Sums of squares in function fields of elliptic curves
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SUMS OF SQUARES IN FUNCTION FIELDS OF ELLIPTIC CURVES
by Geoffrey William Cunningham
A Dissertation Submitted to the Faculty of the DEPARTMENT OF MATHEMATICS In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA
August 3, 2000 UMI Number. 9983868
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TABLE OF CONTENTS
ABSTRACT 6
CHAPTER 1. INTRODUCTION 7
CHAPTER 2. WEIL'S GENERAL RESULTS 14 2.1. A Representation of the Heisenberg Group and Its Normalizer ... 14 2.2. Sections Over Subsets of Sp{G) 19 2.3. The Theta Model 26 2.4. The Theta Function Associated to the Theta Model 29
CHAPTER 3. THE THEORY IN THE LINEAR CASE 30 3.1. A Representation of the Heisenberg Group and Its Normalizer ... 30 3.2. Sections Over Subsets and Subgroups of Ps{X) 32 3.3. The Theta Model 33 3.4. A Theta Function on Weil's Metaplectic Group 34 3.5. How to Compute 7(/) 35
CHAPTER 4. SPECIALIZATION OF WEIL'S THETA FUNCTION 38 4.1. .A Special Case 38 4.2. Invariance Properties of 9 42 4.3. Restriction of ^ 45 4.4. Transformation Law 47
CHAPTER 5. MERRILL AND WALLING'S THETA FUNCTION 53 5.1. Definition and Transformation Laws 53 5.2. Comparing 9mw with © 55 5.3. A Direct Deduction of Merrill and Walling's Inversion Formula ... 56 5
TABLE OF CONTENTS—Continued
5.3.1. A Comparison of Transformation Laws 57
CHAPTER 6. REPRESENTATIONS OF ELEMENTS AS SUMS OF SQUARES . 63 6.1. Fourier Series and Restricted Representation Numbers in the Genus One Case 63 6.1.1. General Formulas for Fourier Coefficients 63 6.1.2. Fourier Coefficients and Representation Numbers 66 6.2. Summary of Results 68 6.3. Exponential Sums 72 6.4. Recurrence Relations 79 6.4.1. The case ^ = 0 81 6.4.2. The Case ^ # 0 92 6.5. Bounds and Asymptotic Formulas 96 6.5.1. The Case ^ = 0 96 6.5.2. The Case ^ # 0 99 6.5.3. Individual Fourier Coefficients 107 6.6. Exact Formulas for Small Valuation Cases 116 6.6.1. w{x) = 2 118 6.6.2. w{x) = 3 120
REFERENCES 124 6
ABSTRACT
In this dissertation we examine the problem of determining restricted representation numbers. Let k be the function field of a curve over a finite field and let i/; be a place of k of degree 1. Then r2/(^, n) denotes the number of representations of ^ as a sum of 2/ squares where the summands are integral away from w and have a pole of order at most n at w. This problem has been studied by Merrill and Walling in the case of the rational function field, F,(T), by relating the restricted representation numbers to the Fourier coefficients of the 2/-th power of a theta function. In this dissertation we are interested in the case where k is the function field of an elliptic curve over a finite field of order q, a power of a prime.
We use a more general theta function defined by Weil. A transformation law or functional equation for this theta function follows almost immediately from Weil's results. The analogue of the classical inversion formula is a special case of this transformation law. In the case of the rational function field we show that Weil's theta function specializes to the theta function used by Merrill and Walling.
In the elliptic curve case, the restricted representation numbers are simply a factor of g"' times the Fourier coefficients of the 2/-th power of Weil's theta function. Thus, we may simply study the Fourier coefficients. We obtain a formula for these Fourier coefficients via classical techniques and then derive a recurrence relation for the Fourier coefficients by applying the transformation law to the formula. We are unable to solve the recurrence relation completely. However, a careful analysis of the recurrence relation and of the values of the theta function itself, along with some lemmas on character sums, lead to asymptotic formulas. We show that the Fourier coefficients are asymptotic to as / —> oo and to a constant times ^n('-2) as n GO. 7
Chapter 1 INTRODUCTION
The problem of representing integers by sums of squares dates back to the time of Diophantus (325-409 A.D.). One of his statements is equivalent to the identity
(a^ + -I- d^) = {ac + bd)^ + {ad — bc)^ = (ac — bd)^ + {ad + bc)"^,
which shows that if two integers are both representable as sums of two squares then so is their product. It was known to Diophjintus that no integer of the form 4n — 1 can be represented as a sum of two squares. The answer to the question of which integers can be represented as sums of two squares is given in the theorem below. It was conjectured by Gerard in 1632. Fermat claimed, in 1641, to have an "irrefutable proof based on his method of descent. However, Fermat did not make his proof public. The first known proof is due to Euler in 1749.
Theorem 1.0.1. A nonnegative integer n can be represented as a sum of two squares if and only if all prime divisors, p, of n with p = 3 (mod 4) occur in n to an even power.
For a proof of this result see [Gro85, pp. 16-17].
The next logical problem to consider would seem to be which integers can be represented as sums of three squares. However, it turned out to be easier to consider representations of integers as sums of four squares than as sums of three squares. It is believed that the following result was known to Diophantus. It was first proved by Lagrange.
Theorem 1.0,2. All nonnegative integers are sums of four integral squares. 8
Soon after Lagrange, Euler published two proofs of this result, both simpler than Lagrange's. Other proofs are due to Legendre and Gauss. A proof similar to one of Euler's can be found in [Gro85, pp. 27-30].
Once it was known that all integers could be represented as a sum of four squares, the number of ways to represent an integer as a sum of squares became of interest. What is meant by the number of representations is made more explicit in the following definition.
Definition 1.0.3. Let n be an integer and I a positive integer. Then we define ri(n) to be the number of l-tuples, (mi,..., mi) € Z' such that = "•
Note that this definition counts two representations as different even if only the order of the squares is changed or if m,- is changed to —m,. For example, 9 has four representations as a sum of 2 squares, namely 3^ + 0^, (—3)^ + 0^, 0^ + 3^, and 02 + (-3)2.
Gauss proved the following result on r2(n) in 1801 using quadratic forms.
Theorem 1.0.4. Let d{n) denote the number of divisors of n, and write da{n) for the number of divisors of n congruent to a modulo 4. Let n = 2'nin2, where "1 = np=i (mod 4)?'" m = n,=3 (mod 4) 9"'• TaCn) = 0 if any of the exponents s is odd. If all s are even, then
r2{n) = Ad{ni) = 4(di(n) - dz{n)).
For a proof of this result see [Gro85, pp. 18-19].
