2e6of5f + 4ei~4f~f46 1e2c6f2f3 2 6 -4e2~ff +4.e2e62ifaA e f4f5 - 12ele e4 f3f6 -3ele 2eff5- 2eef 2 f3 f4 4ele2 4 2 ° + e91f + 3e3fg + e3f33 + 7e2e2 e5f 4f5 -8e3e 2e4f 2f7 + 9efg egfl+ ± 3egf33 f f - 18ele e f2f 7 +9e 2e2e5f3 f6 + 6ele 2e5f 2 f7 + 2ele 3e5 4 5 3 5

C2 ~~~ 22 ~~~~~~~~ -3eceae 4f4 f5 + 9ele2e 4 f3f6-ele 2 e4f2f7 - 2e2e3e4flf

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3f2 2 ~ ~~~~~3e2f +19e+6e]fl'esf2f4-13e9fe8f3f4- fle 2 e5 f2 f 6 ef2e 3e5f22f 45e9f+ 3elfe8f2f5I e 3e 5-f efee2e5f3f53f 5 +5elfl e3e5f 6 - 19e2fe3 e5 f 2f 6 - 5efe 3e5f7 + 5e fi2e6f 3f5 +4ef 2e6f2f - 4efl2 e2e6f22f4 - 4e2fle 2e6 f5 - 19e2f2e 2e6f 3f 5 __864f46165]~ +2f2 2 - 42f2e2 +e2f3e2 3 -8edLf4ele5 fl + efIe2e 4 f 3f 5 -4elf e2e4f 2 f6 + 9elf 1 e24f 7 -4ef e5f2f 6 + e4fe 2e4f 2f 6 - 2ef2e 3e4f 3f5 - ef2e 3e4f 2f6 -21e f2e-21(:,~~i2eaf3 8 +le2 4e2fe-1 8f92 f6 -9e2f2e2eIle2eaf8f 4f8 + 4efe1 e2C6f82 e6f8 2,, 2 2 fC2e 2 -9e fl e8 f~ f4 - 21ef2 e8 f 2f32 + 12eYf2e 8f 3f 5 + 12e12f2e 3e5f 8 +2e'fe4f3f52¢rer2 2+ elfe4fe2f62 2 +2 2fe3e5f4 2 e2efe2e6f2 2 2 -e61j1 e4 f8 + e5f e5f8 + e7f2e3f8 - 6e2f3e2f 7 - 6e3f2 2e7f42 2e 11~-~~~~~- 6elf 4 1e1f +6ef f8 + 6e2f]e8f42 e2e 8f-8 efe 9f - 7elflegf33 -7efle3fg - 31efe 2e7f4f5 - 3elf4e2e7f6 - 10efle2e7f3f6 +3e fe 2e7f 7 - 8elfle 2e7f 2f 7 - 3eif2e 2e7f 8 - 11e2fe 8 f 4f 5 2 -11el fle 8f 3f6 - 3ef,1 e8f2f 7 - 14elf14e9f2 f 4 - elfl4e3 e6 f 2f 4 +17e fle 2e7f 2f 4 - 4e1fle6 f3 f6 - 3e fle 6f 2f 7 - 3elf e3e6f42 4 -2elfl e3e6f3 - 7elfle3e6f4f5 - 4e flae3e6f6 - 4elfle3 e6f3 f 6 +4elf3e 3e6 f - 10elfle 3e6f 2f 7 - elf2e3 e6f 8 + 11e3fle7f 4f5 +4el'fle 7f 3f6 3el f el 7f 2f 7 + 17elf12e 7f42- elfl4e2e7f32 -el j e2e5 f/ - e 2fee5 f4f5 + 5e f4e2e1 5 f6 + 18ef l e2e5 f3 f 6 -5elf ae2e5f 7 + 12elfe2e 5f 2f7 + 5elf2e22e5f8 - 8elf2e4e5f42 +23elfle 4 e5 f 4f 5 - 5elf4e 4e5f 6 - 7elfl e4e5f3 f 6 + 11el fj3e 4e 5f7 3 -31el 2lflff7 -llefe- 4e5f8 - 5ef 1e 6 f4f5 13ef egf3f4 -46e lfl3e of22f3 + 33eif2e 9 f2f32- 12elfl2e9 f3 f5 + 13elfl3ef 2 f5 -42elfle 9 f2 f3 f4 + 15efl5egf 2f 3 - 12efl2egf2 f6 + 33ef egf22f4 +ellee 3f 4f 5 -3elfle2e 3f3 f6 + 5elfle2e 3 f2 f7 -7ef- e 2e3 f8 -3e6fle 4f 2f7 - 19elf3e3e6f 2f 5 + 3elfe 3e6f22f4- 12e3fle7f2f3f4 3 -16el fl e2e7f 3f 4 -5elfl3e 2e 7f22f3 + 9elfl2e 2e7f 2f32 +13e f2e 2e7f 3f 5 - 14efj3e2e7f2f5 + 21elfle 2e7f2f3 f 4 +elf'5e2e7f2f3 + llelfe 2 ee7f 2f 6 - 23elf e2e7f22f4 + 26e fe 8f 2f 3f 4

73 - e f2f -7efel 8f23f-3 + 4elfle1823--3e6f5 - 3elfle3e6f23f3 5ef l 7 5 + 3elfl5e e f + 5elfle 2e7 f23f3 -3elf1 6e2e7f4 + 12elfle 2e7f2f5 2 7 5 2 I~~~~~~~~~~~3 e e ff +5e2fle 8f22f5 + e e,fl4e5f2f5 + 18elf2e4e5f2f 6 -elf1 4 5 4 2efl2e2e f f +5elf3e 3 e6 f3 f4 + el fe, 3 e6 f22f3+ 6efl2e 3 e6 f2 f32+ 5 3 5 -3ef ee 5f2f5- 3efl ee 5f 2f3f4- 17elf2ee 5 f 2f6 e f23f +5elf3e 4e5f3f4- 3ef2 e4e5f2 f32- 12elf2e4e5 f3f5 + elfle 4 5 3 f +5eifle 6 f22f5 + 18elf l e3e6f22f5 - 2e4f1 e f3 f6e 4f1 e3f2 7 f -14e4fle 2e4f9 + 13e3f1 e2e5f9 - 12e fle 3e5 f 9 + 33e2fle2e4 9 ee f -42elfle 2e 3e4f9 - 12e fle 2 e6f9 + 13e3fl e3e4f9 - 46ef l 3 9 f f +33e2 fle 2e2f9 + 15e5fle2e 3f9 - elfl e4e5f22f5 + 20e3e7 f l 4 5 - 4e f e flf +5elf e2e5f7 - 4ele 3f4 e6f6 + 3eloff 3 5f -lOelof2f 4 4 6 5 2 e f f +4e4f2e le5 f2- 4e4f4 ele5f 6 + 9e4f4e6f l -14ef2e 6f2 + 4ef 4 2 l 5 5 e2f - 14ef e f -18e2 f4 el e5f 6 - 8ef 4e6flf 5 - 4e4fl e6f 5 + 9e~f4 6 4 2 6 -8ee3f2e6f6 3lfe 6f7 f-2efe 4f3f5 + 2e2fe4f2f6 -4e2f-e2f2eef2 e6f2f32 + 8e2f e6f22f4j~4 + 1Oe6f2+2fe6f3f5 + 6e2f,2e6f2f6e2 f2esf + e f22ef32 +4e2f2 e8flf 7 + 4e2 f2ele7 f8 + 8e2 f2e8f3f 5 + 6e2 4 2 8 - + 1Oef2ef6 +e'2 2 f2e332f 4 - 22~ef22e3f2e4ff3J4f64 e2f2e f -+ 222e2fef4 ~ ~ 2 -4ele4f 2 f +2efaef 12ff7 +12e± I lhe~~~~~~~~~e4ele3e3 e4 f7 ± 3e-f3ef3 e7f4 -3elf 3e77f7f7 3 7 + 3e6f e f + 3e4f e2f7 - 3ef 3e5 f7 +12e3f 3 e2e 3f7 + ~e4f~ 3e~f7 ~ 3 ~~~~43el7 3 fae 5 2 f -3ele 2f 3e3f7 - 9ele2 f 3e 6f7 - 2ele2f3 e3f7 - 15e4e2 f3 e4 7 23 -9elef 3 e5 f7- 24ee2~~~~~~~2f 3 ef7 + 3ele2f3e3 e4f7 -7eef2e 7f 3 f e f +ele 2f3e 7f 4 - 12e2 e 8f22f3 + 24e 2 f 1e 8f 3f 4 + 2e2 8 8 5e3f e f e f +5elfle 4 e5f33- 21elflegf22f 5 - 7elfie3f 3f6 + 2 2 3 5 5 f e f f -ele 2f3 e7 f7 + 3ele2f32e5f4 - 3e2e2f32e6f4 15ele2 3 6 2 5 2 2 2e f e2f +15e2e2f 3e 6 flf6 - 12e 2 f2 e1 e7f4 + 2e 2f 2 e2f4 + 2 2 8 22 44 4 e f4 +2e2fef 6 - 4e2f22e4f6 + e2f4e 6f32+ 2e2fe 6f5 - 3ele2fle 3 4 + f e f f -15ele 2f3 e3 e4flf6 + 12e2 f 2e8f5f 3 - lOe 2f 2e8fl4'f4 4e2 2 8 3 5 -12e f e e f4 +4e 2f 2e3e5f 8 - 12e2 f 2e4f 3 f5 - 12e 2 f2e4f f7 2 2 3 5 +IOC2e3 4 + 6ef62 f3 e2fe22f2 +1e0~f2e6f2 + 5ef 3e7f2f5 - 5ef 3e 6f2f5- e3f3 ele6f7 3 7 f +1ef 3 elf7 - 11e3f3e4f7 + 6e3f32e7f2 + 11e3f33e7fl - 11e3f32e7 4 2 - 15ef4 f e f -e3 flf 2 e7 f7 - 24e3f f2 e7 f32+ 3e3flf 2 e7 f3 f4 3 2 7 4 -e2flfef,24efl f52f,5 -7e~flf 2e2f 7 - 3e3fl f2e2e5f7 - 3e3f f2e7f3 + e flf 2e4f7 3e fe f32 -2e 3flf23e7 f 3 + 12e3fl e7 f3f 4 + 12e3fle 7f22f3 + 3 7 2 2f 2 2 -5e2ff23f3- 15e2flf4f5 -3e3 f3e7f7 -4e 3 fj2f4e7 + 5e f f4 +lOe2f2f3f5-15e~ff2f6 + 1Oe~flf3f6-5e2f3f2f5 + Oe52flf22f5 +5e2ff2ff32 + Oef12f27 -15ef2f3f5555 - 5e2f3f3f4 - e3e3f2f3e4f5 0e2;2;1 2 2 +~e e2~~~~~~~~~5J10J1f3f+f f 3 e e f -4e e3 f2 f3 e5 f5 ± e3ef 2f 3e2f5 - e~flef5f3ele2f3e 7f2 f5 + 5e3 1 2 5 7 22 3+1 2f3~~~62e~102, ~ ~ 3 +3e fe 4f7 + 18eef 3e4f7 + efe 3fe 4f5 -2eef 2f3e4f5 -13eje2f2 f3e5f5 + 9e2f_1flef5f 5 - e2e3f 1f3e5f4 + e2e3flf3e 5f2 e e f -15e flf 2ele 6 f3 f4 -4e 2 e3 f+f 3 e5 f5 + 18e3 flf22e7f4 + 15e3 ff 2 2 5 6 3 f6 + 5e f 3e f e5f5 + 3e3 f 22e4 f 5 - 3e3flf 2 e4 f6 +15e 3ff 2 ele 6 2 3 2 3 +Oe21e4 28 -Oee 4ff 8 + 12e3f e7 2 5 -3eff 2e4f3f4 +C~2f-e4 + 182fe+ ef7flf5 - U3 f~' ff

74 4 -5e 31f e2 e5f 6 - e3f3 e1f 2 f5 11e3 3 e7 7 + ee33 4fl - 3e3 f3e7 f5 +3e,3f4e f -e3f32e f - 2lf22e f + 2e'af e f + elf e2f i 7~6 ~ 324 4 2 8 62 e2fl3ef2 6 8 2 8 -2ef 2e7f8 + 2elfie8f4 + 7elf22e8f32+ e2fl el3 3-e 33ff 2e5f7 3eafa~eflf6 1ef5 -9e 3 f3 ele 5f 7 32f 3 eff 6 + ef3e lf 4 + 17e 2e3f 2f 3 e5f 5 +5e2e 3 f2 f3 e5 fif 4 - 2e2e3f2 f3 f 5 + 5e2e3 f2f 3e le4f5 - 2e2e3f22f32e5 3 -13,e2e3flf22e5f 5 - 2e2e3f2f2e 5f4 + e6f e4f7 + 4e2f 2ee 4f 8 -6e:'f 2e2e6f8 - 12e f2 e8 f3 f5 - 13 f2ef 8 + 2e'f 2e4f8 - 2e5f2e5 f8 2 -2e!f2e3f8 + e f2e f4- ef~e4f6 + 6ef2e7f4 -6e4f2 f8 -1Oele 2flO- 1Oee4fo + 1Oel7e3flO + 1Oele5flO- 1e2e 4fIo + 1Oee4fIo 2 +10eof 4f6 - 1Oelof32f 4 + lOeloff3 + lOeloflfg + 10elof3 f 7 + 35elof6f 5 -5O0e1,of4f23+ 1Oeo2/ 4 + lOeofl 5 + 1OelOflf7-f 1Oe1of8f 2 - 1Oelof4f -10el~~~ ~~~~o/l24 + ofx+ 6 - Oelof2f ± 15of 2 52loflfI25e 42 + 5elefo4 - 5eefo + 15e2e2flo +10e3e 7 flO - 1Oe2efO + 10e e fl + 1Oele3flo + 5e2e f52- 5ee f52 ~2 2f2f 8 ~ lg3 7 0eff 3f 4 f 4 -5e f4f6+ 5eff45~~ - 5eff5%f ~-5e5f3 ~ f7 ~- 5eff4-5ef7~~~ ~ f755 +5ef 2f 2 +5eflf62f146 5ef2f 2 2 2 + 5e5f1fe2 2 2 2ef9+ 2e52f3f7 +60e(-loflf2f3f4 - 1e 2e8 flf] + ef22ef 6 + 2ef22e 5 f 6 - 2e4 f22e6f6 -2e,]f2e 7f 4 + 20ele4 e5 f2 f8 + o10ee4 e5 f2f 3f 5 - 10e4 e6f2 f8 -8eff 3e6f6 +lOeef 2f42- 15elegf22f32- 1Oeef2f 4 - 10ele9f3 f7 + 1Oelegf3f4 -10e, 1e9f4f6 + 30elof2f 2f 6 - 20el0flf 3 f6 - 40elof3f 2f 5 + 30eloflf22f5 -6Oeof2f2f32 - 20eoflf 2 f 7 + 1OOeloflf22f3 - 20elof f f -60elof5f f 4 2 3 5 2 3 -40elof3f 3f 4 + 50elof1 f2f4 - 40eoflf23f 3 - 20eloflf 4f5 - 60eof 2f22f4

-15e 3e 7 flf ± 1Oe3e7 f2f42- 1Oe3e 7f2f 4 - 10e3e 7f lf9 - 10e3 e7f 4f 6 - 2e2flo 2 2 f22 C -4eif2e 2 ef 6 - 9e3 7f5 - 9 3ffflf22ef2e7f6 -3ef 3 e4ff 5 2 e3 J3e4f2f5 + 13e3fe 4ff 6 3e3 ee 5f4 15e3 3e2e5ff 6 +10e2e3e5f1f4 f5 + 1OeIe8f3 f7 -1Oefe 8 f32f4 + 1Oee8f 4f 6 -Oele 7f4f6 -9e3,'f 3e7 f2 f5 - e3f 3e7f 2f 5 - e3f3e7flf 6 - 19e3f3ele6f2f5 -e3f3e 2e5fl + e3f3e2 e5f 4 - 19e3f3e2e5flf 6 + 3e3f32e1e6f22 +13e3f2e 1 e6f 4 + 15e3f 3ele 6flf 5 + 20e2e8 f lf4 f5 + 20e2e8 flf 3 f6 -5ele 2 e3 e 4 f52 + ef52 + 20ele 3 e6 f f + lOe e6fl2f + 5e2e ef52 ~~~~~2 2 8 4 5 8 2 -15ele 4e5f5 - 15e2eiff - 5e~flf 2f3 f4 + ef25 + 5e~flo + elof10 -2elof2 + 5elof52 - O1el o flo + 1Oe2f52- 1Oe3e4f 3f 7 - 1Oe2e4flf 9 2 +1Oe2e 4 flf 9 + e3fl4e7 f23 + 3e3f6e 7 f4 - 15e 2e3e5f5 + 20ele 4 e5f 3 f 7 +20e4e6flf 2f7 + 20e4e6f 2f3f5 + el°f1o + 20ele2e7f4f6 + 5e2ef -5ee 3 e - 5ee 2e 5 f52 + Oe1e 2e 7 2 + 5e2ee 4f2 - 15e 2 e6 f -5ee2e 3f loee+ 5f 1Oelee+ 6f +2Oel 5f 20e2e3e5flf9 +20e2e3e5f4 f6 + 60ele2e3e4fIo

75 Table of Pm,2 for m > 3

m Pm,2 2 ele e ± e e - el2e + ele7 + 2e2 e6 - e 8 4 e1 e 3e 4 - 3ele 2 e 5 + 5 - 3 5 6 2e e e 5 e4e6 + e2e4 + 3ele 2e7 + 3ele 3e 6 - 4e2e2e6 - 2ele 4e5 - 2 3 5 2 + ee - ele - 2e2e8 ±e2e3e5 + e1 o- e3e7 + 2e - ele7 - 2e4e6 + 22e 6 8 9 e4e + ele e4e5 - e + e4 - e21 elo0 6 elell + 3ee 6 - 3ele2 e3 e6 + e 5e7 - 8 2 1 2 3 ~ ~2 ee 3e e e - ele22 q-e 2 +2e e 2 ±+ e8 e+ 9 e3e9 - 2eI + e1e7 - 3 4 5 2 1 2 ee3e7 +e3 232 5le2e - 22e + 4e e e +±le 3 e6 - ele 4 e6 + 3ele 5e6 - 5ele 2 e7 + 7 8 3 7 e -3e 2e3e 7 - 3ele 4e 7 + 4e2e2e8 - 3ele3 e8 - 3ele 2 9 e2 2 2 3 7le2 2 e2 + e2 2 ee + e e - ee e 7 ele4e5 5 + 3 + 4 e - 2e 6 3 5 6 5 6 e + 4ele3e7 +2 e2e e7 + e4e3 e7 + 5ele 3e8 +3e e 5e7 - 4e 3 e 4 e7 - ele4 7 3 - 4e2le e +9e2e2e8 - 6ele 2e8- 4ele 6e7 - 4e2e5e7 + el4 + 3e2e3e9 3 9 ± 3ele ell + 3ele e 0 -5ele2e9 + 5e3e2e9 + 3ele5e8 + 6e2e4e8 - 5e+e4e8 2 3 1 e ele4e9el+ e e 2 -2e 2 e1 2 - 3- 2e2e8 + e6e8 -4eIe 2 10 3 2 22 + 2 e e + 2ele e e7 -2ele 3e 5 + el2e2e 4e6 - ele 3e4e6 + ele 2e5e6 - 4e2le2 3 7 2 4 2ee - el-e - 2e6 e8 - e9 -9ee 2e3e8 + ee 6 - e 5e 9 + e 1o + 2ee 1o - 4 10 e e6 + 3e2e - e4e e7 - 3e4e5 e7 8 ele2e 5e6 - 2e2e3 e5 e6 - e2e3 5 8 5 - 3ele e e - 6e e elo0 +2ele 6e 7 + elae2e4e7 - 2e2el 2 + ele 3e5e 7 2 6 7 2 4 4e2e e -3ele el - 5ele 2e11 ± 5ele2ell - 3e2e3e11 - 3ele4ell + 2 12 5 0 2 -3ele2e e7- e ee4e7 -3ele 3e12 - 3ele2el3 - 17e e2e3e9 + 12ele2e4e9 4 e e6 - e2le e6 +5 3 7 3ee 2e 5e7 + 2e2 e1 4 + elel 5 + ele 4 5 2 2 e e - -e1e16 + e22e+4 +3ele 3 e - 2eee e 1 - + e 6 9 - e2e + ele e + 5elee 8 -5e 2e3e8 +2e 4e12 - 3e3e6e 7 + 2e 2e7 + e 3 e1 3 14 3 8 e3 2 + e~~e2 2 2e2e 2e6 - 7e5 e e 14elea23e9- 2eg -5e3e2 e3e8 + 5ele2e 3e8 ele4 6 6 e 2 9 3 4 e2 2 ~~~~22 e e2e6 + 9ele e e,10 + 4ee 3ell -7e le2e 9 + 6ele 3e9 + 5e2e3e9 + 3ele3e 9 + 2 3 - 6e e4e - 3e e e9 + 4e e5e9 +e 5e1 1 + e7e9 - ee1 --2o2eee - 9 3 4 - - 5e3e3el0o+ 2e el0 - e4le4e8 -3e 2e 5e 9 - 3ele6 e9 + 6e4le2elo 9ee2elo 6 e +- 3e2e e e - 8ele 3e 4e8 +2e3e 5e8 + 5e3e5e8 - 3e2e6e8 + 2e2e6e8 + 5ele 7 8 2 4 8 -4ele 2e5 e 8 + 5e2e4e1o + 2e3el0 + e3e3 2 e e + 3ele e6- 8ee 2e1o 9 - 12 -2 2e2e5e7 - 3 3e5e7-2ee5e 7 + ele2 4e5 6 5 el3 +20e4e2elo - 16e2e2elo + 7e5le3eo + 7e2e2elo - 3e2e2elo + 3e2e3 6 ~~42 e2e2 e2e 2eeee-ee + ele7 - 2e e7 - 3e3e4e5e6 - e5e8 -6e 2 e7e9 + e61e2+ 3e5e6 e7 + 2ee7 4

2 22e3 2 + - 27ele + e4ell-9ee6e e e - 27ee2e3 e 12e 3+e 2e2e0 1 4 -ele±7ele~e17 +3 64e- 3e3e 6 +ee 6eeseeeeio2e 4 e1 2 + 4e2 + 6ee 5el0 2 2 6 10 - 2e el6 + e6e e9 +19eiee 3eo + 21ee 2e4eo - 8ele3e4e 0 - e3el5 + e2e 2 3 - 1lOe2ee eo -2e2e 3 e9 - 12ele 2e3e + 4ele 2ee9 - 2eee 4 e9 3 4 e2 3 ~~~~~~~~~32 9ele e e + 6ele e6e9 + 3e3e9 + ele4e7 +6e2e3e4e9 - 7ee 2 e5e9 + 3 5 9 2 +e1eee8 +6~~~~~~~~~~e2--5eale+ 9eeel2 1 2 ee 6e8 +3e 3e e7 + el8 - 2e8el 0 - 6e41e2el2 -3eIe e -6e e e eCe36e9 -ele 4e9 + 5ele4e9 + 2ele5e9 + 4e2e5 e9 6 9 3 6 9 - e el3 + ele + 3ele e10 +5e2e 7 e9 -6ee 8e9 + 2e4el 0 - 2e2e3e5 8 5 14 7 +7e5e2e - 14e3e2ell + 7le3el - 6e4e3 e11- 5e2e3ell - 3ele2e

76 +6e;e e 4 1 1 + 3e 3e 4 e11 - 4el2e5 e1 1 + 3e 2 e5 e11l + 3ele 6 e11 22e 4 e8 - 3e3e 4 e 8 -4e 2 e64e8 - ele 5 e8 - 8e2e 4e1 0 + 4e1 e 2e3e5 e 7 - e62ee 6 - e2e5e6 2e+ee- + e2e e6 - 6e e e ±+3e e e +`.:- 2 ~,e-e~C4 6 4 5 9 l 3 14 - 7eeee4 e 1 0 ±+ e2e3 e5 e8 +2e1 e4 e5e8 - ele 2e6e8 - 3el e3e6e8 + 5e l e2e7e8 - 3e1e2e3 6 e64e10 +el4e 2 e 4 e8 - 42ee 4 e8 - ele 3 e4 e8 4le 2 e5 8 - 3e1 26e5e 8 2 -e1el + e3e5 - 2e2e6e8 + 6e4 e6e8 - 3ele7e8 + 3 3e7e8 + 2ele8 - e 8 -2eel4 - el3e15 - 2ee 12 - 12ele2e5e10 - 6ee 2e3e9 + 9e2e 3e9 + e8lel0 -e711 - 5e 6 e 10 + 3e 3e5 e10 + 4e 2 e4 e5 e 7 - ee162 e6 e 7 + e1e2e6 e 7 .5ee 3 6e e7 - 3e2e3e6 7 - 5ee 4 e6 e7 - 32e 2e6 - 3e1e2e42e7 e12e22e5e7 - 2ele3 e5 7 - ee1 3 - 5ee 4e 12 + 3ele5e1 2 + 5el3e2e13 -5ee22e13 - 4e 3e13 + 3ele4e1 3 - 4e62 e14 + 3ele2e15 10 -el4,e2 8 57 ± 46e2- 3e5e 7e8 + 5e2e 5e11 4 +8 e9 2e3e5e9 -3e;e 2e7e8 - e6e5e9 - 7e e3 e4 e9 - 4e l e1e4e 9 + 3ee 5e9 + 2elee 7e8 +e5e e - 3e e e + 6 9 4 7 9 e15e2 e 4 e 9 - 5e2e, 4 e9 2e 2e3e7 e8 - 34e 7 e9 -3e6'e 3e7 e 8 + 4ele 4 e7 e 8 + 3 2e - 32e - 3e 3 e8 e9 - 4e 4e5e 6 - 5e 2 e1 5 +5e e2e + 42e 1 15 3e15 - 3e2e3e1 5 - 3ele4e1 5 + 4e2 2e 16 - 3ele2e17 - 3e 5e14 2 2 + e7 e2 +2eie2e 8 + e 3e5e6 + ee 3e10 + ee 2e3e5e9 + 2ele3e8 +19le 3e 2e4e 0 + 4e 3 e42e9 - 22e 16 - 3ele3e1 6 + 7e1e e1 0 + 16e2e3e 12 -76 e 3e1 2 - 7e3e 12 + 7ele3e1 0 + 4e3e4e5e8 - 4e2e6 e8 + 3e2e 6 e8 2 2 4 +7e2e e o - 4le e e + e + 2 C 3 1 2~~~~~~~~~~-11 3 4 9 4ele 4 e 5e 9 3e--2e4 e 5ee99 - 2ele 2e5e6 - --e6le4e10 +8ele2e5 el0 + 14e2le35e1e - 13e2e3e5 el - 13e1 e4e5e10 -9e7e 2 1e1 + 2e4e16 +e54 - 2e 2 ee 1 0 + 6e6e,1 0 - 7e 4e32e1 0 - e2Le5 e6 e7 + 5e 2e 5 e6 e 7 3 2e3e1 2 +7e4 e4 - 4 1 2 6e3e5 e1 2 - 3e 3e 5e1 2 + 5e2e 6 e12 - 3ele 7e61 2 2e 5e10 + 8e26e4e1 2 -4e2 - 24 6e10 62 10 - 3e 2e7e11 - 3ele8 e11 8e6 2e 2 - 2046e2 12 +4ee71e 1 - 5ee2ee 9 + 4e4e6e10 + 2 8e12 + 4e3e7 10 - 52 8 e610 - 4e4e8 - -e6 1 4 7ele2e3e 10 21 2 ee e9e1 4e 2 e8e10 - 6e3 4 el3 - 812 _2 e2e2e -e 2e e e 1 3 6 9 3e 2e3e6 e9 + 2e le4e6e 9 + 7ele 9e10 - 6 1e2 e5 e9 + e 5e1 5 + 2e6 e1 4 +2e e2 - 3e26e - 6 4e 39e4e 2 e3 e11 45e2e2 e 3 e 1 27e,5e2el1 1 12ele 2e 4e1 3 -1317- e4e12 7e3e7eI - 6e 2e6 e1 2 - 17ele 2e 3e1 3 + 3e 3 e5 e6 -21ele 2 e4 e12 +8e 1 le3 e 4 e1 2 + 12ele 2 5e 12 + 3e3e 3 6 8 - 3e2e 4 e6 e8 - 7e5e 2 e3 e10 +14ee2e 3elo + e,2e 2e4e5 7 - 30e3eell- 25ele2 e4ell - 17ee 3 e4 el +12e 2 e3 e4 e11 - 20le 2 e5e11 + 9e l e3 e 5 e11 - 3e6 3 e5 e 9 - 3e 1e5e9 +4e2e5e7 - 2ee 12 + 6e 2e3e5e8 + 6ee 5e6e8 - 4lee 7 + e3Ce 2 +9ele4e 1 1 - 7e5e 2 el 3 14e3e2e 13 - 7ele3e 13 + 6e4e 3e1 3 + 5e e3e13 d-26~1 2 2 2 e2136 26462 -.1 ele 6361 +3ele3el3 - ele2e7 + 2e e3 e7-e 2e4e7 + e7e5e7 + e3e17 - ~- 7e16267 + -- -3e e2e3 e6 e 7 + 5e4e 2 e4 1e 0-5 2ee4 e1 0 - 7e1e 2e8e9 + 8e6e 3 e11 +12e:3eIel - 7ee e - 8e5e e + 5e,4 e241 3 11 4 e2 e5 e 9 - e1 e3e 4 e5 e65~7 - 4e2e7 e9 +ele1 5 + e2e4e6 - 3le 2e 6e9 + e1e2e6e9 -e 18 262 18 - 4ee 2e4e8 -5ele 2e 3e 6e8 - e3e 5e 7 - 3e 3 e5e7 - 3e 3e4 el3 + 42e 5 e1 3 - 3e2 e5e1 3 2-- ~~~~~~6461 e -3e(e-6 e13 6e'e 2e14 - 9ele2eM - 5ele3 e14 5e646 14 - 6e 2e4e14 + e17e13 9ee 2 e7 e9 - ele 3 e7e 9 - 3e 5e6 e 9 e7 e13 12ele e e + ele e e e 3 2 2 6 11 2 4 5 8 +4ele3e~e8 + e e ee - 3ee 3ee - 2e4eee + 3C2e +4e('C~e8 2 _ 120582 e 165568 + 326558