Jacobi introduced the use of elliptic functions and theta functions to the study of representation numbers in his Fundamenta Nova in 1829. He used these new techniques to give another proof of Theorem 1.0.4 and also to prove Theorem 1.0.5 below. 9
Theorem 1.0.5. Let Ti{n) = 8cr'(n). Two different proofs of this theorem can be found in [Gro85]. Jacobi also gave explicit formulas for rein) and rgin) in terms of divisor functions. However, his methods failed for odd values of I and for I > 8. At this point it is useful to explain to the reader how theta functions are related to representation numbers. The classical theta function on the complex upper half plane is given by nez n6Z for g = The Zth power of © can be written ri(m)g'". 7'=ln,€Z ni,...,ni€Z mgz m>0 Thus determining r[(n) is equivalent to determining the coefficient of g" in the Fourier expansion of ©'(-z). General formulas for rj(n) for any value of I were given by Mordell in [Mori7b, MorlZa, Morl9] using modular functions, and Hardy in [Har20] using the circle method. However, for / > 8 these formulas were not explicit like those given for I = 2,4,6,8 by Jacobi. They involved other functions whose complexity increased with I. For an introduction to the circle method and an overview of the methods used by Mordell see [Gro85]. Ramanujan continued the work of Hardy and gave aisymptotic formulas for r/(n) with / > 8 by showing that the Fourier coefficients of powers of the theta function were asymptotic to the Fourier coefficients of Eisenstein series (see [Har59]). A recent announcement by Milne [Mil96] gave explicit formulas 10 for some infinite families of I. While trying to state the main results, this brief synopsis has left out many contributions to the theory. For a more detailed history, see [Dic66, pp. 225-324]. Work has been done on a number of problems analogous to the classical problem described above. These problems include the number of representations of integers by various other quadratic forms, and the number of representations of elements in the ring of integers of an algebraic number field by sums of squares of elements in that ring. The analogue of interest in this paper is that of a one variable algebraic function field of characteristic p > 2. Let Fq be a finite field of characteristic p > 2 and consider the one variable algebraic function field F,(T). In this case, the polynomial ring F<,[r] will play the role of Z. Given ^ G F, [T], we want to know how many ways we can write ^ for 6 FI;[T]. Unlike the classical case, the answer can often be infinity. Consider the following two examples: • Suppose —1 is a square in F,. Then zero has an infinite number of representations as a sum of two squares. • Zero always has an infinite number of representations as a sum of p squares because adding up the same square p times will always give zero. We want the numbers we are trying to compute to be finite, so we refine the problem to be that of determining restricted representation numbers. Define ri{a,n) to be the number of ways we can write a = for each € F,[T] of degree at most n. We will refer to these restricted representation numbers just as representation numbers. Carlitz and Cohen studied this problem in the 1940's in the case where n = and n = 2 deg (a) — 4. Cohen obtained exact formulas in [Coh47, Coh48] using the 11 circle method, but it was not clesir from these formulas whether the representation numbers were nonzero. Serre showed the representation numbers were nonzero in the case 1 = 3. For a reference, the reader should see [EH91]. In a paper in 1993 [MW93], Merrill and Walling considered the case 9 = p a prime. They used a theta function introduced by Hoffstein and Rosen [HR92] on a function field analogue of the upper half plane. Merrill and Walling showed that this theta function transforms as a half weight modular form. As in the classical case, the representation numbers are related to the Fourier coefficients of powers of this theta function. Merrill and Walling used their transformation laws and some elementary techniques to find an explicit formula for r2/(0, n) and in the case where a ^ 0, a, formula for r2i{a,n) involving Kloosterman sums. They used the latter formula to determine the order of magnitude of r2i{ct,n) and compute an asymptotic formula as I 00. Hoffstein, Merrill, and Walling recently extended and improved the results of Merrill and Walling's 1993 paper by comparing the Fourier coefficients of powers of their theta function to those of Eisenstein series [MW99]. They removed the restriction that q = p and showed that, as in the classical case, the representation numbers are asymptotic to Fourier coefficients of Eisenstein series. Moreover, they computed formulas for ri{a,n) for all values of I. In this paper we want to consider the more general problem of computing rep resentation numbers in the function field setting. In order to accomplish this, we need a more general statement of the problem. Let k be an algebraic function field of dimension 1 over a finite field and suppose char{k) ^ 2. Given ^ E k and a positive integer I, we want to determine the number of ways ^ can be written as a sum of 21 squares, ^ such that the have bounded poles. In order to make the problem statement more precise, we will need some notation. 12 For each place v oi k lei denote the completion of k with respect to the absolute value associated to v. Let C?„ denote the ring of integers of Now let 5 be a finite set of places of k zmd for each place u 6 5 choose a nonnegative integer n„. For a.\\ V ^ S set n,, = 0. Define n to be the tuple of the riy, i.e. n = (u^). Set Now given ^ G X, we define the restricted representation number r2i{^,n) to be the cardinality of the set {(A, —,^21) € '• ^ = Pi 1- Our problem is to compute r2/(^,n). The problem studied by HofFstein, Merrill, and Walling is the case where k = ¥g{T) and S consists of just the place at infinity. Our methods will mimic the methods in [MW93], so we will restrict ourselves to representations by even numbers of squares. The theta function we will use and the theory behind it is described in detail in "Sur Certains Groupes D'Operateurs Unitaires", by Andre Weil [Wei64]. Let G be a locally compact abelian group and let Sp{G) denote the symplectic group. We want to consider Mp{G), an extension of Sp(G) by T, the multiplicative group of complex numbers of absolute value 1. Weil defines a representation of this extension as unitary operators on L^(G). We examine the above extension in more detail in the linear case. Let k be either an algebraic number field or an algebraic function field of dimension 1 over a finite field. Let Xk be a finite dimensional vector space over k and let A denote the adele ring over k. Weil is primarily interested in the cases G = A', where X is either a finite dimensional vector space over a local field or X = Xk 0 A. In these cases Weil defines the metaplectic group, Mp{X). He shows that this group can be realized as an extension of the pseudosymplectic group, Ps{X), by {±1} where we have Ps{X) = Sp{X) for char(k) ^ 2. Thus the above mentioned unitary 13 representation gives a unitary representation of Mp{X). In the case X = Xk 0 A Weil uses this uniteuy representation to define theta functions on Mp{X) and shows that these theta functions are left invariant under Ps(Xk). We will summarize the relevant results from Weil's paper and specialize to the following case. Let X = ® A for k a finite extension of Fp(T) and p ^ 2 a prime. We will show, by interpreting Weil's results in this special case, that our specialized theta function transforms as a half weight modular form on an appropriate analogue of the upper half plane. We