77 -7ele2e 5 e8 ele 4e 2eee 8 4e ell + e9ell + 2ele2e4e6e7 2 - +2e4e 6 + 2e e - 2e2e6 + 9ele 2e3e14- 2ee 3e 4e + 9e2e2e3e4e9 e4elO - 9ee e elO .±elel9- 151e2e 2 ell + 32e3e2-3e4e1 + 7e4e5ell - 12e3e3 2 5~~~~~~~~~~~~~~~~~~2 -e20 -- 2c2e4 5 + 2eee7 2ee7 e2e14 - 4e2e e8 + 5ele2e4e9 - ele2e3e7 e7elO- e4le16 -6e3e 6 ell - 3e4 e5ell + 5ee2 -e6elo-8ele 3e 6el0o- 8ele2 +27e3le2e3el2 - 19ele2e3e2 + ele2e6 e7 - e2e3e6e7 + 5le32e6e7 ±ee 4e 6 e7- 5e3e4e 6e7 - 2e2e 6e8 - 3e3e6 ell

78 Table of Pm,3 for m > 3

m Pm,3 3 4 -el 1 + 3e e6 - 3e le 2e 3e6 - e 5e 7 + e2e6 + ele 2e4 e5 + el2 + e4 2 +elelo - e2el0 - e2e4e6 + 3e4e8 - ee 9 - 3e3 e4e5 e1 25 + e2e5 ±ele 4e6 + ele 5e6 - eCe2e7 - e2e8 + 2ele 3e7 + e 2e3e 7 - 2ele 4e7 +e2e 2 e8 - 3ele 3e8 + 2eze2 e9 2ele2elo - 4e2e3e10 - 3ele4elo0 - 3ee2ell + ele3ell + 2ele2e12 -3ele3 e4 e7 + 5ele2e5 e7 + e2e42e5 + e4ell + e6e8 + 2e3e2 - e3e9 + 2eje 3e5e 6 - 4ele 2 e6 - 2ele2e4 e8 ± ee 3e8 + 3ele2e6 -2e1 e4e5 + 3e2e5e6 + ele2e 9 - 2e3le3e9 + 2e e4e9 + 3e2e4 e9 -4ele 32 + C22 + e~2 e2 5 e 9 - ele 2elo0 + 3e 3e6 - 4e4e 5 e 6 + e el - e2el 3 e+1 5 + ele 3 e5 -2e 2e3e5 ee 2e3e9 + e2e7 - 4ee2e3e7 + 3e 2 e4e7 + e4e7 -e3e2 - e3e4e8 - e7e8 e2e11 + ele2e4 e6 - 2e2e3e4e6 - e 2e3e4e6 2 2 +4eej0 + 2e2e2e7 + 4e ee7- e4e, + 3e eeo + el+ e e2e6e7 -2el>e6e7 - 4e3e5e7 - 4e2e4e7 - ele14 - elee 8 - 2e 2 ee5e6 32 a232 -2ece 5e7 - 5ee + 5e e e + 3ele23e - 3e e e + 3ele e e + 2e,5 3 3 3_~~~22e2 8 3 4 8 5 8 e 2ee_eleaese5 8 2 3 8 6 e e5 + ele5 - - e e e e - 22e e e - e 3 3 ~ 4~ 3 25 ~226 e23 5 6 eee 5 6 -3e e4e5e6 + 4e2e2+6C3e,10 ~eleeeee 6 + 4e 4e 6- 82+e2elo8e2e4e6-2 4eee4eee5e6 -2ele2e3e62± e3 3e9 - 2ee 9 - 2ele3e~ + ee 0 4e2ee 6 +3e]e 3e6e8 - 5ele 2e7 e8 + e8 e2e8e2e 5e7 - 5ee4 5e7 5e5e6e7 ~422 2 2 -2e e7 e 2e7 ele 2 e 5e1 0 - 3ele 2e 3e11 -ele 2e4 el l + 2e26e +4ece+4eje~ea e+2 2 _ 2e2e2 a3 e2 leC 2e3 e5 e7 3 e6 + 6e6 el12 - ee2e q 6 e6 le 2e4 e7 _2 52C 4 -e 2e 3e4e7 + 8ele2e3e12 e3e4e11 - el5e1 13 + 52ee 8 + ee 6e8 +2e,;e 6e8 - e 4 e6e 8 + e3 e 7e8 ee 8 ± e2 e + 5e2e - 3e e el 4 2 3 +5eleel 2 3 0 +e41 e4el0 + 2e2e4 el0 - 3ee5e 10 6e3e5el0 e12e16 - e2e1 6 - elel7 +5ee2e 4e 7 - 3ele 2e2e 7 + 4e2ee 4e8 - 3eae 3e4e 8 - 3ee 2e5e8 -3e e 2e5 + 2elee 1 1 + 4e2e 5e1 1 + 2e 2e 5ell - 5ele6e1 1 e4Le2el2 -3e'e2e1 2 - 3ele3e12 3ee 4 e2 - 3e2e4 1e2 - 5ele5e 2 - 2ele3e4e6 -e3e 15 + ee 1 2 - ee2 e3ele4e 7 + ele 3 e4e - e24el0 + e18 + e4e5e 9 -e7 ell + 6ele 2e 4e 5e 6 - 2e3e4e7 + e2e14 - 3e2e 2e1 4 - e4e14 - eel0 +3e3e 7e8 + ele 8e9 + eeel 0 3ege9 - 5ele3e8 - 5e~e4e8 I 1 2 2ee333 2 e4e 23 e2ea -5e 2e 6eI0 + 2ele7e10 - ele2ell + 2ele2e1 + 2ee 3el -2e2e 3 el +ee 8 + 7e2e3e6 e7 + 4ele 4e6e 7 + 5e]e2e 2 e2e5e7 - 3e3e5e7 +4elee 7 - 2ee4e 9 + 2e4e 5e 9 - 2e2e 5e - 3e3le6e9 - 3e3e 6e9 -2ele 7e9 + e2e7e9 - 3ele2e4e9 + e2e3e13 - 3e3e12 - 5ee2e 3e8 +5e12e2e e8 + 3ele2e 7 - e5 e 13 e4 14 7ele 2 e 6 e9 -4ee 3e 5 e9 +4e13e2 e 3 - 3ee2e 13 - 2e2e3e13 + 2ele4e13 ee 3el4 2ele2e15 2 +e4e + 2ee 6 2ee2e 4e 9 + 8e2e3e4e9 + e2e3e4e9 - 2e2e2e5e9 -3ele 2e 3e4 e8 - 2e3e4ell - l1ee 9 + 2e2e4e 8 e2e 8 + 6e2e2e8 3 e2C2 7 ee2eq q-+e9 +3e6 + 4ee2e 3 l9 - 7ee2ee2 e e2e3e 5e6 - 3ele 2e3el0 2 2 32 ±2ele2eel 0 ee 2 e 4e10 - 9ele 3e 4e10 + 3e1 e2e6e7 - 6le2e 6e 7

79 - -5e e 3 e6 e 7 + 6elee6 5 e8 + 6e 3 5e8 - 9e2e3 e5 e8 2e1e 4e 5e8 -4e2e 2e6 e8 33 3 2 35e 3 7 -ee62e12 - 5e3 e2 -2e2e 3e12 + 2elee12 - 2e51e3e1 3 - 3612el 3 - - eie4e12 +ele 8 el2 + 5e2e 7el12+ 5e2 5 12 +1365e 12 2e1e, 12 2ee e - e e62e6 +e 2 e6 e13 + 5e21e6 e13 + e3 e5 e613- 43e 5e613 + 4 13 2 13 2 2 433641636e- + 5e7 + 4e3e2 - 9e e e7 -4e13e-5e,67eM~~11 3 +--- 3e1e 3 e14 3ejee4 3e2 1 4 -32 5 -3e3e e -5e 1 e 7 e1 3 -e 2e19 + 6e1e32e14 + 4e3le2e4e8 + 5e26e3e1 4 4 8 - e e - 33e e -6e 6 e 7 e8 - 5e2e5e614 + 5e26e5e14 - 9ee 6e2 6e3 4 14 4 14 e e -3e e 4e15 + 6e2e2e25 + 2ee 2 e3 e5 e8 - ee3615 + 2e le2e3 4 9 --e461 21 1212 31' e4e -5e1e 2e15 + e2le5e9 + 5e2e5e9 + 4e2e6e 9 - 2e5e7e9 + 6e1 8 +3e2e4e615+ 2e1e4e16+ 8e2e4e9- e6ee 0- 3ee1e6 + 4e0e2e 16 - 32e e +7ele2e3e 4e 8 + 2ee 2 e18 3e4162 e3 1e 2 - 4e1 6e2 e 3e 15 2 17 e +e42e5e8 + 7e1e2e 3e6 8 - 5ele2e 3 12 - 5ce3e36e 9 + ee 4 6e9 -6e 2e4 6 e9- 2e3eeo + 5e3ee210 + e4ie17 + ee 13 -6e 5e7e9 + 8ee3e5e -44le 8e9 - 42e 8e9 + e1e9 - 6e4e5e8 + lele 2e3el12 + 9616 e e +3e e3e5 ell -4ee 4 e5e11 - 4e2e2 e6 e11l - 4ele 3e6 e611 2 7 11 2 e3e2 +3e6e5e8- 263e5e8 + 162e3e4e5e6-- 412631ll + 3el263611 +5c6e2e312 l-3e62 31 6 15e,12646732e2 + 6e2e3e9-9 33 2 -4 le2e7e8 + I12e7e8 + 8el3e367e8- ele356-4e1 4 _11 2 e2 34~266910 26260 3651 -1 6364C8 + e3e465e9 + 3e-162e69 - -4e2e-5e0 -7l 5110e~~e 2 9-12le32e36ee261623 2 836462+ ,le3e4e28 --12clC23el412231

+7ele 2e 4e14 - 4e2e 3e 4e612 + e2e17 + ele3e17 - 10ele 2 4e 5e69 2 +4e 2 e4e5e8 -e e4e5 621 - 52e 3e4 6-4ee2e - 3 e4e8 _C5 4 223 -e65e2e14 + el62e13 + 7e2e4 e5e10 6 2 465 610 - 616e2e6 e610 2 2 812618e6ee6e10 + 6136106e e3e6e0 - 16e2 e3e6 1 0 - 2e1e44 e6 e120 - 11ee~ 2 e7,7le10 e2 3~~~~ 2 4 ~~~3 6261 e -2ele 3e16 - 6e 2 e7e12 - 2e2e3 e 5e7 - 2e36ee 5 e67-4e 2 e4 1 3 12,e e e e6 -4e e2e5e12 + 11ee 2e7 e9 - 12e1e2e7e - 11e 3e 7e 9 + 2 3 7 9 + jje2 2 e2e3el 436~1 +6ele 4e 7e9 + 11lee2e8e 9 + 6e1 6e6e8 + 7ee3e1 0 + 4e2eel 0 +2ee 3e4e6e7 + 4e1e2 365 7 - 2e2e 3e4ell + 15ele2e3e5e10 - e - llele e el2 +5e3e+5 4 ei1 + 6e32e4ell3 + 5e2leell1 44 - 4e2eell+ 2ele5 11 3 5 6e e3e e -5e126e 5e9 + 161e32e5e + 66le 3e5e9 -13e2e 3e 5e 9 + 1 5 9

22 - ± e e -3e+56266671e 2 e5 e+6 3e61e6e7 ee 3 ee+6 6eee-3ee0+ + e2e 3e5e6 + 4ele4e6e leee6 + 10ele3 7 10 7e2e e e - 8ee2e e + 6e3e 4 e6 e8 + 2e1e 2e 5e 13 -5ee 2 e 8 e10 + 3 6 8 2 ~~~~~~~~~~~~3 6 8 + 2ee2el - 4e2ee +6e 6147 - 6e166 614 + e2e19 + ee- 2eCe7e8 + e4e + 19e 6 e e e9 -3ele 3 el - 8C13 + 2ele 2 e6el62- e2e3 26 e7 2 13 1 2 3 6 e -2e e e e +2e 2e e6 e 7 + 3e4e 4 e6 e7 -9e2e 4 e6 e7 - 3ele 5e6 7 3 5 6 7 +5e e2e4 e6- 36ee6e9 - ee5 + 6e3e7 8e -6ee 4e8 - 17 e3 -e2e3e e +e 4 e 5 e1 2 + e2 e 9e610 + e2e24e11 -6e 4 e5 e -6eje 2 5 8 +ee 4 -2 e 2e4 5 eee 3e5e7 - 5l61-6e 1e3ee 9 52 e~~~~~~~~22" 4 2+ 2e5e6 C2 +3e5le82- 4eee 4e7e8 ± 4e2e4e7e8 e61646e- ee5e6 3e4e e +e2ed-613e646-3 e4 e-5ee 5e6162646682 e4 e6e8- 101- --6516661ee-6e 0 1,+ 66,566106e5e6e 0 ,1 364167e107

80 +5e26e e - 5e e e +61:1o 7 1 0 e1 4 7 1 0 + 3e3e8el1 0 + e3e8e10 - 2e2e 9el 0 - 3e 2 3 2e7 3 7 4 +el620 + 24C7 + 3 624 e7+ 3e364e 7 -3e 2e43 7 - 6 5 1e 6 2 2 ~4 C2C2 6 +12e6e 2e 3e 4e10 + el15e 7 + 4e2e5 7 - 6e24e4e 9 + e6e1 5 3 -3e e12 + 5ee26 4e9 - ee 2e4e5e7 - 7 e2e3e5e9 + 3ee 3e1 5 +6 eee + 3ee - 3e e e e - 3 ee + ---- C3 13(:5 ,26q 3 621 4 5 6 5e6 5e2 e2e5e6 -102e 5e6 e8 - 2e2e 5e6 e8 + 8e3e4e6 e8 + 5e3e2e3e13 + e5e8 -9e;3e e e e + 35e e e + 11e~~~~2621eIe3e - 22 2 3+4 +9~ e4 8 9 -+e3-2514 e9 e2 3e4 e9 + 2ele 2 e4e 9 +3e'6e2e4e11 - 5ee2e 4e11 -8eIe 3e4e61 + 43e 2e5e11 -4e1 2e15e - 33e5e7 + 52 3e4e5e7 -2 - 9 3e4e7 +4e'!e5e - 2 2ee5 8e - e2e6 5 8 -4e 4 e5 8 -2e2e6 e11 +9e 3e6 e612 + e 2 e3 e1 6 + 34e9 + 5e4e2e 7 + 2ee2e 3 eI3 + ee6 3 3 ~ ~~~~~~~~~26361 2 1 +3e3e -ee2 - 9elee 5e7 - 4e 2 2e7 - 61e3e2e7 + 7e1e3e4e5e8 6 4~~1 5