will then show that this specialized theta function is in fact a generalization of the theta function used in Merrill and Walling's 1993 paper. Finally, we will specialize to the case where k is the function field of an elliptic curve and S = {ly} for some degree 1 place, w, of k. Since riu = 0 for all V ^ w in this case, we will abuse notation slightly and just write n for tIu;. We obtain bounds on r2i{a,n) and asymptotic formulas as I —> oo. We obtain exact formulas for r2i{a, n) in the case n < 3 for q G F, the field of constants of k. 14 Chapter 2 WEIL'S GENERAL RESULTS 2.1 A Representation of the Heisenberg Group and Its Nor- malizer Given a locally compact abelian group, G, we want to define the Heisenberg group, A{G). We then want to show how to realize the Heisenberg group and its normalizer as subgroups of Aut{L'^{G)), thereby giving us a representation of both of these groups. First we need a little background and some notation. Let T be the multiplicative group of complex numbers of absolute value 1. If G and H are two locally compact abelian groups, then a bicharacter on G x i/ is a continuous function f : G x H ^ T such that for all x G i/, /(x, •) : G —> T is a character and for all y e G, y) : H T is a. character. A continuous function f : G T will be called a degree two character or second degree character if the function (x.y) i—> /(x + is a bicharacter of G x G. We denote the Pontryagin dual of G by G' and let (x,x*), for x e G, x* G G*, denote the value on x of the character x* of G. If / is a second degree character of G, then f {x + y)f {x)~^f {y)~^ = {x,yg) for some symmetric morphism g : G G*. We say that / and g are associated to one another. If g is an isomorphism, then / will be called non-degenerate. If the map x 2x is an automorphism of G, which we will assume throughout this paper, and ^ : G —^ G* is a symmetric morphism, then g is associated to the second degree character fg{x) = (x,2~^x^). Choose a Haar measure dx on G. Given a function $ on G, we define the Fourier 15 transform ^ relative to dx by $(x*) = J $(x) • {x,x')dx when the integral exists. Let ^ € L'^{G) and let / be a second degree character of G. We define the convolution of $ £ind /, denoted $ • /, by ($ * /)(x) = / ^{u)f{x — u)du when the integral exists. Definition 2.1.1. Let G and H be two locally compact abelian groups with Haar measures dx and dy respectively. Let a : G H be an isomorphism. Then we define |Q:| by the formula f F{y)dy = la| f F{xa)dx, Jh JG where F 6 L^{H). Theorem 2.1.2. [Wei64, P- 161] Let f be a non-degenerate second degree charac ter of G, associated to the symmetric isomorphism g : G G' and let G L^{G). Then where f is the second degree character of G' given by J{x') = -r{f)\Q\~'^ f{x' for y{f) a scalar factor of absolute value 1. Despite the notation, note that / is not the Fourier transform of / in the usual sense. We will see later how to compute 7(/). Let F denote the bicharacter on (G x G*) x (G x C) defined by F{zi,Z2) = {xi,xl) (2.1) for zi = (xi,xi), 22 = {^2,^2) &G y.G'. 16 Definition 2.1.3. An automorphism a of G x G' is symplectic if it leaves the hicharacter F{z\,Z2)F{z2,Z\)~^ invariant, i.e., if it satisfies the relation F{zia, Z2(T)F{z2cr, zia) = F{zi, Z2)F(Z2, Zi) for all Zy, Z2 G G x G'. IVe denote the group of symplectic automorphisms by Sp{G). Let a be an automorphism of G x G'. If we take {x.x') € G x G*, then a can be written in the matrix form (x, X*) (x, X*) • ^ ^ ^ ~ ^ where we write a = ^ ^ ^ ^ for morphisms a : G G, 0 : G G', -y : G' G, and 6 : G' —> G'. Let cr* = ^ J. ^ denote the automorphism of G' x G dual to cr and let t) : G x G' G' x G he the isomorphism given by ^ 0 ) define I . _i (5' -P' \ ^ a' j' and note that cr^ defines an automorphism of G x G'. For an automorphism cr to be symplectic. it is necessary and sufficient that era' = 1. The Heisenberg group, A{G), is defined to be the set G x G' x T with group law given by • (uJ2,t2) = (wi + ^2, F{uJi, (J2)tit2) where uji, uj2 E G x G' and ti, t2 E T. Let u = {u,u') € G X G' and t £T. Then for $ G L'^{G) the unitar\- operator t • U{u) is defined by t • U{uj)^{x) = t • <^(x + u) • (x, u'). Let A(G) be the set of operators of the form t -[/{u). Note that the set A(G) is in fact a group and if we give A(G) the topology induced by the "strong" topology on 17 .4uf(L^(G)), then the map {u:,t) t-U{u)) from i4(G) to A(G) is an isomorphism of topological groups. For a proof of this result, see [Wei64, p. 149]. Let B{G) denote the group of automorphisms of .4(G) acting on the right, and define BO{G) to be the set of all elements of B{G) which induce the identity on T, the center of A{G). Let s 6 Bq{G) and suppose that s induces a on G y. G'. The action of s on A{G) can be written {cjJ, t) • S = (W(7, f{w)t) where f : G x G' —> T is continuous. As s is an automorphism of A(G) we must have f{ui +u;2)/(WI)~V('^2)~^ = F{uJia,uJ2cr)F{u)i,cj2)~^ = F{uj2a,uicr)F{uj2,uji)~^ (2.2) for all UI\,IJJ2 € G X G*. From (2.2) it is clear that / is a second degree character of G X G* and cr is symplectic. Thus we can write elements of Bq{G) in the form (<7,/) where A is symplectic and / is a second degree character of G X G'. For (cr, /), (o-', /') e BO{G) the group law on Bq{G) is given by {aJ)-{a',n = {oc',n where /"(uj) = f (cj) f {uicr) for all a; e G x G*. The map s •-> cr defines a homomor- phism from Bq{G) to Sp{G). The kernel is {(1,/) 6 BO{G)}. From (2.2) we have that / must be of the form f{u,u') = {u,a') • {a,u') for a € G and a' 6 G*. We conclude that (1,/) is the inner automorphism of A{G) associated to the element {-a. a', 1), and so the kernel of the map s cr is the set of inner automorphisms of .4(G). This gives ker{s ^ G x G\ and we have the exact sequence 0 —^ G X G* —^ Bo{G) —^ Sp{G) —> 0. 18 We are assuming that x H-> 2x is £in automorphism of G so we can be more spe cific. There exists asection s : Sp(G) Bq(G) given as follows. Ifcr=^^ Sp{G), then S( Bq{G) ^ (G X G*) X SpiG). We compute the section, s, explicitly on a generating set of Sp{G) as follows: (; r )-<«•((; -v)->) where f{u,u') — {u,u'), {10'^) where / is the second degree character of G associated to p : G —>• G*, and where /' is the second degree character of G* associated to g' : G' G. Recall that A{G) = A(G). Let Bo(G) be the normalizerof A(G) in Aut{L^{G)). Let T be the centralizer of A(G) in Aut(L^{G)) and note that T is the group T thought of as a group of operators. Given a € A(G) and b 6 Bo(G), a i—> b~^ab defines an automorphism of A(G), and therefore an automorphism of A{G). Moreover, this automorphism fixes T, and the elements of Bo(G) that induce the identity automorphism are precisely the elements of T = T. Thus we have a monomorphism 5^ So(G). 19 Showing that this map is in fact an isomorphism is more difficult. However, the result is given by the following theorem. Theorem 2.1.4. [Wei64, P- 157] Every automorphism of A(G) which induces the identity onT is induced on A(G) by an interior automorphism determined by an element o/Bo(G). Furthermore, is isomorphic to Bq{G), the group of automorphisms of A{G) inducing the identity on T. Using this theorem we obtain an exact sequence 0 —> X —^ Bo(G') —^ Bq{G) —^ 0. Recall that there is a monomorphism i : Sp{G) —> Bq{G). We will denote by TTQ the surjection Bo(G) = Bq{G). The metaplectic extension of Sp{G) is the pullback of Bo(G) by i. In other words, Mp{G) = {(s,s) € Sp{G) x BqCG) : i{s) = 7ro(s)}, and we have the following exact sequence; 0 ^ T -)• Mp{G) ->• 5p(G) ^ 0. Given an element S = (s, s) € Mp{G), it acts on $ 6 L^(G) via 5 • $ = s • In general the map TTQ does not admit a section, i.e. Mp{G) is not a semi-direct product. However, we can define sections over certain subgroups and subsets of Bo{G). 