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83 ~62 e~3 2 5 2 466 -2e2e8 e~~e+ 8e3 e7e14 + 6e2e8 2e1e3e -ee 3e4e6e7 - l1e e e +4ele2 e 2e 6e -13e e e4e eeC3-4 e7el 0 3 4 7 10 8ee + 6eee8 - 6ee62e5e13 2 _362656761O_-3ee5e7eio - - 2ee2eee6C7-4 4e6e7 - 2 23 eee + 3e e2e3 e7 +6e2Ce2e-6162656135 e 3-6e6e -6l3~,3+3 5 e13 + 4e41e13 8 e12 - 2e1 2 15 1 e e1012 - 2e elOe12 +3ee3e e 5 e9 - 4e32e44,9--e5e9 61111+ 2 -[-612636,5e6 2 ~ e 23 e12 -3e e e - 6e e4e9 -5e 1 e 2 e - 3e3e--52 e3e 5 e8 + 2ee 9 4 36128 12 3 635 + 5e1e e3e e + e6e8e10 +3e 3 e6 e5 - 3ee 2e5e1 2 - 7el1e2e 4e 17 2 4 9 -4e e e e e + 8e2e2e 7 e9 +12ele 3e4e ±+ 4 4 16 5e4e 2e 12 1 2 5 7 9 3ee e e15 + e e5e6 +2e5e 3e7e9 - 2ee3e 7e9 + 14e2e63e7e69+ 2 4 2 e2 e2 8 3651 e32 +4e+26ee6 2_1~~ - 9ee e ee9- 18ee3e5eo -17ee5e0 2 - e5e19 2 3 - 18el3ee4e +6e3 e 4e e 7 + 3e1 4 ee 7 - 3e2e 3e4e11 -9ee 2e4e 4ee5 6 7 - 8e 5e6 7 + Oeee 6e7 + e24ee -3e06 e e e - 5e2e 3e 4e6 e9 e27e1 0 - 12e26e4e6 e 10 8 e2 2 e33 e66 11 ee,9+e2 22 4 +14e12e3eI0 + 616ele5el68 - 3ee2ee 6 e7-18eee 8e e - 21e 23e e + 6e6e 4e 8e9 +1Oele2e 8 e9- 11e4e3 e8 e9- 13e2e 3 e8 9 1 8 9 C4 ~ 3 2 - 2e -~236e4612 e - ele2e e e + 6e5e9el0 - 2ee1 0 - e2e2 0 + 32e 3e4e13 -2e4e 5 11 4 5 8 Ce 4C -3c, C~~~~~22 -2c2 - e7 -3e1 23e4e13 + 5e1e2 ee 1 3 - 2e2e 3e 19 8e2e2e 4 + 3e3e e e e +5e2e2e 3e7e8 + 2e2 e5 e17 ± ee5 e59 + 6ee22e4e15 2 3 7 9 +2e3e2e7e8 + 9e 1e2 e4e5e8 + 43e 4e6e8 + 8e2ee 6e8 - 4e e3e4e9 - 72e3e4e15 + ele2e 3e14 -3e5e 5e 6e8 - 7e2e3e4e9 222 -2 2 _4 e4e0 -1961e2 3ee14-- e1 e264 e14 + 3e6e 2 e16-2e~2ee5, +7C~~~2 2 el2 e- - e e +7e22e3e4e5e8 + 6ele 2 e 3 e5e 7 - _ 2ee13e 4 5 7 3 +e2 52 e o +ee 2 e4 e5 ell - 3e5e 2 e3 e 14 + 8e1e2e 3 e 14 + 1Oe1e4e5 10 6ee e e -4ele2e 4e6e7 + 1961e2 63 e6 e12 -4, e 1e46e7e 8 + 5 7 8 21eje2e e e11 -9e2e 5e7e8 + 8e4e5e7e8 - 4ee 6 e7e8 + 4e3 e6e7e8 + 3 5 3e32eeo -e3e 2e413- 4e2e2e40 - 5e2e3e4el5 +ele52e4e9- 8 e2e 4 _ 2 +7e8e16 -4e2e 2e 3 5e6 + 3e3e 5e8 + 15e1e2e3e4e6e8 + 7e3e 10 e4e o0+ 4ele e7e - 3e6e16 - 5e3 16 5 e9 + 7e2e8e10 +7e2 1 3~~~ 5 73 I 4351 22 + 2ele1e13 - 5e3e4e613- 73e 2 e5e9 -6e62e2e5e15 - 5ee2e 3e12 22 _4ee2e2e 9 - 3e2e4 e14 + e1 e4 e14-3e5e 5e14 - 7e6 e8e10 C4 2 e2e3 -3 2 2 31636e9 +4ee2e 3e13 - 7612e3e13 -- 3e 2 2 313e6 e9 - 7eje~e e -4le42e 6e9 - 3ee 2e4e13 + 8e0e e4e1 3 4 13 9e2e e e e +8e4e 3e 4e 13 + 6ee6e 9 - 5ele 3 ee4 10 2 5 6 9 - 62271 __- - e e- e2e e7e +7e.e2 e3e4e6e75e2e 2e3e 5 6 e8 6e4e 2 7 11 2 16ee ee612 - 8e3e4e +e 3 e37e11 - 3e2e47e1ll - 6e2e4 e7ell - 3 ± ee el2 +2e6e3 e1~~~5e6e7 ± e292 9 ± 3e1e3 e5e15 ± 4e12ee3e122 3 612 6 e e e6 +2e2e25e26+ 4ele4 e 3e12 + 4e1 e4e4 e1 1 + 15ee 2 4 7 10 2 5 7e2e3e~~~~~~~~42 -e 2 e3e 4e 5e7 + 5e5e9e10 ± 13ele 2e 3e 7e11 + 10 e e8 -14ele 2e4e6 e7 + 4e2e3e4e6 e9 - 6e3e6e 9 - 3e3e 2 3 4 2 2 e2e2e2 4622 2 1 8 264567 +e4e13e7 + 2e1e3e4e8 + 4e2e3e4e8-12e e4 e5e6e7 7e 2e e e9 -9ece3e9 + 8e e4e - e2e4e9 - 6 le1 5e + 4 6 -e e1346e7e9 1 9 - 2ee+ ee1o0 -6ee-6355e7 -- 61e6364e769-- 6126367e3e2 e2 21124512 +e451 7 -- 64~,

84 -4e'le 2e3e6e9 - 6e2e4 e8 el0 + 6e2e4e8 e10 + 7ele 5 e8 el0 - 3ee2Ce20 -e4e. + 3ee2e3e5e6e7 - 1oee 2ee 8 + eC3ee 8 + 4e.e2e2elO C2 2 e2 2 -16ele 2 2e4 ell + 45ele 2e3ee 9 - 4ele2e 3e4e8 - 4ele 3 e4e5e6 -7ele2 eell - 3e5e2e6 ell + 6e3ee 6ell + 4elee 6e11 -155ee2e 4e8 ± 4e1e8e22 -1 82 e 2

85 Table of Pm,4 for m > 3 m Pm,4 2 + e7 4 2e4e6 ± e1 e3e6 + 3el e4e7 + ee 7 - e2e + e4e5e7 e2e5 7 - + 2e2e - 2e e e -e 2 eCe8 + e6 e10 + ele 7 e8 + 3e 2 e6 e8 ee 6 e 8 4 8 3 6 7 e3e +e2e3e9 + 3e2e 5e9 + 4e 3e4e9 - e 9 leee- 2e 6 e 9 1 3 -2e 2e5e9 + e3el 3 - ee10 - 2ee 5e10 - 3e2e3 e4 e7 - 2e2e3 e11 -e2e3ell + ee2ell + 2ele3e12 + ele4e11 + e2e4e10 -2e 1e2 e 3 -2e e - e e e e7 + ele2e6e7 - e1 6 - e 2 e12 + ele15 - 3el1e2e4e9 4 12 l 3 5 C2 _ 2e 23 22 - 5e -3 4e4eo- 8 + ele5 + e5e11 - e8 + e3 e5e,8 e2e+12 e2e14 6 e elo0 e7+ e e e e5e 6 -3e 1e4e 5e6 +ele 3e4e8 - e,22142 + 2ele 2 3 7 9 2 3 -2e1 e 2e5 e8 -e 2e6 5 -3e2le6el2 - 2e3e5 e12 - ele5 e2 + 3ele 7e12 + e2 e6 e1 2 - le17 - 3ee3el + 2e2e el3 +2ele 6 e1 3 - 3e2e5 e13 + 2e2le5e13 + 3e 3e 4e 13 3 3 1 -e le2el3 - 2ele4e15 - 3e2e 3el5 22 +3 el5ie+ 2e1 ~e 1 5 - ele 2e15 1 32 2 -ele 3 e4 -4e~e 8 ± elee4 ± 4ele2 e5e2 -3e~le 2e16 ± e6e27 -4e1e2e3e2 + e4el6 - ee 19 + 4e2ee1 0- 2e4e6 e1 0 - 3e3e7e 0 +21262e31l3 + 3ele 2e27 - 3ele2e4l3 + e2e16 + 2ele 2e17 - e +3e2e 3e5 e8 - 4ele6e5e8 2e- 5ele 2e6 ell 3e2e7 9 e1e +-e5- 2e4e12 + 3ele2e4ell - 4e2e3 e4 e11 - 3e2e3e4e1 1 - 3e2e2 5 11 - e e e 4 +3ele 3e5 ell - e 2e 18 e1 5e8 -4e 2e5e8 + 5e3e4e5e8 e 1 2 3 1 e e - 2e e e +5ele 2 63 e6 e8 + e1 32ee4 e9 - 5ele 2 ee9 - 2e2e4 5 7 5 7 8 + e eel2 +e4e2 + 2ele2e4e6e7 - e9ell + 3ele8e11 - 2ee 4 12 2 -ele3elo - 2e 4e 0 - 4e4ee 6 + 2e2e2e8 + 6eelo0 + ee 4e 0 - 3e e e + 3e e e -2e 4e 16 - ele 3e4 e5e7 + e362614- 2ele 2e 5e6 5 6 9 4 7 9 + e - 3e e5e -eale 8e 9 + ee 9 + e2 e9 - e32e5e 9 4e3e5 9 1 9 l22 - +2 +2e1e 2 e 5el0 + 3ele 3 e5 e0 - 2e8e12 + 5e2e 5e 6 e7 - e3e4e9 e + 3ele2e7ei0 -7e 2e3e 5 elo - 8ele 4 e5eI0o- 4e26e2e6e1o+ 4ele 3 6 e10 2 e2 + -2e e e e e2le e e 562636768 - 3ele 4e 7e 8 + e21le2 e8 1 2 7 8 3 7 8 +2 C2 - - 3eW1 e3el2 e + el e e e +4e 4 e5 e7 - 3e 5e - 3 23 12 e 1 2ee4 1 2 1 3 4 1 2 -e12e- 2e2e 6 e8 - 2e2e 3 e6 e 7 + 3eee 6 e7 + 3e2le 4 e7 -4e e4e7 2e12- e3e17 - 3 3e 4 e6 e7 - 46le 5e6 e7 + ele 2 e3 e4el0 2 24 C2 +2 2 + e e +3e 2e 6eo - 2ee6e7 + e1e5e7 + 3e2e 4e14 + e2e3e5e7 3 8 e -2eje 3ei- e3elei - 4ee 4e6e8 + e2e4e6e8 + 5ele5e6e8 - 4ee 6 8 e2 2 23 22 -4e 3e 5e 7 - 4e2e3e4e8 + 4ele 3e4e8 - 2e3e11 + e2e3e 0 2 2 + 2 e5e6 + e2e e6 + 21le e16 - e7el13 ± e8e1 0- 2e2e8e10 +ele3 2e24 3 32a2ee -3ee9e0 + ee + 3e3e5e26 -e6e 1e 4e14 - 2e1le2e3e5e9 e9 + 2e e e e9 + 3ele2e e 9 - 3e2e 3e 6 e9 - 3e 2e3e6 e 9 +5e26e4e 5 2 4 5 2266 ~~~~~~2616 4511 2e4e6 2e e - 2e e e- +2ele 4 e6 e9 - e3e14 + 4e5e15 2ee + 2 4 5 e1 +4eee1 1 -2e51 ++ 4e 6e11 42e2e3 6e11 -+ 7e11 2e2e7 e e +2e 0 + ee 18 + e20 - 3el1e2e4 e5e8 - 2ee 3e4e + ee 2 7 9 +ele6367e9 - e2e3e9 + 2ele2ee1 + 32e 3e4e9

86 6 e4e2 - e2e20 + 4ele 5 e 8 + 3ee 11e 1 2 - 2e5e 9el10 + 3e2e10el 2 + 3e 2 e4 e18 -4ee 2 e 19 + e3e21 - 2ee 2 e2 1 - 2e1e3e20 + 3e e2 2 0- 3e1e4 e19

-2e 2e 3 e 19 + 3e2e 3 e 19 + 3ele2el9 + e4e3 el3 - 5e 1 e 3 e4e 7 e 9 -3ele 3e4e2 - 2e3e6 e9 + 8e3e62eg9+ 4ele 2 e3e2 - 3ele3e 6 e11 +2e:2e3e13 + 2e2e4e5e61 + e3el 1 3 + 3e2e 7e1 3 -e 2e3e 0

+e24e7 e 9 - 3e 1 e2 e3 e5e8 + 3e2e5e 8 - 2e2 e3 e9 e1 0 + 3e2e2e 5 e13 22 -61_ 2 3 -3ee 3e 53e 3 + 8ele2e8e13 - e3e5e8 - e7e 0 e23e11 -e,1e 2e3 e 4 e1 4 + e2 e2 2 + 3ele 4e9e10 + 2e 8 e1 6 - 7e162e 3ee1 0 -3e' e e - e e4ell + lOe2e e eg + 2e2e2e e + 4e ee e 7 8 9 l 3 3 4 5 ~~2164695 9 2 4- 5 9 -2c, - 2e2e- 32 -2e2e 9 e13 - ele 2 5e 9 - 3e2e 4e5e9 + 6ele 2e3e6 e9 + 7ele 2e 7e14 +2ee 2 e5 e21 4 - 6ee2e 5el 4 + 3e e 3e 7e8 e e + 3e e -2ee 1 6 - 14ee 2e3e6 e12 - 3e3e2e,13 + ee3e 8 + 5e3e2e13 +7e:,e 2e8 1 2 9e1e3e8e1 2 -7ee 2e9 1 2 2e e 0 + 3e1 2ee8e9 e2 2 2 6 46111269 +4e 5 e6 e 13 + ee1 1 13 + 5ee 7e 9 - 4~~~~~42e6e1182 e4e ee 5ell + e42e2e9ell -2ei[e 3e9 e11 + ele2e,10eii - 8e2e 3 e 5e + e 5 e1 9 - e,2e22 + ele3e 4 e1 0 - e22 -e 2e 1 4 + 3ele 3e20 + 4el e e9 - 2e2e4e92+ e2e4e -7ere 3 e6e1 3 - 4ee 4e6e 13 + 4e 2e2e 7 11e - 4e e2e 6e2 -2e]e2e 3 e6 eo0 + 5e 5ee 7 - 5ele2e 5 e6 e8 - 9el 2 + 4ee 3e6e7 -5e 2,e3e26e7 + llele 5e6 el12 - 5ele 2e7e12 + 7e1e2e 7e 12 + 8ee 3e7e1 2 2 e2 2 2 -7ee 3e7 12 - 4el1e 4e 7e 12 + e3e5el3 - 2e2ee + 4e2e4e 8 ,-e23e+ 2eje2e e -462e e e -3e 2e3 e 4e 15 + e2 e3e5 e8 -- ee 8 21 2- 4 15 3 4 1 5 3-- +ele 2 3 + 5ele 2e5e11 + 2ee 3e 8e11 + 8e2e 3e8 e11 +- 2ele 4e 8e11 -3e 1e2e 13 + e3e4 e11 - e3 5e7e 9 + 4ee 6 e7e9 + 4e6e7e11 .22 2 3 2 2 e2 +2ee3e 7e9 + 6ele3e4e 5e1 1 - 2ele3e4 e6 e7 + e2e4 e7 - 2e3e 4e7 22 2 2e 22ee +3ee4e7 - 3 2e4e7 - 2e1 e2e3e5e,13 - 4e6 4e6 e8 - 32,e42e6 e8 -2ei'e 4 e52e8 4ee2e 3e 7e9 + ele2e 3e 5e11 - 2e46e ±-4ee2 e5e7e8 + 2e12e1e5 e1 - 8e2ee 511e - 4e3e4 e6e11 -5e"e 5 6 e1e 1 - 7e2 e5 e6 e11 + 4ele 2 e3 e4 e6 e8 - e3e4 2e7

+ele23e5 e27 + 5e l e26e11 + e3 e 7e8 + e3e53e6 + e24e16 - e2e 2e2 +ele 3 e9 - 3e2e3e6 e9 + 11ele2e6 e7e8 - 362e4 e 7 e9 - 23e7e 4-36¥62646769~ 2 2el 2 24 +3ee 2 e4 e7 e9 e2 e5 e20 + 9e4 e2 - 3e e e12 + 6ee 7 e13 2 -22e2ee 12 + e1e4el4 + 2e2e4e14 - e2e2el4 + 2e,36el5 + 8ee e6el3 +6ee 3 e6 e 3 - 3e3e 8el3 -e2e 2 e 3 e5 el2 - 5e3e3 e7 e11 + 2e2e 4 e7 el -3le2e 3 e 4el12 + 2e 1 e 2 e2e 4 e11 l- 3e3e6 e 1 2 + 7e~e6 e1 2 - 7e 2e6 e 7e9 -5e3e 8e 13 - 3e4e7e13 + 4e1 e2e3 e1 4 - 3ee 2 e3e14 +- 7e2e3e5e1 2 +2ele2e5e12 + e63e4 e5el2 - 12e3e4e5el 2 - ee + 7e 2e2e3e 7e10 +4e 2 e 4 e5 e6 e 7 - 2e2e2 e5 e6 e 9 - 2ee 10 el13 + 5e l e2e3e2 -5e1 e2e4e7 e 8 - 22e3e 4e12 +- 2e2e3e4e12 + 2ele 2e4el2 +5e 1e3 e6e1 2 + 4e2e6e5e9 - 5ele3e5 e9 + 3e3e 4 el1 4 - 2ele3e 5 e1 2 2 2 2 2 2 e~2 e2 + -2e_2- e5e6 -e 2 e5e6 - 4ee17e8 + 4e2 e7e8 e2,e3 e4 e5 e 0 -14e2 e2 e7 e13 + 7e1 e3 e7 e1 3 - 7e2 e3 e4 e6 e 9 + 2e2e 18 - e4e20 -66162e4611-6e3e2e4e6e1 -2e~e~el-9ele2e- 2321 - - 3ee4el34e 3 -ee3- 29 ± 7e1e4e5e27

87 4e e6e - e e e 5e14 +2ee e6e eo - 5e3e 2 e6 e1 3 - 3e2ee 8 + 4 8 3 2 5+2ele23 1 2 6 3 2 4612 - e3e e eee+ 4e3e1ell +7e 2 e3 e 5e14 - 3ele 4e5 e 14 5e2 4 9 ~~~~~~~~263362 e3el2 - ele33~~~ e7e9 + ele e4e5e62 + 2ele2 e4 e17 -e3e4el 0 +2ele2 5 2 + 2ee e el2 +3e24elo + 6e2 e4elo0- 4e2e4ee8 + 4e12e6eel2 3 6 2 12ele4e e o -ele 2 e4 elC3 - 9ee 4 e6e12 + 15e 2e 4 e6e12- 5 1 -26l1e256102 2 -2ele3 510 + 7ele2e6e14 - 6ele13e6614 + e1e2el5

3 2 -2e3 1-e4ee7 4 ~~32 + ele2e - 12 17 - e4e13e 7 -e 3 e1 2 - 2ele 9e 14 + e 1e6e9 1 7 e17 -3e2e el7- 4e e2 e4 e6 elO0 +3ee 3 el7 + e3e4e17 -3e 3e 4 5 -2e e e 1eloel4 - 3e2e3e4e5es - 6eie 2e3 e1s + 4e2e6 e7 4 6 7 - 2e2e23631e -5ele_51232e7 +_] e63 -2e624 -e2ee_2 2 6e8 -__6126676ele5e6 e7 3 15 e + 4elee e7 -e2e3eei1 ± 4e32e8e10 + 6ele2e3e5e6e7 + 2e4 20 5 e e - e4e e6 - 3e2e 8 e 12 + 4e 4 e8 el2 +2e6 e 1 1 466 9 + 3e 5 7 1 2 8 12 el3-ell +2ee9e12-3ee13e16 + 2e2e4e2 -2ee 5 5 + 8ele e e6 e8 - 3e2e2e3e15 +8e2 63e 4e7e8 - ele2e4e5e8 - 2e3e1 4 5 + 6e e e e e +6ele2 e3e15 + el3e2e 4el5 - 2ee3e 6e10 1 3 5 7 8 2363 +562e36492 + 5e 134e9 - 5ele2e6e + 3ele7 + e3e7 - e24 -e e ±e22 3e4e6 e7 - 3e3e4 5 6 + ee 9 - e6e14 2 20 e2 032 - 3ele e e6 + 5ele 2e4e 5e8 + e9e15 -2ee1 62 e 3e4e 7 - 4e 3e6 e768 2 3 1 e ell + 4ee2 -4ele 2e 5e + 8e2e3e6e -ee 3 e6e1 + 6e3e4 6 - 2 2 5221 2 ' 2 2 3 2 - 3ele2e3e13 + 5e1e2e4ei +4e 2e63e7e9 + 4ele 2e6e8 + ele3e15 e6e +8e2e5e6e9 + 11e2e3e5e6 e8 -Oeee 7e8 - 5le 4 7 e2e e e -ele2 e3 e8 e10 + ele 2e3 e6ell - 3e1e2e7e10 + 3 7 10 3 3 86le4e5e63 -7ele3e7e10 - 3e e4e7el10+ e le21 -3e2e4e5el3 + 2e e el14 + 4e1e 8 e14 - 3e2e 8e14 -4e2e 6 e14 + 3e4 e6e14 - 5el3e7e14- 3 7 - 3 e e e8 - 5e3e6e7e8 -5e 3 ee9 + 4ele 2e 4e 5e2 - 8e2e5 e7 e8 4 5 7 e + 2e e e e +e e4e5e8 - 3e 6e16 -8el 1e3e6e268 - 9e4e5e6 9 2 4 7 11 3 e e el -4ele 5e7elI + 2el3e2e8ell - 8eee86e11- 9e 2 4 5 - 3el2ee e9 + 8e2e5e8e9 -3ele 2e 3 e4e5e9 + ele 4e8e9 - 4e3e4e8e9 5 8 1 e - e e e769 -4ele 6e8 e9-2ele2e4e1412 ± e-6e3 e4e14 ±+ 2 5 e6eg + ele9 - 3e2e5e6 e7 - 3e3e5 e6e7 - 3e2e3e5e0 -5ele 2 2 - e5e - 3e e e9 ± e e4e6 -4ee 3 e5 e6 e9 + 6ele 3e5e 9 6e2e 3 9 1 4 3e e e - 3e5e8e11 +3ele 5e + 2e le3e1 4- 2e1e 2e 4e2e7- 3 4 11 2e e e - e e2ll + 5e3e4e6 e7 -e41le9ell + 3e2e9ell + eel0oell - 3 1 0 11 2 e e0 ee e e10 +e e9 + 8e1 2e4e5e10 ee 5 e0 - 4ee 4 6 5 6 2 2 -- 5366660-2e2 662666 -- ~ 10 +462e1 4e2 e e0 -5e3 e5 e6elo - 6e2e4el2 - 6ee6e10 2e2lei - ele e 5 + 3e2e5 e15 +2e2e3e4 e3 + 16ele3e 4e6 eo0 + 3e2e9el13 5 1 e - 2e e e6 - 3ee2e 4e1 3 + 2e2e 6 e10 +2e 4 e5e 15 + 4e3e6e5 + 4e2e 7 1 5 2 7 15 O + e e e e - 5eje6 e 7e1 0 +2e2e3e 7 + 3e3 e 4 e7 e1 0 + 3e2e 5 e7 e1 20 5 7 1

3 2 e2 e2 C2 e3D e- ee22e e +2ee3e7-2e2 e4 e6 e + -4ele9e1e63e8e4e6e9 - 4e2e - 6e 2eel2+ ele 5+e8 3e - 6 e5ee8 e 03 8 92 3 7 - 2 e e6e7 - ele2e16 +5ele3e 8e9 + ele3 e5el5 - 7eje2e6el5 e 4 5

88 2 -3ele 8 el5 - el e2ee 6e7 + 4ele3e6e ± e3e8e 0 + 8ee 2e3e4el3 -5e'e2e3e8e9 + ee 2e9e 0 + ele2e9el0 - 4ee 3e9e 0 + 6e3e9e 2 ~~~~2 e2 -2e:eloel2- eel 4 + 3ele2e5e6elo0 - 3e2e3e1 4- 4e2e3e4e5el0 -2le2e+ 8e1 - eie0 3e8eIO + e 4e8 e1 0 -9e 2e4e8 e10 + 2ele5 e8e10 +e2e3 7 - 5e6e18 - eel 8 + e3e 6e8 + 3ele2e4e7e10 - 3e4e9 ell +2e:3e 2e3el 6 - 3e2e 2e4e 16 + 8ele 3 e4 e16 + 7l2e 3e 4e7e8 -ele 2 e3 e7e11 + 6ele2e 4e 6e9 - 2e2e4 el6 + 2el3e5e16 - 3e3e 5e 16 +5e 2e 6e1 6 -3ele 7 el 6 -2e 2 e5 e17 + 4ele 6 e1 7 -el4e 2 el 8 + 3e e2el 8 2ee3 e -2e e e -2ee6e8 +2cje~e,184 8 - - 2elC4Cl8 3elCelS

89 Table of Pm,5 for m > 3 m Pm,5 - 73 4 -ee26e7+ 4e5e6e9 + 3e6 e10 + 4e1e4e5e1 0 -2e 1e3e6e10 - e6e + e e e6e7 +2ee e - 4eee 9 e4e8 -- le19- e2e8e10 8 3 4 5 9 - 3e e e8 +3eel2 -e2e ele 2e2 el2 + e6 e -2ele 7e612+ 2e3 e6ell 5 7 e e e5 2e2e3e5 e2 4 e14- 2e2e4e14 + ele5el4- elee15 + 1 3 - e e e -2ele4e5 + ee 21e6- 2ee6 3e1 + 2e1e2e17 + e2e4e5e9 3e2 5 6 7 - e ± e2e e0 +e eel2 -e2e 4el2 + e2e6 el2 - 2e2 e6 1e2 e6e 4 1 2 e 2e e e4el + e e3e5 e11 -2ee 62e e111 -3e 3e 4e 5 8 -e 2 e6e2~~~~1 0 ± 3e8e12 2 3 1 --6129 + 21e6261463 -2ele263e64 + 362e26e8+ 62e6e8 -2e378-26164678--326365610-- 6664-223669 3ee6e9+- - e5e - 2e e e4e12- e3e1 + 2e1e5e + 3e4 1e 6 3e 4 -e 2 e1622 + e2e4e2 1 3 e - e e e9 - e 4e6 e10 ±e21e7e1 ±-2e 267 21 - ee 8e1 + e1e9 1 0 3 4 el-2e e el 3 -e 3 e7e1 + 0e8e8 1 + e 3 e4-e 2 e3e 3 elle 3 3 4 e 4 + ee5e e8 - e2 e6 -e3e 17 -2e95e13 ± e2 e5el3 - 2ele 6 e13 + e2 1 6 e7e9 + 6 e e e9 - e9ei1 -e2 e7 e9 - 2e4e7e9 e26- 2 + e20 3e 3 1 2 8

e2 - e 2e-8 - e e e ± 4e e e9 2e e2 e4 e-- 5 -3e3e4e + e 75 e2 2e± 5e5el 0 1 4 8 4 2 2~e~~e~io+ ee78+ e e2eseel +e2 e16e9 -ee 3 e6- ± 3e2 2 4 e e e768 4elee6ei0 -- eee -3e2 e3e20 - 36ele4e2 - +4e 2e3e4e2e+eeeIe+ 5e1 3 6 _C2~~~~~~- + e16e e -2e6e2 e3e8- 5e2e7e8 + +243+ 5e1e 2e6e7e9 2 6 8 e e - e ee - 2ee4e 7e -e3le2e20 + e2e3e-61 -+ 5 e 2 4e1e e e5 - e 2ee 14 + 3e2~~~i2 e4 e 9 e1 0 2- 5e1e5e9e10 +4ele223 e6e 53 + 2 7 - 63- 4e6 3e6e9-223 +3e3e26e1 0-_61364617 + 4217- 1 - e e13- 3e 2e2e4e13 -4e5e6e14- 6e2e5e7ell± 4ee3e2e112e6e 4 6 ± 6e ee6ei 4 2e2e 4 e7e12 - 6e 2e 4e72e12 +-2ee 7e17 e~~~2ele4e16 _ 32~~2 42632 2 -6 le 6ee e e ± ee 9 +2e-e5 e7e12 ± 3e+e4e5e1 + 5e 5 6eeo- 6 7 1 2e3e e e e -e 2 e2 1 +2e-e3 e4 el4 ± 2eoe8 e5_+ 4e32e6e 6 + 2 5 7 0 + ee 22e +~~72e3i - 1'eeloel2 e e3e% -4e 2 e 3el 0 2 1 0 ee616 +4ele 2 e2 4e6e +e-5e 3e3e+e8 + 2e3e6 e164 21ee2 + 3e2e5e14 - 3e 2e 3e5 e6 e9 -5e1e2 e3 e7e12 + 2e2e7e16 - 3e26e7e16 - 3e e2 4 - 4elee 8e12 + 2e2e3e8e12± 2ele3e4e13 -9e4ese6ei0 3 1 2 ±2e2e4e5e14 + 3ee6e8e0 ± 2e1e86 ± ee23 + e ee - 4 2 e e - 3e e e9e12 +e2e e - e63e742 3e2 4e6e9 + 6e1e5e61e 3- 3e 3 6 7 9 1 3 3~~~~~~ ee e9 +3ele2eloel2-3ee2eeio ± e4e1 3 - 3e-e8e9-2e2 8 2ele2 e16 - 4e2e6 e7e8 +3e265e8e1 0- 6e 2e5e8el60± 3e1e 4e6 e + 4 e e14 3e2e e 4 2e2e9 e14 -5e3ee2 + 2e2e 4e5- 4elel0el4 + 2e3- 8 9 1 3e2e e e1 6 - 6ele 3e81e 3 -4e 2ee 6 e 9 - 2e6 611 ± e2e4el69- 3 4 el - 3ele e5 e7 68 - 2e 4e6 e7e8 - 3e2e 3e6 el2 -4e5e 9 e1 4e2e7 1 2 4 2 ee2 2elee6e2-eel9 6 2e3e8e2 3e +4-216 61- 31 4- 6~26l+3681 -- 2e6161 4-312C612