2.2 Sections Over Subsets of Sp{G) We want to consider the set of Schwartz functions, 5(G), inside L^(G). The set S{G) is defined as follows. If G = R" x Z'' x T' x F for integers n, p, and q and F a finite group, then we say that G is an elementarj' group. A polynomial function on an elementary group G is a function that can be written as a polynomial 20 in the coordinates of the factors R" and I? with no conditions on the factors T' and F. For an elementary group G, define S{G) to be the set of infinitely difFerentiable functions ^ such that P • is bounded on G for any translation invariant differential operator D 2ind any polynomial P. In the general case we introduce pairs {H, H') with the following properties: • H is generated by a compact neighborhood of 0 in G. Note that H is therefore both open and closed in G. • H' is a. compact subgroup of H, and H/H' is isomorphic to an elementary group. To each such pair corresponds the family S{H,H') of continuous functions on G, with support contained in H, constant on the classes of H', and such that the function on H/H' given by restriction to H and then passing to the quotient is in S{H/H'). We then define S(G) to be the union of the S{H, H'). Examples: • Suppose G = R. Then S(G) = {/ € C°° :suPX6r(1 + |x|)^|5°/(a:)| < oo for all nonnegative integers N.a}. • Suppose G is a non-archemedian local field. Then S{G) is the set of compactly supported, locally constant functions on G. • Suppose G is the adele group over a function field k. Then a Schwartz function on G is a linear combination of functions = Ht, where the are subject to the following conditions. (i) <^1, is a locally constant compactly supported function on A:„, the com pletion of k with respect to the valuation u; 21 (ii) is the characteristic function of the ring of integers of for almost all places v. Recall that the definition of ^ at the beginning of section 2.1 does not make sense for all ^ € L'^{G). This definition does however make sense for all $ € 5(G). We want to define maps from subsets of Sp{G) to Bo(G) by giving the action of the image on a function $ G L'^{G). In cases where the action involves a Fourier transform, we will define the action of the image on a function $ € S{G) and then extend the definition to all of L^(G). Let $ e L^(G). We define the following maps from subsets of Sp{G) to Bo(G). Q ^ do (a) which is given by do(a)^(x) = |a|2$(xQ:). ^ Q J ^ to(/) which is given by to(/)^'(x) = Hx)f{x) where / is the second degree character associated to q. Note that do and to are monomorphisms into Bo(G) from the subgroups of Sp(G) generated by ^ ^ and ^ Q J ^ respectively. We also have a map ^ ^ do(7) which is given by do(7)$(2;) = |7|~5$(-X7*~^). (2.3) The definition (2.3) is only valid when the definition of $ is valid. As stated before, the definition of If / is a second degree character of G, a an automorphism of G, and x € G, then we will define f° by /"(x) = f{xa~^). To avoid confusion, for a = —1 we will write f~{x) to mean f{—x). We have the following relations: do(a)-4o(/)do(a) = to(r) (2.4) dj,(7a) = d^(7)do(a) (2.5) dJ,(a-W) = do(a)dJ,(7)- (2.6) Recall from section 2.1 that given a G Sp{G), we may write cr in the matrix form ^ ^ ^ ^o(G) be the set of all (cr,/) 6 Bq{G) such that 7 is an isomorphism. Given s = {cr, f) € f2o(G), a can be written uniquely in the form Vt^J-VO 1 j"U 0 jU 1 )• From this decomposition we see that /? = Q7~^5 — 7*"^, or equivalently 7<5* = 5Y- Define a map FQ : f2o(G) —> Bo(G) by ro(-s) = to(/i)do(7)to(/2) where fi and /2 are second degree characters of G associated to Q;7~^ and 7~^5 respectively. The following theorem shows that the map defined above is not quite a homomorphism. Theorem 2.2.1. [Wei64, P- 163] Let s = {cr,f), s' = (cr',/'), and s" = (cr",/") be three elements of Bq{G) such that s" = ss'. Suppose that a, a', and a" can be written in the form where 7, 7', and 7" are isomorphisms from G* to G. Then the formula fo{u) = /(O, u7"')/'(«: —uq'7'~^) defines a non-degenerate second degree character of G 23 associated to the symmetric morphism 7 ^7"7' ^ : G —> G', and the operators ro(.s), ro(s'), and ro(s") respectively associated to s, s', and s" satisfy the relation Tq{s)tq{s') =7(/o)ro(s"). where 7(/o) is defined by Theorem 2.1.2. We want to extend the map ro to a map r which will be defined on all elements fa 3\ V 7 S J ^ 7 = 0 or 7 is an isomorphism. Of course r will coincide with ro on elements of Clo{G). The extension will be made under the following assumptions. We will assume G = G*. Fix an isomorphism (f) : G G' and identify G with G' via 0. We then make the further assumption that for all ^7 S ^ ^ have a6 — = 1. These assumptions will hold for sdl our applications. We now need to define r on the set of upper triangular matrices in Sp{G). Suppose we have an upper triangular matrix, a. Then we can write a in the form ^ 0 ^a~^' ) for a, 6 € G and a an isomorphism. Define <0 A)- Given elements cri,a2 we waiit to show how to multiply r(cri) and r(cr2). Let f a a~^'b \ , { c c~^'d \ , . . . = I and a2 = I Q I be two upper triangular matrices m Sp{G). Then using the relations defined in (2.4),(2.5), and (2.6), and the fact that to and do are homomorphisms, we have \{ a a~^'b \ f c c~^'d \1 f ac a~^'c~^*{a?d + b) \ V 0 a-i* J 'vo c-i* ) ~ 0 ) = to(/a2(i+6)do(ac) = to(/b)to(/a2d)do(a)do(c) = to(/6)do(a)to(/d)d(c) / a a~^'b \ f c c~^'d \ ~ 0 a-i* j V 0 c"^' ) ' 24 and we conclude that the map r is a homomorphism on the set of upper triangular matrices in Sp{G). Now we want to consider the product of a matrix ^ ^ ^ ^ Qq{G) with an upper triangular matrix ^ ^ '(t OKo " '»(/<"-'K(7)to(A-.i)to(/6)d„(a) = to(/a7-i)do(7)to(/-y-ti+6)clo(a) ~ )do(')')do('^)to(y7~i(5a-2+6a~^) ~ ^o(/"a7~' )do('y®)^o(/7-'<5a~2+6a~^) -'[(•; 5) Now we want to consider the product of an upper triangular matrix with an element of QO(G')- We have "•(o s) = to(/>)doWt„(/„-0di(7)to(/,-.,) = to(/6)to(/Q-,-ia2)do(a)do(7)to(/-T-i5) to(/<)+a-|r-iQ2 )cdg(o 'y)^o(y^~'(5) = r f a a~^'b\ f a p \ .1, 0 ; • V 7 S )_ Finally we consider the case where we multiply two elements of no( / a 6 \ / Q ^ \ ^ aa + bj a0 + bS \ \ c d J y 5 J ~ y ca + d'Y c0 + d6 J ' 0 ^ + m "pjbs y V aa+by / We want to write the matrix above in a more convenient form. In addition to our assumptions above, we have the relation: ca + ^7 = 0. 25 We deduce that aa Moreover. /aS —l\ /ad —1\ a^ + b5 = af / \ ) _ aaSc — ac + Sadj — Sj C7 ac + (J7 C7 _ c f ac + S'y\ / 2 £ f a£±s2_\ Thus the matrix (2.8) can be written as 1 7 v ; 0 If we apply r to the matrices on the left hand side of (2.7), then we have '•(c d)'"(7 s) = to(/ac-OdJ,(c)to(/.-:,)to(/^-Od(,(7)to(A-d) = to(/ac-0do(-^)d[,(-l)d(,(-l)do(-7)to(/-r->tf) = to(/ac-0do(--)do(-l)do(-7)to(/-r-'j) c = to(/ac-Odo( —^)to(/T-i<5) ^0(yiac~' )^0(yi57c~^)do( ~) T ^o(/c~2(ac+iy7))do( ~) 2 £ (OC^)( °C+<57 = r c 7 0 7 It is clear from Theorem 2.2.1 that r is not a homomorphism. However, the fact that it is a homomorphism up to a product of factors 7(/) will prove useful later. 26 2.3 The Theta Model We want to define a space of theta functions on G x G*. This space of theta functions will be isomorphic to L^{G). Using this isomorphism, we will obtain actions of subgroups of A(G) and Bo(G} on the space of theta functions. In other words, we will have a theta model for these actions. Let r be a closed subgroup of G and let F. = {x' € G*|(^, x*) = 1 for all ^ € T}. .