90 -4ele 3e 5 82+ 6e26e611 -3e 2 e6e1 1 + 17 - 2ele2 e8e1 4 - 2e3e 4e16 +6ele 2e 5e 7 + 2e3e5e8 e9 + ele2e4e5e13 + 2ele3e 21 + 3e2e10 e11 2 2 22 -3e 1e 2 e5 e 8 e9 - 3e3 e6 e + 81 7e + e3 e4 e6 e12 - 262e 5 e6 e1 2 2 2 12_21268 5 221 0 +2e2e5e6e 2 ± 3+5e17+5 2eee7 -2ee 2 el e 8 -5e 2e3ee 0 -ele, 4 e5 e,10 + 2e le l le1 3 + e7e + ele42e5ell - 3e2e 7 e,1 4 - ee 8e 14 -e ee 12 - 2e3e7e9 + 5ele3e4 e8 e9 + 2e1e 1 - e6e9 e1 0 - 4e4e10 e11 +e3e 1 - 3ele4e8ell .- 4e3e7el5 23e 9 - 4e1 e9 + e25 -8617 -3e61e 9 e12 + 2e 4 e9 e1 2 -3ele e11 1 2 + 3ele 2e 4e6 e1 2 - e2e4e5e6 e 8 l 2 e2 -3ee 3e4 e7e1 0 - 3e1e36e8e1 o - el 24- - 2e 3e4e 7e8 - 2e2e4e 8e1 1 -4e 2 e1 0 e1 3 + 5e2e3 e7e 3 - 1e 4e 7e1 3 -e ee 14 -2ele 2e5e616 2 + e e5 e8ell + 3 4 5 e 0 - 4e3e4e5 e9 + 2ele 3e5 e1 6 22e 4e1 4 o2 . 2 2 2 -2e:[e 5e13 + e 6e1 3 + e 31e 9- e9e 16 + 5e4e5e 8 - le 2e3e19 +3e:e 2 e81e 3 - 4e7e8 e10 + e4 21 -e-le 2e3e4e1 5 -7el 1e 3e4e6 e11 +5ee 5e1 2 - e6 e9 - 3e 2e 3e 5e 14 - e2ee 13 - 42e 3e 7e11 -6e]le2e6 e16 ee2e' 9 el- 3e1e2ee 4 ± 2 2l5 15 4e6e15 -e3e 7e1 5 + 2e e81e 5- e2e8el5 - 4e1e9 e15 - 2e 3e9 e11 +4e]e 3e5e 7e 9 - 2 e 3e 9e11 - le1 2e4e 18 + 3 2e23e5e1 2 + ee3e 1 7 2 2 1 2 1 ±2+ ± 2 -2e,e 3e6 e7 - 42e 4 e5 e12 - e12613+ e26e3 e5e1 3 + 2e4e6 e7 +3e 1,e 4e6 e614 + 2e 2 e7e14 + e 6 1e 1 - e7e18 + 5 2e3e4e7e9 +e 3ee 6 e 8 + 3e3e9e10 + 4e2e 4e25e9 + 5ele 4 e9e11 + 2e2e3e6e11 -e3e 6e310 + ele 2e4e8e 10 2le 2e 22 + 2ele2e 6e14 - 3e2e3e6e 14 -e 2e3e6 e14 + 3e2e10e13 + 2e2e 17 - 2e2e 5e13 + 2e2e3e5e7e8

2 +3e1 e 2e 3e 8e11 - e6 e 19 + 4ele 2 e3e6 e1 3 - 3ee626e12 + ele e65e5 9 , 2 ~ 3 +3e'e 3e 5e1 5 3e2e2e7e10 + 3eee 10 + 4ele2e4e7e1 1 + 2e e,2 +2e2ellel2 - 2e3 e5el 7 + ele22 - 2e6 e1 3 - 8e 2e5e6 e11 + 5e10 e15

+2e e e4el + 2e 2~~~~~~~~~~e e e + 3ee el ell - 4e e el22 + e e e8 ±e'e2 3e3 8e10 2+ e5ee3 84 +5 2e134e8e 3 -3e3 03e8eII +3 5e55e720 + e2e2 74e8e9 -3e 2 e7e 9 - e3el0 e12 - 32ee4 e7eC0 + 5 1e5 - 3e 6el7 - 3ele2 e 1 -3e6e 6 e13 -e e 22 + 2e2e6 e1 7 + 3 4e6 e11 ± ee 41 5 + 2e2e4e215 -e4e2 - 2e 6e 8e 9 + e e e6e + 2e e e + 3e e e2 - 3ee2e e el2e2 3 3c2 6 e28 9 3 9 ±2-13 2 5 3 7 1 3 ±e1 ,ee8 elee28 + 258- ee4e 0 + 2e6e7e2 - e3e4el0 +3e 3C4e8 el0 + 3 2e3ee 8 - 4ee 4ee 8 - e3el6 -4eee 7e9

-e 2 e34e15 - 2e2e5e6e8 + 4e1 2e6e8 + 2 4 e7e14 + 5le 4e 5e6 e9 +2e2e 3e18 + 2e3e4e5e11 - 3e2e 18 - 4e1e 5e26e7 + 2ele 3 e5e 6 e1 0 +ele + 22e 2 e4e 7 - 3e 4 e17 + e e5e 6- 5e3e5 e7 e 0 - 4e2e3e5e 15 -8el 4e 56 15 - 3e 2e6 e15 + 2e3e 4el8 ee2 21 + 2e2e - e4e7 -2e2 e4 e6 el3 + 32e 5e18 - e3e4e12 - 4e5e7el3 + 3 5e6e2- 2ele3e3

+2e2e 4 e1 9 + 2ele2e 20 + 2e2e 3e20 - 2e3e 9e13 - e6e3 e19 -4e 2 e5e18 2 2 +le2e - 4ele5e1 9 - 3ee 3 e4e 5e1 2 + 3e 2e 6 e7e10 + 2e4 e 5e 7e 9 -3e2e 6e 7el 0 - 3e1 2e3 e17 - 4e2e5e 7e9 2ele 6e18 2 -el32el0- 2e2 e 4e5 e

91 92 Table of Pm,6 for m = 4,5 m Pm,6 4 2e1 e3e8e12 ee+9e 1 e4e6el4 - 27-22e6el4 + 3el5 +2e5e8e11 + 2e4e9e11 + e3e 5e6 e10 - e-el0- 3ee 5e6 e12 - 2e3e10e11 -2e 3e7e 4 - 2 2e 5e 17 + ele 3 e6 e1 4 - e2e21 + 2e 2 e8 e1 4 + e4 e5 e7 e8

-e 12e8 e1 4 3e3 e6 e7 e8 + e3e5e6 - 2- 3ee 6 e1 7 - 3e 4e 5e6 e9 +2e1 e7e1 6 - 2ee 9e14 + 2e162e3e1 8 + e1e7e, + 2e3e4e17 + 3e2e5e1 7 -ee 4el8+ 2e2e6e8 - 2e2e3e8ell + e2e4e6 + 2e3e5e16 + ele4e8e +2e 1e 2e7e1 4 - 2e2e 4e 5e 3 + 2e2e3e5e 14 - 2el e4 e5e1 4 - 4e2e6e10 e-e-3 0e2 2 - 2e2e e e - 3ele e e -2e e e e +e3~2 2 5 8 9 6 8 9 1 2 431e,2-2lCe4117 +e2 C7 13 - ele3e9 e61 1 - e3 e 4e614 + e 2e2e14 + 2e6e18 -- 412 +e2 3 ~ ~~2 2 2 2 +e2e2 0 + ele21 + ele 2e10ell 3 8 10 - e84e8 - ele 7e9 + ele 13 -e1e5 e7e611 - e3e5e7e9 + e2e6 e7e9 + 2le 4e7e12 + e2e3e9e610 +3eel 2 - 3 5e6 e1 3 - + 2e 3e8e1 3 + e61e2 3 + e 2e6 e1 6 - 2le 2e9 e 12 -3ee 1 6 - e2e2 0 - 2e5 e9 e10 + 2e 4e 7e 13 + 3e2e 9 e1 3 -2e 2e 9 e1 3 -2elel 0 e 1 3

+e10 e1 4 - 2ele 3e 7 e1 3 - e24 + e2 e5 e7 e10 + 4e1e6e 7 el0 + 2e 2e 4 e1 8 2 2_ 2 -2ee 3e4 el5- e5e14 + ele2el 9 - ele3e19 - 2ee 3el9 - ee 2e20 -2e61 e 2 e8 e 3 + e3 e3 - 2ele 2 e 21 - 2el1e 3e5 15 - 2e2 e3 e7 e1 2 - 2e1e 4e 9 e10 +e3 '4el1 3 + ele2e6 e15 + ele4e1 5 - 3ee 16 - ele22 + e4e5e11 232 -e 3e5 e11 + e7e617 + 2e3e 4e 5e12 + e5e 19 + e5e9 + 3e3e669 + e5e 7el2 +2 22eee e 2e65e1 5 - 2e2e4e7e11 + 2e7e89 - e 2e3e616 + e2 e3 e6 e1 3 - e2el8

-2e2e7e15 + 2e1e8e15 + e1e3e60+ e3e4e8e9 + e-e7e9 - e2ee 8 +2e6e3e 4e1 6 e+2 e22 -ee 5e1 3 + 3eee 1 3 - 3e 3e4 e6 e11 + 4e2 e5e6 e11 32l- 2e1e 5e18 + e2 e4 6e 1e 2 - 2e 1e2e5 el16 + ee 11e12 + 3e21e 0 e1 2 -e 2e5e1 2 + e6 e8e1 0 + e9e15 + 3e6e7 11e - 3e6 e7 e11 + 3e2e4e8e1 0 -2e1 e5e8 e10+ 2e1e4e1 9+ 2e1e3e20 5 -e 1 e7e8 e14 + 4e6e- 9e15 3e1 e2 e6e 11 ± 2e3 e6 e1 0e11 - 5e1e 2e6 e10 e11

-e7a 9 -e 1e 2e 3e 24 - e6e 9e18 - e1e3e4e5e1 7 5e1 26ee5e1 0 e1 2 -5e3e 51e 0 1e 2 - 3e6ee19 ± 3e3e9 e1 8 - 3e5e8 e17 + e8e21 - e4e7e11 +2e'~eoe 8 + 3ee 6 e - 4e2 e10e 8 + 2e3 e5 e - ege eo ± 3e6e7 e+7 +3e6e11 e1 8 +22±-2 e6e7e8e9+ 3e2 e1 2-e16 e4e 01 ± 2e6e5el19 - 2ee 10 e 14 -e 3 e 17- e 2e1 7 e61 0e 1 1 + 5e 2 e 7 e1 0 1e 1 -2e4e 8 e1 0 -7e 2 e4e 5e 7 e612 -2e5 e7 el 8 ± ee2 ± 3eee 6 e 5 + 2e1 e2e9e1 6 + e3e5e6e8 -4e4 ee6 el0 - 6e4e5e 01e elee + 2e6e4+ 1l 1+3ee6e7el

+e2 e7 - e4e2 6 - e4 e18 + 4e1e3e5e21 + 2e2e5e21 - 2e3e4e9e10 -4e 3e6 e8 e13 + 2e2e7e8e1 3 ± 2e8 e9e1 3 -2ee 6 e18 ± 2e64e7e 15 ±2el 3 e6 e7 e 3 -4e3 e4 e8 e 2 ± 2e2e4 ee 1 4 -4e 2e6e 1 ± e2ele 7-2e3 e 4 e7 e8 _ 6e1e5 e9 e1 5 -7e 4 e6 e8 e -8e e e e ± 4e e e22 e -- 23646762, -- 6617s691 -- 76466861261-- 2 86s45+ 6 696o9 10 4 6C93 346106,144 1 0 13

93 ±+2e 3 e +2e2e3 e7el8 + 2e2le4e5e19 -4ee 6 e8e 4 + e2 e6 e8e14 10 1 4 124511 - 6j--612636-7617+ e636466617 +5e3e+563646686104e5e8el0 - -4ele461727 ell - lee2e4e5 e18 -6ee 2e3e7e17 ± e3e4e6e17 +2e2e 4e7e17 + 263 e4e6 e8 e9 - 2e6e22 - 4e4 e5e6 el5 + 12e5el0oel5 22 2 + ele7e~~~~~~~~~22 4 +3e3ee 6 e13 + ele3e6e17- 3 2 e5el8 - 7ele2ellel6 1 C~--381 - ele128 e el9 +2ele+2124e666172 e4e8el7-- eelle136611613 ++ 416,264e67e64ele2e4e7el-ee - 8Ci6 2 8 + 2e e e e el -4e3e 4e 5e2 + 2e2e+e7e13 + 4e2e4e6 e8e10 2 3 6 8 el + 4e e + e e3 e 5 e9 e1 1 -22e 5 e 7 el1 4 + e30 + 2e1 2e4 e21 - 43e 5e6 1 3 10 20 2

e03ee e +5e 4 e5 e 9 el12 + 2e 5e8e9 - 3e3e7e20 ee 8 4 2 2 e0e1 + 8e 6 e e12 +4ee1 65 e8e6 - 3e1 1o3 -3 4 7 13 3ee 8 5 6 7 2~ ~ e2262~~~, e12,5~l 2 22 1 +6e25e20 + 3e26 e e16 + 6e2 e 12 - e 1 2e5e9 e13 -2e2e6e4 6e 4 6 e2e le2 ~ 2 - e e e e -e 3e 5e6 e 7e9 + 2ee7e615- ee 6e7e10 + e6e7e10 2 3 4 21 e +2e2e e e - 4e2e8e9 ell + 2e2e4e10e14 - 2e8e 9e11 + e2e5e6 12 2 3 23 3 e21 + 2e l0 - e3e4e15 _2e3e 3e2e 7 e21 2e 2 e 7e21 3ee 5 e2 2 + 2ele 8 2 2 e e e e14 - 4ee e e -3e 3e 4e6 e7el10 + 4e 4e6e14 + e4 e2 2 + 2 4 8 4 6 8 - 6e e e e - 4e e e29 +5e3le6 e21 - 3ele 2e 3e22 + 3e 3 e5e9 e1 3 2 4 7 17 4 8 2 2 22 -3e1e4e5e20 ± 3e6ee3e2 ± 3e1e e e-4eee 4ee2e10e16 2e2le e e15 - 2e4e 5 e8 e9 +26 2e 5e6 e1 7 - 32e 3e 7 e16 + 4e le2e5e2 2 + 4 9 2~~222 2 2 2e ee + ele264 -463e25 e7e1 + 7e6- -3ee 7e8 ell + e3e8el2 + 3 13 4 e~2 + e2e2 e 2 +2e1le2e6 e8e13 + 3ele2e4 ellel12+ e4e1 4 +-2e 2 e3ee17 6 0 e 7 + 3ee5e el0 -3e 3e 5 e22 - 32e 9e17 + 2e4e 9e17 + 2e2e2e 9 1 7 e + e4e25e -e3341- 3e2e 3e6 e 19 + 3ele 3e6e2 0 - 22 e3 e6 1 7 12 e e6 e 4 -4e26e22- 3e2e 3e2 5 - 3 2 3e25e1 5 + 2e65 + 5e 2 4 10 1 e 2 -e-e e -6le 3e6e 9 e11 + 2ele 3e5e6 e15 - 4e2e4e 6e ± 3e2e6 2 5 19 e15 -e8 22-2 - 2e e7e13 -3e 7e 0 el 3 + 2e21lle15 + 42e 7e17 - ele 2 6 e5eo0e -2e2e e e16 + 2ele 2e321 + 2e 4e 7 e19 + 5ele 5 e6 e2 + 3ele 3 31 9 3 9 e - 4 le4e8e16 +2ele 7 e22 - e2e3e21 +3e2e 3e 9e1 4 + 2e le3e 7 1 6 el2e4e +2ele2 e8el8 - 2e2e 3e 5e20 - e2ee 13 - 4e2e4e2e10 + 20 ep2 2 e 2e e e e2e 4el8 + 3e2e4e8e12+ 4e2e365e68e12- ele3e 6el14+ l 13 16 e e e7e62- 6ele5el10el4 -6el62e 3e4e20 - 4e3e5e7el5 + 5e4e5 e8e13 + 2 3 -4ee 2 8e10- 3e25e8ell - 3ele3e27e12+ 4ee527e10- ee37e8 62 e22 2e e e -4e2le6 e7e15 - 6ee 3e5e 8e 13 + e7e + e2e22 - e1 4 11 13 6e e e e e -e3e 8e1 9 + e2e3620 - el e4e 7e 18 + 4e2e3e 7e18 + l 3 4 9 1 3 -3ele2e7els + e4e6e7e13 + 2e6e4e20 + e2e4e6 el8 + 2ee 5e6e18 -3eeel4 + e2e2e +-4e 1e2e3e8e6 - 2e4elle3 - ele3e4eloel2 -- + 2e e e -3e 5e7e - 2e4 e6 e2 ± 02 1 ± e2e13 e 1ee 20 3 8 19 -3e e 4 - 8e e e e e1 6 +2e2e9 e1 9 + 2e2e9e19 ± 656se 8 e 2e6e 8 1 1 2 5 6 e e e2 3 -e 1e5 e ±- 4e2e4e5e19 2e3e4e2e1 3 + e3-+e4e 5e6 1 2 4 e e e5 - e +2e 3ee6 -4e2e 12 e16 -4e6e 2 e1e1 5 ± 2e2 6 7 3 2 7 ± 2ele eel8 ± 3ele2ele14 -2e2e+2 32e 8 e17- 3e2e4e5e6613 3 ±ee4e61-2ele6eel22 + 2ele25e120+ 2e2e4e5e15- 3e3ee1 - 2 3 22 2 2e- -e 3e 467e16 56C17-- -ee 2 623e ± 2ee 99 3e16561e63

94 -3ele 2ee 13 + ele 0e12 + 4e 2 e7 e8 e1 3 - 2e 5e61 2 e1 3 - 3eee 13

-[46;ii68¢10'2 -+ 661636e8612- 2 ±e2e3e10e12 ± 2e~e3e4e 9 -3e86263667612-323e6696103e7e20 ± 2el~e8e20 + ele6e7e9 -2ee 5e 7e1 6 - 3elel 0 e19 - 3e3e 10 eo17 + e-e 24 + 6e1e 4e6e 7e 12 +5e,e 2el2 el5 + 2e2 e 3e5e 2 0 - 4e2 e 2el 4 + 5e2ee 1 6 - e2e62e16 +2e 1e 5 6 e 8 e1 0 - 2e3e 4e 5e 7 e1 1 - e33e5e 16 + 5e 2e 3e7e8e10 +2e:1e 4 e6 e9 e10- 3e1ee 12 613 + 2e2e 4 e 9e1 3 - el1e3 e7e 19 -6e2,e 3 e 5 e20 - 32e 7 e 19 + 2e2e 3e 23 + 4e4 e7 e9 e1 0 - 6ele 2e 4 e9 e 14 +4e, 5e2 5 + 4e 2e3e4 e7el1 4 + 3e 2e5e7e1 6 - 3ee4e 7e1 4 - 4e2e 3ele 1 4 -5e,,e e el 5 1 l0 1 3 + 2ele 6 e10 eo1 3 + 3e 4 ee21 0 + 3le 2 e3e611e 13 + 3e 2e 3 e5 e7 e1 3 -4ee 7e8e9 + 3el ee 1l0el - 2e8 l0 e12 - 5ee 4e2e13 + 3e4e12 el4 +ele, 3e7e9e1 0- 3ele 4e1 5 -321 - 3e 5e9e 12 + 3ele4e5e9e11 3 p~~2e2 2 2 6269 -el,'2 - 6e1 e4 e5e 20 + 2e3e11 + 2e4e6e 1 0 + e4 e5e 7 e9 +e 2e5e9 - 2e6 e3 + ele4e5e7e13 - eel 3e14 - 2e2e4e 10e12 -3e,!e699el3 - ele7e96l3 + e26e5e162 2 -3e465616 22e3e8e9 -5e,e e 3 4e 8 el1 3 - 3e 36 2 3 e2e3e5e6 1+ 4 1 e4e 23 2e 3e1 2 e 13 +2e 2,e 51e 1e 12 + 5elee 6 e613 - 2e2 e4 e - ele4e617 - 2ee 3e 5e 9e 12 +2e el 3 e2 ~~2 3 3e 14 + 2e4e - 3e 2 ee 91e 1 - e2e 3e61 1e 14 + e4e 6 e8 + 2e 7 e 9 e14 -4e e e e - 4e3e e el 1 6 7 1 6 6 8 20 + 32e 210eo16 - 3 2e 4e 5 e 8 e11 ++ e~~__2e2e4e11e2 2 2 + e~2C2 2 2 +e2 24 + 3ele9el1 + 3ele 4ele 14 + e6e5 18 + elel 4 + 5e2e3e 2e13 -e e e - 2 5 6 e 1 1 2 - 0 -- 4e 2ee 5 e 2e 2e66e + 2ele 4e 5e6 e 1 4 -5ele5 e6e12 + 6e52e0 - 4e 6e618 + 3 3e8e9el0 - 3e 4e7 9 +3eaee 8e9 + 2 3e4e2 3 + 4 2e6e7e9 - 56e 7e11 + 3 4e5e7e14 -3e1 e2e 4e 0 eo1 3 + 3ele5e7el1 7 + 3le 3e7e8e1 1 + 4e3e8e1 3 -3e 2e32e4e18 + 5 2e4e5e9e10 + 6ele6 e8e1 5 + 2 2ee 5 1 7 - 1e2e4e 23 +e3e18 + 2e2e3e5e 8 + 4e2 e5e 9e14 + 2e 6e 9e 14 + 4e2e5e7e11 -ele 2 e 5e8 e 14 + 2e2e 3 e25 - 3ele 4e2 5 + 6ele 2e6 e7 e61 4 - ele 3 e4e 7e1 5 -3ee 2 e226 - 4ele 3 e4e21 - 3e 3e6 e 7e14 - 3ele 4e1 2el 3 + e163 e8e2 3 2 2 C -ele 2 7 + e8e14 + 3e3e4 e8 el - 2e 6e 7e9 + 6e4 e6e 9el - 2e2e3e9 el6 +3ele 2e5e 7e15 + 5ele3 e4 e8e14 - 3ee 4e5e17 + 2ee 5e17 +2ee 2e12 e 4 + 2e2 5e9e14 - 3 2e5 e23 - 2 5e6 e19 - e4ee 1 -3e2e 3 elle,12 + 2e2e5e23 + 2ee 7e,17 + 3e2e3el0 e13 + 2e2e4 e6e 7e -3e 2e 3e 4e6 el5 + le2e7e12 + 4e8618 + 4ee 15 + 3 1e6e 23 +2e 3e4e7e 12 - 4ee 2e6 e20 + 2e lee 4 e19 + 3el 1 2e17 -le29 C 2 - + 22 +2e2e 3e1 0 e1 5 2e2e8 e18 - 3eele 13 + e2e3e6 el6 2e 5e 0 e1 3 -5e 3e1 2 el5 - 3e 2 8e20 + e2e26 - 2e3e5e6 e - 3e 9 e20 + 4e3 e4e5el8 le2e - 2e2e6 e8e 12 + 2ele3e e 16 + 2e2e e20 - 4ee 6 el0e12 e2 2" e 3 e3e 7el - e4e5e613 - 4e 5e611e12 + 3e 5e7e 2 -4e 2e63e0 +6ele 3 e6 e 20 - 7e5e7e8 e 10 + 2ele 7elo0 e 2 + 2e2e3e7e15 + 6 2e6 e10 e12 +2ele 2ee 17 -3e 3 ee 1 7 - ele 2e13e14 - 3ee2e 3e 22 -9e21 -6ele 3 e 8e 8 - 6e1e 4e10 el 5 -2e e6 e, 2 3 117- 4e2e 11e 17 +e-2ee 2 4 2e 2 e11 e1 7 - e2 e 2 8 -3ee323 --2el 1e3e 5 e 7 e1 4 - 3e2e9 e13

95 el - 4e e3e 13 -37614 -7e 3e 4e9 14 - 3ele2 eel9 - e6e 8 6 2 9 - 2e ee e 4 - 3e3 4e 5e 14 + 4ele 4e9e16 -8e 2e 3e1o6105 + e6 4 6e14 2 6 1 - 2e e e ell + 2e2e3e8e15 -367e 8e 15 + e3e5 e6ell - 5e3e4e6ell 3 7 9 - 3e2lee e21+ e5ee -2e6e 8 e14 + 2e6e2 e4e22 - ee 3e24 2 5 e12+ 4ele e7e20 - 3e4ee15 -2e1 5e7e8e9 - 3e3e4e19 + 2e6e7e9 2 - 3e e1eo + 2e2e22 +3e3e6 e21 - 3ee 3e6eo +±ee 5e9eo 9 4ee e6e3 -3e2e 8e15± 3e1e2e7e9e1 - 3e2e3e4e21 + 2 3 1 +-5e2e e e11 - e3 ee1 2 -6ele3 e el5 -3ele 4 15 -2e 1e2e31e9e 5 6 7 -3ee + 2ee4e24 +-3e2e5e7e ±3ee 4ee9-e7e23-2e2ee 0 9e 0 5ele el0el7 + 4e3 5e 6 e0+ 2e6el -2e6 e9e615+ 2e1e 2e6e9e12 + 2 2ee e e 9 - e2ee 19 + 2e3e4e8e5 +2 3e561636117e1e12 -5e2 e7 e9e12 + 2 7 1 e2 36164676861o e e + 3ee 2e6el9 ±4e2e4e9e5-4e-esel2 ± 2ee e21 ± 2e2 4 24 -3eie e eseio ± 5eie3e -3eie1e 9 e1o + 2e2ee 22 + e3e4e 0 4 7 - 3e6ese16- 3 1e 5e24 +e2ee612- 3e2e13e15- 51e 4e6 e8e11 - e5el9 - 6e 1e2 e 7e 8e12 +5e6el16 + ee 28 + 3e26e5 e6el15 ele2 e3 1 +6e3e46 1-23e4e5e61 2 - 2 -l3961 + 52e4e76e8e9 -3e5e96 --226769e0 + 4e2e46e62-e616e64- 462636619-e639615 e - 2e e 4e 2 2 -e26 24 ± 3e2e6 e20- e1e2e5e21 - 3e1ee 9 1 2 2 eee17+ + 323492 e 3eee ++ 4 31eeel0el6 +3e1e5e 22 eleee96469 + 423872 - 8e6e e6 e21 - 3e 4e 5e21 +2ee6e 5e616- 3e1e2e 8e17 - 3e2e3ee1 12 + 2e2el e15 -eiie - 4e6e7e13 - 4e2e6 ee8 + 3ele4eei1 3 - + 3e6e e6e1 - e6e2e25 -3e2e5 e6 e8e9 + 3ele1ee112 e14e16 5 - 2e e e e 17 + 6ee 2 e3e6e1s -6e 4e6 e2 0 - 4e1e3 e4e6 e16 16 1 3 9 -3ele2e e8-ee7e2 43e~ee9ei-~ 5 e6 e7e61 + 2e62e62 4e539e6ee15 5

- el0el6 e27 - 6e e e8e15 + 6e 3e5 e6 e16 3e 4 -4ee 6e14 + 2e1e 3e26 + 2e1e2 2 5 2eee25 + 5e2e5e6e7e + eee 9e 4 -2e 6e24 3ee 4e7e +26162625e1+ 56C285+ 371 2e,56015 - 6662e4ee16j2467e61