^'ote that r» is a closed subgroup of G'. We will identify F. with (^)* and F with Later in this paper we will be particularly interested in the case where G is of the form X = Xk 0 A^. Here Xk is a finite dimensional vector space over a global field k and Ak is the adele ring over k. For X = Xk (gi Afc we will consider two different choices of F. If F = Xk, then F and F, are discrete with compact quotient [Wei64, p. 164]. .A.S before, ky will denote the completion of k with respect to the valuation, v. Define X^, = Xk 0jfc ky and choose a basis V of Xk over k. We then define X° to be the group of points of X^ such that all the coordinates, with respect to the basis V, are integers of ky. Set F = X°, where X° = flv^v- The second choice of F we will consider is a slight modification of this one, in a way that will be explained in detail later. In this case F and F, are compact with discrete quotient. For either choice of F, there exist isomorphisms from G to G' taking F to F.. We now return to the general case. Let dx be a Haar measure on G and let dx' be its dual. Choose Haar measures d^ on F and dx on ^ such that for all € 0{G), J^^{x)dx = m [Wei64, p. 164]. We denote the dual to dx by dx'. 27 Let $ be a function on G. We will assume that $ is continuous with compact support. For all x € G let be the function on F defined for ^ e F by ^i(^) = $(x + ^). We define a function 0 on G x G* by 0(x, X') = ^ ^(x + 0 • As $ is continuous with compact support, the above integral converges and 9 is continuous. For all ^ € F, 6 F, we have 0(x +^,x' +r) = e{x,x') • (e, -X*). If we set 2 = (x, X*) and C = then the above relation can be written as 6{z +Q)=e{z)F{C,z)-' (2.10) for all 2 G G X G*, C £ T x T.) and F as in (2.1). In particular, for all € F., 9 is invariant under the map x* x* +^*. Moreover, \6\ is invariant under z ^ z + Q for all G F X F.. Now define Q = ] ~ Y ^ ^ define the norm of 9 by II e ll^= f m^'dz (2.11) JQ where dz = dxdx'. We define H{G,T) to be the Hilbert space of all solutions, 9, to (2.10) with the norm given by (2.11). The map ^ ^ 9 was defined by (2.9) for functions which are continuous with compact support, a dense subset of L^{G). This map extends by continuity to a linear map Z : L2(G) H{G, F) which preserves the norm. In particular, the map Z is an isomorphism. The map Z~^ : H{G, F) —> Li^iG) is given by <3^(x) = f 9{x,x')dx'. JG'/T. For more details the reader should see [Wei64, p. 166]. We will use Z to identify H{G, F) with L^(G). To simplify notation, we will write U{u}) instead of Z • U(uj) • 28 Proposition 2.3.1. Using the notation above with u = {u,u') and z — {x,x') in G X G', the action ofU{uj) on H{G,T) is given by U{ui)9{z) = d{z +u;)F{z,u}). Proof: Let ^ G L^{G) and suppose 6 = Z{^). It will suffice to show that Z{U{uj) • $)(2:) = d(z +uj)F{z,uj). By definition we have {U{u) • ^)(x) = ^(x -h u) • (x,u'). Now = J ^{x + It + 0 • (?> a;*) • (^, u') • (x, u')d^ = y ^{x + u+ ^) • {^,x'+ u')d^ • {x,u') = 6(z +u))F{z,uj). A Recall that A(G) denotes the set of all operators t • U{cj) for i G T, and Bo(G) is the normalizer of A{G) in Aut{H{G, F)). Let Bq{G, F) be the subgroup of Bo{G) formed by elements s = (a, /) G Bq{G) such that / takes the value 1 on all elements of r X F. and a induces an automorphism on F x F.. If cr = ^ ^ ^ all s = {a, f) G Bq{G, F) we define the operator rr(s) on H{G, F) by Tr{s)9{z) = 0{za)f{z) = J ^(xa + x'j + ^) • {^,x/3+ x'S)d^ f„{x,x'), (2.12) where we use to emphasize that / is the second degree character of G x G* associated to a. The a will be dropped from the notation in cases where it will not cause confusion. Proposition 2.3.2. [Wei64, p- 166] The map rp : Bo(G, F) BQCG) is a homo- morphism. 29 Define to be the set of elements s = (cr,/) € F) where cr is given in matrix form by ^ ^ ^ and 7 :(?*—> G is an isomorphism which induces an isomorphism from F, to F. In his paper, Weil shows that rr coincides with TQ on QO(G, F). He does so by showing that rr( 2.4 The Theta Function Associated to the Theta Model Let F be a closed subgroup of a locally compact group G and let $ € S{G). We want to define a theta function on Sq{G). This theta function will be closely related to Z($) and it will have properties related to the properties of the action of rp on Z( Let Bo( Theorem 2.4.1. [Wei64, P- 168] For all € ^(G) and all s e Bo(Cj, F), we have Proof: Let ' = s - Then set 0 = Z( Now given a closed subgroup F of a locally compact abelian group G and € S{G), we define a theta function on Bo(G) by fl(s) = J(s • = Z(s • $)(0,0) for all s € Bo(G). By Theorem 2.4.1, 9 is invariant under left translations by elements of Bo(G, F), i.e., if s G Bo(G) and s' € Bo(G, F), then 9{s' • s) = 0(s). 30 Chapter 3 THE THEORY IN THE LINEAR CASE 3.1 A Representation of the Heisenberg Group and Its Nor- malizer Let X be a finite dimensional vector space over a field k with char{k) ^ 2. Let X denote the linear dual of X. For x € X and x G X we will denote by [x, x] the value on X of the linear form x on A". Let Y be another finite dimensional vector space over k. Ever\' bilinear form on X xY can be written in the form [x, ya] where a is a morphism from Y to X. Now consider the case where X = Y. We say that a is symmetric if [x, ya] is symmetric in x and y, or in other words if a = a*. We recall the definition of a quadratic form. Definition 3.1.1. Suppose X is a vector space over a field k. A function Q : X k is called a quadratic form on X if: (i) Q{ax) = a^Q{x) for a €. k and x E X, and (ii) the function (x, y) t->. Q(x + y) — Q{x) — Q{y) is a bilinear form. Remark: Condition (z) above is an extra condition imposed in the linear case. It is this condition that makes the theory in the linear case different from the general theory. Suppose / is a quadratic form on X. Then for all x, y G X, /(x + y) - /(x) - /(y) = [x, yg] for some symmetric morphism g : X —^X. We will call / non-degenerate if g is an isomorphism. 31 We want to consider automorphisms a of X x X. Using the notation of sec tion 2.1, we may write these automorphisms in matrix form ascr=^" ^^.As before, we set I - Consider the bilinear form on (X x X) x (X x X) \ -1 a J given by B(zi, Z2) = [xi, X2] for zi = (xi, xi) and Z2 = {x2, ^2)- An automorphism a of X X X will be called symplectic if it leaves the bilinejir form B{zi,z2) — B{z2, zi) invariant, or equivalently if = 1, where 1 denotes the identity automorphism. The set of such automorphisms is in fact a group, which we will denote by 5p(X). Let IA{X) denote the group which is equal toXxXxfcasa set and has a group law given by (^1) ^i) • (22, <2) = (21 + z2, b{zi, z2) + ti + ij) where Zi, Z2 ^ X x X and ti,t2 € k. If cr is an automorphism of X x X and / is a quadratic form on X x X, then for z € X x X and t & k the map {z,t) ^ {za,f{z)+t) (3.1) defines an automorphism of U{X) if and only if cr and / satisfy the relation f{zi + 22) - f{zi) -f{z2) = b{zia, z2(j) -b{zi, z2). We call the set of isomorphisms of U{X) defined by equation 3.1 the pseudosym- plectic group of X, denoted Ps{X). The group law on Ps{X) is given by where f"{z) = f{z)+f'{za) for all 2 6 X x X. As we have assumed that char{k) ^ 2, the map (/, cr) 1-^ cr from Ps{X) to Sp{X) is an isomorphism [Wei64, p. 181]. Now we consider the case where fc is a local field. Fix a nontrivial character X of the additive group of k. Note that if / is a quadratic form on X x X, then X o / is a second degree character on X x X. We will identify X with X* via 32 x([2:,x]) = (x,x*). Let the groups A{X), the Heisenberg group, and Bo{X), the normalizer of A(X), be defined as in section 2.1. There is a homomorphism (a;, t) (u;,x(i)) from U(X) to A{X). Define a homomorphism /i : Ps{X) Bq{X) via = (x ° /i*^) note that this homomorphism is injective. Let A(X), T, and BoC-V) be defined as they were in section 2.1 for a general locally compact abelian group, and as before let TTQ : Bo (A") —>• BQ{X) denote the canonical projection. Again following the definition in section 2.1, we define the metaplectic group of X to be the pullback of Bo(X) by /i, i.e. Mp{X) = {5 = (s,s) e Ps{X) X Bo(Ar) : ix{s) = 7ro(s)}. We have the following commutative diagram: 0 > T Mp{X) > Ps{X) >• 0 0 > 'x > ) * ^o[.X) > 0. .A.S char{k) ^ 2, we have Sp(X) = Ps(X) and we could have used Sp{X) instead of Ps(X) in the definition above. We note that the action of Mp{X) on L'^(X) is defined exactly as it was in section 2.1 and that this action preserves the set of Schwartz functions, 5(^) [Wei64, p. 186]. 