96 Table of P4,n,for n = 7,8 n P4,n 7 -2ee e - 2ee e ll - 1-26,16~13 ~ ~9 1 ~+ ~13626123ee2e23 e2e2 3891-2e 3 e12e 3 e52~-9Ie9e1 4-2ee-+ 4e2 3 -3e +2ee 3 e2 3 5e7 e16 + e2e6 e16 + e2e3e 23 - 2e6 e8 e12 + e 361 1e 14 2 2 2 +e -ele 2 e 23 - e2e4 e 8 - 2ee 3e 2 4 - 2e8e 16 e6e2e24 + 2ele 2e2 5 +6e4Ie8e1 6 + 2e1 e2e5e2 0 + 3e4e2 4 + ee 1 0e1 6 + 2e1e5e6 e16 + e2e3 1e1e12 -2e:1e 4 e1e12 - 3e1 e7e 8e 12 - 2e2e6 e 9e11 - 2elelle 1 6 + e7e10e11 +3e 2 e6 el20- 2e 2e1 0 e16 + e 2e 7e619 + el1e2 e1 2 e 13 - 3e2 e 7e9 e 10 +2e [e8 e 9elo0 + 2e 3 e4 e6 e1 5 + e e22 + e4 e6 e7 e11 - e3e e 1 7 -3e 4e 5e 7e 12 e5e8 - 3e4e 7e8e9 + e1 e7e 9e11 - 2e1e2 10 e 15 - 2e4e9el1 5 - 3e1 5e 17 +3e,"!6 e18 + ee 16 - 2e4e6 e18 + e 12 e214 - e62 15 - ee313 +4e,3e 7e18 + 2e 2 e3 e4 1e 9 - e6e22-3ee- 7e 2 0 - 2e2e8e 18 -23 2 C2 2 2 +e2e8e1 8 + e5e8e10 + e3e8e14 - 61627- 2 2e 5e1 0 e11 - 2e2e6e 20 +2e ],e 1 9 e1 8 - e2 e 4 e1 0 e1 2 - 6 e8 e 14 + e2e1 0e12 + e2 e4 e7 e 15 - 2e6 e10e 12 _ 2 + I 2 2e -e2e3 e2 + ee 3e21 - 2 3e4e 21 - 32e 5e2 1 - e 8e120 + e3e4e7e14 22 - 2 -2624 - 3 4 e7e17 -2e 3e8e 17 + e2e 3e 20 + 2e5e1 0e13 - e2e11e 13 e62e + 2 6 e636e1 0 - 2e2 e4 e6 e 16 + ee 4 e1 0 e 13 + 2e 1e 3 e 5 e 9 - 31e 9 e 17 -el1315+ 2ee - 2 e 8 e1 7 e 3e 2 5 - e 9e 19 + e 4 e83- 2e3 e5e8e 12 + e2e 3 e7e 16 +4e 2e 7e8 e1 1 - 3eje 6 e 7e 14 - e 3 e122 + e25e6 e12 + 628 + 3 4e7e1 0 2 - 2 2 2 +-e4e 5e14 + 2e 2e 5e2 1 -e 3 6e 13 + 3e4e 2 + 2e4e 5e6e1 3 - 2e3e7 e-15 +2e 1e 4e 5e18 + 2e 2 e4 e 5 e1 7 - 2e2 e 4e2 2 + 2le 5 e 22 - 261e 4e8 e1 5 -6469611 + 4 6 e 7 e1 5 - 11 617 - 3e+1 e6 19 2e 5 9 +2e5 e6 e 17 + 3e le2e13 + 3e3e5e 9e,11 2e3e6 e19 + e2e961 3 + 2e6 e9 e13 +2e2,e7e1 1 l - 2ee 2 e11 e14 - 2e4e 5e19 + e262 16 - 2e1e3el0 e 14 +2ee e e 5 1 0 1 2 - 2e5e11e 12 + 3e4e 20 - 2ee 3e 8 e16 - 2e3e 5e 20 - 2e3 6 e9e10 -2e 2 e5e6 e1 5 - 2le 3e6 e18 -3e 7e8 e13 - e5e2 3 + 382e12 e2e22 +2e 2e 5e8 e13 + ele6 e8 e61 3 62le64622 - 2e3e4e9 e12 + e 3 e4 e10e11 -ele6e11 - 2e 3e4e20 - ele9 - e2e12e14 + 2ele3e9e15 + 2e1e6621 +ee,5el 7 + 2e2 e 3 e8 e15 + ee1 61 3 e14 - 3e5 e6 e 7e1 0 + 2ee 2e 4e 2 ~+2eV - e10e18 - e35e 0 + + 2 +2e3e7ed + e2e8e2 -2ele 6 e 0 e1 1 + 3e12e1 6 - 2e1e2e6 e 19 - 2e3e5e6 e 14 + 4 3e6 e7e12 -e 2 2 4e5 e 15 + e3e 15 6625e3- 2ele2e 3e 22 - 2 3e4e 5e 16 + 3e1e 3 elle13 -2e2 C3e6 e1 7 2ee 4e6e17 - 3e1e 2e 7e18 - e6 1 6 -- 2e 2e 3e10 e 3 -2e 5e8e1 5 - 2 2e3e9e1 4 2 1e4e9e14 + e2e65e9e12 + 2e2e 11e15 e2C + 2 e2el_2 -e 2 e8 e 0 6611615- 2 3e10 e 15 - 2ee 12e 15 - ee5 6 e11 -e42e 6e14 -2e3 e4 e 8 el3 + e 3e6 e8e11 - 2e4e 5e8 e11 -e 4el 0 el 4 + 2ele 6e 9eI2 +ele 5 e7e15 - 2e2 4e8e 14 - 64e6e8e10o+ 3e2e6e14 - e3e2e9 +3e e e e2 8 2 0 2e4 1 2e 2 e9 e1 7 - ele2el9 2e 1e 10oe 17 e 3 e7 e 8e 10o +26 26 4 e9 e1 3 - 5le616569613 + 2e616568614 + ele616562- e2e 5 -41e2 -2 62-2656e192 ++36~~26 3 210614 ±e+34e18 l620ee - 2666,5,1223518 +e616367617 + 2e2e7e619+ 6e2elo0- e2e26 +- 636567613 - 36266676e13

97 e e +3ee 8 e1 2 -e 2e4e2 0 - 4e3e2ell - 2ele8e 19 + e4e5e9 e1 0 - 2e3 9 16 3e5e e 8 ee 11e 13 + 3e3e7e21 + 2e1e7e24 - e2e 12 e14 + e3e1 3el6 + 9 13 +3e ee15 - 2e e e17 + e e~~~~~~~~~22ellel2 + e e e8e1 2 - e26e9e1 7 - 1011 3 6 9 4 5 5 7 ± e22e e -2e2 e4e26 - e2el4e16+2e + e8ellel3 ± 2e6 e1 2e14 + 2e1 e2 e1 1e18 e e,~~~~~~~~~~~~2136153 15 3e e 4el6 +-2e9 elle12 + 2ele2e6 23 - e3 e5 elle13 - e2ele19 - 2e6elle15 + 2 1 -2ele4 6 e21 - 2e3e10e19 - 4 4 e12 - 2e5e9 e18 - 2 3e6 e7e616 +1615616 - 2e2e116e19 - 2e3ele 18 - 2e1e2e2 9 + 3e 2e4 e612e1 4 - 3 2e 7e 8e15 - e e e +e-5 e8e1 9 - e2e5e2 0 + 2e l e12 e61 9 - 2e3e6 e23 - 2e3e4e6 19 1 9 11 +2e 4elOel8 -elel 2 e18 + e2e 7e 21 - 2ele5e2 6 + e3e4e8e17 e +-2ele6 elle14 + 2e2 el2el8 - 2e2e4 e9e17- 2e5e10 el7- 21e 7 23 e2 + e3le -2e 3e 7el0oel2- 2e5e6 e21 - 2e3e6 elle 12 - ele 4e 26 - e2e4 3 29 - 2ele e - 2e e e +e 6 e26 + 2e 2 e5 e6 e1 9 + 2e 2e6 ele 3 - 2el1e5 e1 2 e1 4 13 18 2 7 2 3 e e e -+ele6e19 + 2ele 7 e1 0 614 + e2 e 11 e1 7 + 3e l e9e 13 + 2e 3 4 9 1 6 -+e201e 2 - 2e3 e5 e9 e15 -- 2el1e2e13e16 + 6e165e9 e17 e e - 2e4e e e13 +ele 8 e23 - 3e2e 24 - 2e2e7 el6le12 + 4e 3e 8 e21 - 3ele 2 8 21 6 9 - ee +2e 4 e1e17 +- 2ele2e 3e26 + el3e19 + 2e4e 5e 7e16 + 2e 3e1 2e17 2ele7 11 13 -2e 5 el 3e14 - 3ee 10 e61 8 - 2e2e13el 7 + ele 2 e 14e15 ' e1e6e2 3 + -2e 2e 3e4e23 + 3e21el3e17+ e2e5elel4 1le2627 -2e 1e2e4e25 +2e2 e 5e7e8 + 2e3 e4 e25 + e2 e 28 - e 6 - 2elel 4e 17 + 2e 4e8e2l0 +ele ellel2 + 2ele3eloe8-ee- 4 -4e e 6 - 2e7eloelh+ 2676-126132e7e12e13 8 + 263108- - 212 1424- 46216 -27610e15 -[[68162 e e 7- 2e e e e -ee 3 e2 7 + e2e 9 e19 + e4 e 4 -2e 4 e5 e 9e14- 3ele 6 8 1 2 4 7 19 + e ee6 -3e4e22 - e3e7e9el1 3 + e9e23 + 2e 3e 5e61 0 el14 - 2e 5e 6 e9 e1 2 2 2 2

A-ee2 0 e61 + e26e4e 24 + e 14 e1 8 -2e 4e5 e6 e1 7 - 4e 20 + e3e1 7 e62e615+ e e e e 4 -e 9 2e14 + e3e5 e8e16 - 2e 3 e4 elOel15 + ele 4 1 2 3 1 3 1 - 2 2 - + e4e5e8e15 - 26e e 7 + e e e21 +-2e3e6 e9e14 - e6 e7e12 e5e 6 el2 6 2 3 2 22 3 4 +-2ele3e4e24 - 3e2e8e9e613 le 6 e24 + e3e6e10e13 + e3e6 e17 2 22 +-2e2e3elo0e17+ 2ele4e2 7 - e6e20 + 220 - e eel6-e e 2 - 2e e - 2ele e e -e 4e6e 1 8 + e2e 6 e618 + 3e1 1 + 2e 2 e4 e6 20 8 24 4 1 3 14 - e e + 2ele e -ele 2 e28 - e3e26 +- e 26 7 e9 e14 + 2e 2 e7 el0el3 3 6 20 3 2 8 - e - 2ele el0el7 +-e2 e6 e8 16 - e2 e29e12 -- e63e14+ 2e 2 e6 e7 e17 + ele 31 3 2 4 - ell +-2e3e5e6 el18+ 6ee9e1 - e6e23 + e611e2 1 - 2e8e12 e5e7e9 - 2ele e e +6e4eel6 - 3e1e8e9 14 + 3ee 1 1e1 5 + 2e 2 e5 e1 0e1 5 5 6 2 0

+e265e23+2elC667618 + 2e465623- 2le,4C9e8 -2elel0e21 -2 3e3e7e8el4 + e4e5el9 + e~e6l + 4ele768616 2e2e3e9el8 - 3e61220 + 2e e e e66 -3e5 e 8 e9 elo + e7e 9el6 e 5e22 - 3e2e6el 8 - 2ele 5e 7el9 l 3 12 1 +-ele6 elOel5 + e2e4e8e18 + e15e17 +- e2e30 + e5e27 + e2e8elo0el2 -2ele3ele 17 + 2e2e3e5e22 + e5e6 e1 0e11 +-2e 4e9e19 + 2e5e1 2e15 e + ele e2l4 +-2e4e7621 + e10 e 22 - 2ele 4 e 5e 22 - 2ele 9e10 e612 + 4ele 4e8 1 9 3 2 _ 2 - e A- 4e e e el A-e3 ege1 0 - 2ele 3e7e2 l - 2e 1e 2e 12 e1 7 e 10 12 3 8 9 2 +-2e26e8e12 A eel0e14 + 3e2e 7 e19 +- 3e 4e6 elo0el2 - e2e5 e2 1 22 2 -+3e5 e6e26 - 12e 5 e7e17 6- 10 2e 3e5 e2 4 -e 3e 5e 9 -2e 2e3 67e2o -- 2e e e10ei1 -2e 2 e 5 e9 e66- 2e3e 4e11e14 - 3e266e22-62630 4 7

98 +62e 7e 13 + 2e5e7e 20 + 3e 5 25 - 2e 2e5e25 - 2e3e 14e615 + 4e8e 9e615 +3e e2e 21 + 4e6 e 8e 18 + 2e 2e6 e9e1 5 - e 7e9 e 15 + 2ele 4 e 7e2 0 -2e1e6 3 e 5e 2 3 - 4 6 e21 + e4 e8 e 9 11e - 2e2 e4 e 5 e2 - 2e1e6 e 2 5 -e2 e625 + 2e2e4e 0 e16 + e8e1 0e14 - 5e2e6 e10 614 + 2e4e6 e22 - 2 4e5e10 e 13 2 2 +e 23 +e 3e 4e61 2e 3 - e4e 9e10 + 3ele 5 613 ± e2e 3e25 + e4e7el1 7 + e 5e9 +e5e 7e 0 -3e 3e 8 e1o0e11 - 26e7el9-e 3 e-10 e 16 - 2e9e1 0e 13 - 2 3e 7e 22 +e 7e 25 - 2e 2e 5e1 2e1 3 - e3e9 e11 + 3e6 e1 0 e 16 - 3le 5e8 e1 8 - 3e 5e6e 8e613 +e4 6 e 8e1 4 ± ee 9e15 + 2 8 20 + 2e 2e9 e61 0 e61 1 - 7e 4 8e 20 -ee 3e 3eI 5 - 2ele3e9e19 + e2e8e22 - 2e2 e3e12e1 5 - 2e3e4e7e1 8 -2e 2 e 4e 1e1 5 - 2ee 2e5e24 + e3e23 - 2e4e6e 7el15 - 3e6 e7e8e1 1 2 2 e -2ee 6 e 2e 3 -e 7e618 + 2e 3e9e 20 - 2ele 9 e 22 - el e3e2 5 32e 9e21 2 2 e2 -2e=,e9 e2 l elele20 -e 5e7eB3 + 2e2e 0e20 -e 2e7e16 + 2ee 2 e7e22 - 2 +e5e:11 e16 e 2e 3e24 + 2e le l l e20 - 2 2e3ele 6 + e6 e7e9e10 -ee 5e611 6e15 - 2e26 + ele3e8e20 - 2le 6 e9e1 6 - 2 7ellel4 2 3 ~~~~~~2 -2ele 5e 10 e 16 + 2e1 e3e6 e22 -e 5e 7eI 5 - 2 30 + e3e5e122 - e2 +2ele 4ellel6- 7e7 - 2ee2 e9e2 0 + 2e4el3el5 - 3e4e28 + 2 2e6 e 24 +e 4e 7 e 9 e1 2 + e3 e2 9 + 2ele 2 el 0 el9 - 5 4e1 2 e 16 + 2e 5e6 e 7e1 4 + 3e2 e e1 4 62 - ~22 +e 2e6 e2 2 + 4 4e7e8e13 -e 4e 7e 14 + ele7e22 + e 7e8e1 7 -3e 2e3e 8 e 19 + 2e2e 3e6 e2

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