3.2 Sections Over Subsets and Subgroups of Ps{X) Let TT denote the projection map from Mp{X) to Ps{X). We want to define a section of this map over a subset of Ps{X). In analogy with section 2.2 we define G Ps{X) such that 7 is invertible Using the results in section 2.2, we can define a map tqo fj, : Cl{X) —> BQCA"). This map can then be extended to a map from the set of s G Ps{X) such that 7 = 0 or 7 is invertible to BqCA"). We can make this map into a map to Mp{X) via 5 (5, r(/i(s))). Note that in the case where X = k, i.e. X is a one-dimensional 33 vector space over k, this construction defines a section r : Ps{X) —> Mp{X) over all of PsiX). We now consider the adelic case. Let A: be a global field eind let A denote the adele ring over k. Let Xk, ky, Xy, and X° be as defined in section 2.3 and set Xa = Xk ® A. Interpreting the definition from section 2.2, we see that 5(XA), the space of Schwartz functions on A, is given by the following. Let $ € 5(XA). Then is a linear combination of functions such that for x = (x„) € XA we have n where 6 5(X„) for all places v and is the characteristic V function of X° for almost all v. The local case above gives us for each place v a map, which we will denote by r„, from elements s G Ps{Xy) with either 7 = 0 or 7 invertible to Mp{Xy). If we have an element s € PS{XA,) such that <7„ is the identity for all v outside 5, a finite set of places of k, and for all u € 5 we have either 7„ = 0 or 7„ is an isomorphism, then we can define r(s) Note that this is really onh' a finite product as r„(s) is the identity operator for all v ^ S. 3.3 The Theta Model Suppose k is a. local field with a discrete valuation. Let L be a lattice in A'. Set L. = (x* € X' : {x.x') = 1 for all x G L}. In the notation of section 2.3, set G = X, G' = X', r = L, auid F, = L.. Define Bq{X) and Bo{X, L) as in chapter 2. We then define Ps{X, L) to be the set of all s € Ps{X) such that x ° / = 1 on L X L. and a induces an automorphism on L x L,. Note that /j. : Ps{X) Bo{X) takes Ps{X,L) to BQ{X,L). Using the construction outlined in section 2.3, we obtain a representation r^, : BQ{X, L) -r Aut{H{X, L)) and a closely associated representation n • 34 Ps{X,L) —> Aut{H{X,L)). Using the isomorphism Z defined in section 2.3, we can think of both of these representations as maps to Aut{L'^{X)). We may then define a homomorphism from Ps{X,L) to Mp{X) via s (s, r£,(/x(5))). Now consider the adelic case. We have Xa = A"*® A. Set F = X^ and note that in this case we fix a character x defined on the additive group of ATA, trivial on Xk- Let if, a) e Ps{Xf,). The map ^A : Ps(Xf,) -> BQCAA) is given by HKiif,cr)) = {x° /•.(^) 3-nd this map takes Ps{Xk) to BQ{Xj^,Xk) [Wei64, p. 188]. In analogy with the case above, we can define a representation r^o/i: Ps(Xk) —> Aut{L^{Xx)). We will abuse notation and say that for s E Ps(Xk), rk(s) acts on L^(Xx). Given $ E SCXA), we define d{x,x') = ^ $(x-l-0(^,x*) for all X E X^. and x' E X}^. For s E Ps{Xk), the action of Tk{s) on 6 is given by rk(s)0(z) = d(za)x(f(z)) where 2 = {x.x'). The above action is equivalent to the action of Tk{s) on given by where ip is the second degree character of XA X X^'y defined by ip{x,x'j) = x{f{x,x') — [0,X*]) for x E Xx and x' E For the derivation of this equiv alence, the reader should see [Wei64, pp.190-193]. 3.4 A Theta Function on Weil's Metaplectic Group In analogy with the general case of a locally compact abelian group, given E S(X\), we want to define a theta function on Mp{Xx) that is closely related to the 35 theta model discussed above. For S = (s, s) e Mp(X\) and $ € we will write in place of s$. Proposition 3.4.1. [Wei64, P- i89] For all $ G S(Xa) the map S i-> 5$ is a continuous map from Mp{X/^) to S{X/^) Given € S{Xf^) we define a theta function on Mp{Xs) by the formula 9(S) = Y. Note that 9{S) is just 0(0,0) in the theta model above for 0 = Z{S • ^). Theorem 3.4.2. [Wei64, P- 193] Let Xk be a finite dimensional vector space over k. and let $ € 5(A'a). Let 9 be the function on Mp{X/^) defined for all S G Mp{Xf^) by the formula 9(S) = Y, ceA't Then 6 is a continuous function on Mp{Xk) invariant under left translations by elements of Mp{Xfi) of the form Tk{s), with s G Ps{Xk)- This theorem follows from the general case of a locally compact abelian group, which was proved in section 2.4. 3.5 How to Compute 7(/) Having fixed a character x of the additive group of k, to each quadratic form / on X there is an associated second degree character on X given by x° f- We will abuse notation slightly and write / for this second degree character. We quote the following theorem: 36 Theorem 3.5.1. [Wei64, p- 169] Let f be a second degree character of X, taking the value 1 on all elements of a closed subgroup F. Let X' be the dual of X, F, the subgroup of X' corresponding to F, and suppose that the symmetric morphism, Q : X ^ X', associated to f is an isomorphism which induces an isomorphism from F to T,. Then 7(/) = 1. As in the discussion above, the most important application of this theorem will be in the case where x is of the form xk ® A. For the remainder of this section we will assume AT is a finite dimensional vector space over a local field k. Given a symmetric morphism, g : X X', associated to a second degree character of X, fg, we want to find a formula for the scalar factor y{fg). Choose L C. X such that fg = \ on L. Let L, be the subset of X' given by the set of ail x' e X' such that (^, x*) = 1 for all ^ e L. We have the following diagram: X —^ X' U U L )> L.. Let Lg denote the image of L in L, under g. We define L' = g~^{L.), the image of L, under . Note that L C L'. For all lattices M C X define g{fg, M) = j f{x)dx. J M If M contains L, then we have 9{f0,M)= ^ f{x) f{y,-xg)dy. x^m/l Now let Af = M n L' and let m{L) denote the measure of L. Then we have fix). xemil Note that g{fg, M) is independent of M if M contains L' and M is sufficiently large. From Theorem 2.1.2, we have that 7(/) and the integral oi x° f over L' differ by 37 a positive real number. We know that 7(/) has absolute value 1. Thus, we have 38 Chapter 4 SPECIALIZATION OF WEIL'S THETA FUNCTION 4.1 A Special Case WeiFs theory defines a theta function in a very general situation. We want to specialize Weil's theta function to a more specific case and deduce a transformation law for this specialized theta function. Let k he a. one variable algebraic function field of characteristic p ^ 2. In other words, A: is a finite extension of Fp(T). We let Xk = k and for all places v we set Xv = k.^. As before A will denote the adele ring over k and we have A'A = Xk Theorem 4.1.1. [Wei95, p. 291] If k is a one dimensional algebraic function field over a finite field, then there is a divisor class of k whose square is the canonical class of k. Proposition 4.1.2. Let k be the function field of a curve over a finite field F. Then every k-rational divisor class is represented by a k-rational divisor. Proof: We will write D for the group of divisors of the curve defined over F, P for the subgroup of principal divisors, and CI = ^ for the group of divisor classes. We have the following exact sequence. o^p^d-^cl^o 39 Let G = GaZ(F/F) = GaZ(ifc/fc). Then the set of A:-rational divisors is given by H^{G,D) and the set of A;-rational classes is given by h°{g,Cl). Thus the state ment of the proposition is equivalent to the claim that the map h°{g, d) —> H°{G,Cl) is surjective. This claim will be true if h^(g,p) = 0. Consider the exact sequence 0 —> F'' —)• P 0. This short exact sequence gives the following exact sequence in cohomology. h^g,k'') h\g,p) -> h^{g,¥'') We know that H^{G, k^) = 0 from Hilbert's Theorem 90. We note that F*) is the Brauer group of F*^. The result then follows from the fact that the Brauer group of a finite field is trivial. [Ser79, p. 162]. A Theorem 4.1.3. [Sha86, p. 158] For any divisor D on a nonsingular curve C, and any finite number of points xi,..., Xm € C, there exists a divisor D' in the class of D such that x, is not in the support of D' for 2 = 1,.... m. Let 5 be a non-empty, finite set of places of k. Using the theorems above we may choose an idele / such that the divisor of / is not supported on 5 and the divisor of is in the canonical class. We may then choose a differential cj on k such that the divisor of u is equal to the divisor of f^. Let d be a differential idele, i.e., d is an idele such that div(d) = div{u) = div(f^). We will use F to denote the field of constants of k. At each place i; of fc let F„ be the residue field. Definition 4.1.4. Let Zy G and choose a uniformiser 7r„. Then we may write UJ = E,vdnv for € ky. We may write ^ and ai E F„ for all i. We define the residue of ZyUj at the place v by Resv{zuj) = Trf^/f{a^i) where Tr^^,/^ denotes the trace from F„ down to F. 40 Note that the residue at v is is not an element of the residue field at -u, but rather an element of F. Theorem 4.1.5 (Residue Theorem). [Ser88, p.15] If ^ e k, then y^ReSyj^cj) = 0. V Define an additive character ipy : T hy Note that for all places v, ipv is continuous. We define an additive character ip : A T for all z = (zy) € A by •ip{z) = Wv'^vi.Zy). We see immediately that ^ is trivial on Ylv Moreover, we have the following result. Proposition 4.1.6. xb is trivial on k. Proof: Suppose ^ E k. Then V _ J^g2wi(Trr/rj,(Kej„{{„a;)))/p V _ g23rt(5;„rrr/rp(«c5v(€uu;)))/p _ g27rirrr/rp(o) Residue Theorem) = 1. A Define a quadratic form, Q, on X^. by Q{x) = x^. The inner product on XA associated to Q is given by x.y = ^(Q{x + y) — Q{x) — Q{y)) = xy. Recall from section 3.1 that there exists a symmetric morphism g : X X such that for all 41 x,y e X, Qix + y) -Q{x) -Q{y) = [x, gy] = 2xy. We have fixed a nontrivial additive character ip of ATA which is trivial on k. Thus we may identify XA with XJ, via Tp{[x,x]) = {x,x'). Moreover, we may identify XA with XJ^ via the map X I-)- (t/ (x,y)) for all y € Xx- Given a symmetric morphism g : X X, this morphism may be represented as multiplication by an element of A, which we again call g. The morphism g is then associated to the second degree chciracter of X defined for x E X hy (x, 2~^gx) = ip{gx^). A similar relation is then induced on Xu for all places v. In order to define a theta function, we need to choose a Schwartz function. We recall the definition of a Schwartz function in this case. Defiinition 4.1.7. A Schwartz function on A is a linear combination of functions $ = such that (i) is a locally constant compactly supported function on k^; (M) is the characteristic function of the ring of integers of k-u for almost all places V. We now choose our local Schwartz functions £is follows. For all places v let = , the characteristic function on f~^Ov. We define $ on A by 9(5) = ^ 5$(f). ceXfc We need to know how to take the Fourier transform of our adelic Schwartz func tion. We will use the definition of the Fourier transform from section 2.1 for each of the local Schwartz functions and then apply the following theorem. 42 Theorem 4.1.8. [TatSO, p. 327] Let $ € S[Xk) be of the form Then V On each localization ky of k we need to choose an additive Haar measure. So that we may make this choice in an invariant way, we will choose the Haar measure, m„, such that = l/r|7^- Note that using these local Haar measures, we can define a measure m = T7^v on A. The choices we have made gives this measure the property that m(A/fc) = 1. Let e S{ky). Using the definition of the Fourier transform in section 2.1, we have the following formula. ^v{x') = / <^„(x)(x,i*)dm„(x). Jk„ In the computations to follow we will write dx instead of rfm„(x) for simplicity. Now we compute the Fourier transforms of the local Schwartz functions. Above we chose For x.x' € k-u we have J kv = j {x,x')dx 4.2 Invariance Properties of 6 Theorem 3.1 shows that our theta function is left invariant under global matrices. In other words for S € Afp(A) and g € SLaik) we have e{TM-s) = e{s). 43 We now consider some right invariance properties of i.e., invariance properties of If-io under operators coming from elements in certJiin subgroups and subsets of SL2{0). Lemma 4.2.1. Let T G {f~^0, and let $ = Ip- Then (Z($))(x,x-) = lr(x)lr(x'). Proof: (Z($))(x,x*) = 0(x,x') = J^lr{x +0-{^,x')-d^ Note that for ^ G F we have lr(3: + ^) = IrC^^)- Thus e{x,x') = J^lr{x)-{^,x)-d^ = lr(x)J{^,x')-d^ = lr(x)lr(x'). Lemma 4.2.2. Let O = C A and let K = ^ ^ G 312(0). Then ry-iol/tT)!/-!© = ni.(r/-io„(^v)l/-io„)- 44 Proof; Let 0 = Z(l/-icj). Then by equation 2.12 {Tf-io{K)d){x,x') =^(xa + j:*7,X/S + X* =\F-IO{XA + X*7)1/-IC?(X/9 + x*5)(x*7,x/3)(x, 2~^xa0) {2-'x'y6,x') = + X*5)(X*7, X/3>„ V (x, 2~^xa^)„(2~^x*75, x*)„ V A By Lemma 4.2.2, in order to prove that rf-io{K)lf-io = l/-ic7 ^ = Ky € 51-2(0), it will suffice to show for all places, v, that r= Proposition 4.2.3. Given Ky € SLi{Oy), we have Proof: In the notation of section 3.3, we have in this case G = ky and F = f'^Oy. We showed above that M,u) = Note that {(o 0 fo'-'Ssa, (J)} is agenerating set for SLziOv)- As r^-»o„ is a homomorphism, it will suffice to check for generators. Take ~ ^ Q 1 ^' Using equation (2.12), we have as ,3^^ 6 fy^^v and xjjy I on this set. Now consider ^ Q Again by equation (2.12), 45 A Proposition 4.2.4. Let ~ ^ g ^ ^ •5-^2(C?v)- Then Proof: ~ ^f;r^Ov(0- A Let Ky be as in Proposition 4.2.4 and let € SL^ik^) be an upper triangular matrix. It was shown in section 2.2 that r„ is a homomorphism on upper triangular matrices. If we combine this result with the above proposition, then Ty{ByKy)lj-lQ^ " Tv{Ky)l^-lQ^ — Ty (By)1 J-I . 4.3 Restriction of 6 Let S be a non-empty, finite set of places of k. Using the invariance properties of our theta function we see that 0 is a well-defined function on the double coset space Tk{SL2ik))\ Mp{A)/rf-io{SL2iO)). We defined Mp{A) as an extension of 51-2(A) by T. Let M € SL2{A). By strong approximation for SL2, there exists an element A e SL2{k) such that AM € SL-iiOy) for all v ^ S. Now let v E S. Using the local Iwasawa decomposition, also known as a BK-decomposition, we may write {AM)v — where Ky G SL2{Ov) and By is upper triangular. Thus, 6 is determined completely by its values on elements of livesof the form )) . We want to 46 restrict the domain of our theta function to the set of elements in Mp(A) of the form r(g) where g € flues SL^ik^). This restricted function is right invariant under Tf-i^^{SL2{Ov)) for all t; 6 5. The coset space given by SL2iKy)/SL2{Ov) will serve as an analogue of the upper half plane. From now on we will use the following notation. Given a Schwartz function We now have everything we need to specialize our theta function. Assume we have chosen a differential w on fc as in section 4.1. Set = tt"" for all t; G 5 where TTt, is a uniformiser at v and ti,, is a nonnegative integer. Let y € flues • ^'ote that X can be extended to sin idele if we set = 1 for all u ^ 5 and y can be extended to an adele if we put y„ = 0 for all v ^ S. We want to evaluate our theta function on elements of nt,es*"f('^'^2(A:u)) of the / jr yx~^ \ form r ( Q 1 - Recall that 9 is completely determined by its values on these elements. 47 '.-•)) - g('(: ?•) (nv,-..)) (o - g('(; cefc \ t) / €eitn(n„«5/v~'o„) V"e5 / ^eL where L = {^ e fc n (llugs/tT^^v) • ^(?) ^ ^ ^ '^}- 4.4 Transformation Law We will use the following setup throughout this section. Let S, x, and j/ be as above and let B = ( ^ ^ € SLsCA). Let A = ( M G SLaCfc n fj C>„). ^ ^ ^ vgS If (cy + d)v = 0 for any place i; € 5, then adjust y„ by an element of x^C?„ to force (cy + d)y ^ 0. To see that such a modification of y will not change r(B)lf-ic>, we consider the following. Suppose (cy + (i),^ = 0. Choose t]^ 6 Ou; and then form T] e O hy setting rjy = 0 for all v ^ w. Set A' = ^ J ^. By Proposition 4.2.4 and our discussion of invariance at the end of section 4.2 we have r 48 Definition 4.4.1. Let K = ^ ^ ^ ^ SL2{0) be given as follows. For v ^ S, /I 0 \ Kv = A~^. For V e S we let Ky — { ^ . Note that c = 0 implies 7„ = 0 \ cy+d ^ J for all V and c ^ 0 implies 7„ nonzero for all v. By Lemma 4.2.2 and Proposition 4.2.3, rk{A)t{B)tf-io{K)lf-io = tk{A)r{B)lf-io- Combining this observation with the left invariance result from Weil's paper given in Theorem 3.4.2 we have eiT,{A)T{B)Tf-^om)=0{r{B)). Note that ABK = AyB^K^ as elements of 5L2(A). Since Mp(A) is by defi- U6S nition an extension ofSLjCA) by T, Tk{A)r(B)Tf-io{K)lf-io and T{ABK)lf-io = {AyBvKv)l must differ by a complex number of absolute value 1. Finding this complex number of absolute value 1 will give us a transformation formula for 0. 1/2 Lemma 4.4.2. t(ABK)1 f-io{0) = n^es cy+d Proof: Note that ABK is the identity matrix for z; ^ S and for v E S we have / z cy+d (ay+6\ \ A^ByK^ = ( ^ )• ^^ote further that this matrix always makes sense because we have chosen y 6 A such that cy + d ^ 0 for any v G S. We compute iT{ABK)lf-:om 1/2 / \ / , I, •nv..» n|^L / ^ r\ i (" 1/2 If we take if = 0, then we get cy+d A 49 Lemma 4.4.3. Tk{A)T{B)Tf-^o{K)lf-.om t rl/2 lu6S I cy-rd iTi"^ (n.« (E„,^ ^ (-fn^)) He * 0. Proof: We first consider the case where c = 0. We showed at the be^nning of this section that Tf-io{K)lf-io = l/-'o- Thus, we may simply compute rk{A)T{B)\f-io- We have Now we apply rk{A) using (3.2) and obtain (rk(A)r(B)lf-io)(^) = 0), where /a denotes the character corresponding to the symplectic automorphism A. Now take ^ = 0 and the above expression becomes Since c = 0 we have 1/2 d e for u ^ S which implies nve5 Thus = nv65 and cy+d V Tk{A)T{B)T f-io{K)\f-io = T{ABK)lf-io- Now consider the case where c ^ 0. (rfc(/l)r(B)r^-.o(/^)l/-io)(0 = (rfc(A)r(B)l/-io)(0 = /" l/-io(a:a^ + xz)i/)(y(a^ + 2)^)/^(?,c~^2)c/£ JcA again by (3.2). As c # 0, cA = A. Take ^ = 0 and we have f lf-io{xz)rp{yz^)fAiO,c ^z)dz. J A 50 We make the change of variable 2 r ^z. We have d{x ^z) = \x ^\dz which gives f lf-io{z)ip{yx~^z^)fA(0,c~^x~^z)dz J A = lf-io{z)ip{yx~'^z^)ij{dc~^x~^z^)dz = \x\~^''^ f Tp{{y + -)x~'^z^)dz Jf-iO c = 1x1-1/2 Jf-i-IQ CX' = 1x1-1/2 [ 'ilj{——z^)dz Jf-^o T = 1x1-1/2 y] f ip(--iz + nf)dz .ric J-yf-'o 7 = 1x1-1/2 t{}{-—n^) f ilj{--{2zn))ijj{--z^)dz .^10 T 7 7 Thus riCAjrCBjrj-ioCA-jly-.oCO) = |xr""|7l l!'(--n'). r /~'o Recall from Definition 4.4.1 that for all t; ^ 5 we have 7„ = c. Thus 1/2^ lx| 1/2|7| = |7(V2 Yl cy + d n iTii'' VUGS Kvis = ("N cy + d Vues because n fn l7iy^) = = 1 (Note that ce ves / Vugs / 51 We conclude that TM)t(B)T,-,a(K)\,-,o{0) = |7|"' Theorem 4.4.4 (Transformation Law). where ui \ 1 t/ c = 0 Proof: Recall the definitions of .4, B, and K from the beginning of this section. By Lemmas 4.4.2 and 4.4.3, Tk{A)T{B)Tf-io{K)lf-io = w • T{ABK)lf-io- We have observed that 6{Tk{A)T{B)Tj-io{K)) = 0(r(B)). Thus d{T{B)) = u - 9{T{ABK)), which is exactly what we wanted to prove. A Consider an analogue of the upper half plane given by the set of cosets n = Y[SL2{k,)/SL2{0.). v£S Recall that given an element g G flues *^^2(^1;) i we may do a B/C-decomposition at each place to obtain g = flues w^here € SLiik-u) is of the form Q ^ and e SL2{0„). Let B = Fives and let K = Ilues^v- In analogy with the classical case, we multiply by a factor to convert 6, the "invariant" form of our theta function, to a theta function, 0, on the "upper half plane". The function © is defined by e(s) = li|-"^«(r(B)r/-.o(A-)) 52 for all g 6 IIUGS SL-z{ky). As 9 is right invariant under tj-io and r and r/-!© agree on upper triangular elements of Sl^^Oy), we have that © is a well-defined function of K. We use our Transformation Law to obtain a transformation law for this new theta function as follows. Let A, B, eind A" be as before. Recall that ABK = I f av+6\ \ cy+d X \cy+d) | fQc all u G 5 and is the identity matrix for all v ^ S. Then (0 ^ J e(B) = |l|-"^9{r(B)) = • u • 6{T{ABK)) ( by Theorem 4.4.4) = (N l Remark: To obtain what is classically called the "inversion formula", apply the transformation law with A = 53 Chapter 5 MERRILL AND WALLING'S THETA FUNCTION 5.1 Definition and Transformation Laws In this section we want to define the theta function used by Merrill and Walling in their 1993 paper [MW93] and give the transformation laws they proved for their theta function. Merrill and Walling's theta function, which we will denote O^w, is defined in the following special case. Choose k = Fp{T), ui = —and S = {oo}. Then choose X so that = 1 for all v ^ oo. Merrill and Walling define their theta function on H = PSL2{koo)/PSL2{Ooo) where PSL denotes the projective special linear group, i.e., the special linear group modulo its center. Thus, by definition, Merrill and Walling's theta function is right invariant under PSL^iOoo)- As the subset of upper triangular matrices in PSLiik^o) contains a set of coset representatives for H, Merrill and Walling's theta function is completely determined by its values on upper triangular matrices. Definition 5.1.1. Let B Then OMW{B) = ^ lc»ocMi/'oo(y^^)- teTFp[r] Merrill and Walling prove the two transformation laws below thereby showing that transforms as a half weight modular form for SL^i^k O Theorem 5.1.2 (Inversion Formula). [MW93, p. 668] Let x = T~'^ and let y € fcoo be of degree n such that v^{x'^) > Voo{y)- Let denote the coefficient ofT^ in 54 the oo-adic expansion of y. Then 9mw{Z) = -^OMW ^"2) f X -^ \ { i/y = 0 .herez^ (O " - J' (^)"/y/(^ Hv^O ' given by [ )fory^O and ( ° ) /ory = 0. 55 Theorem 5.1.3 (Transformation Formula). [MW93, p. 669] Let ( ^ d ) ^ ^^ n d # 0. Then we have ...re. = (J ^;>). = (: ^)^. ^' = ( ! 2)^' = \d\z.' u6rFp[T]M7Tp[r]) 5.2 Comparing with © We would like to show that O^w and 0 are equal as functions. We showed in section 4.4 that 0, and therefore ©, is determined by its values on r{B) where B is an adelic matrix which is the identity at all places outside s and upper triangular for all places in s. We want to specialize to the case in which d^w is defined. Recall that 0 and di4w are both well-defined functions on H. Thus, to show that they are equal ELS functions, it will suffice to show that they agree on matrices of the form ^ = ( q ^ *^^2(A:oo)- We have ©(B) = \X\-J'^9{T{B)) = 10ao(a;C)^(y€^) = XT io=cM^oo(y?^) i67Tp(r] = 9i4W{B). We conclude that 0 specialized to this case is in fact O^w- Since these two functions are equal on H, they must obey the same transformation laws. 56 5.3 A Direct Deduction of Merrill and Walling's Inversion Formula Again we will restrict ourselves to the case in which is defined, outlined at the beginning of section 5.1. From the results of the previous section, we already know that 0 must satisfy Theorems 5.1.2 and 5.1.3, and that Omw must satisfy the transformation law for 0 given at the end of section 4.4. However, it still seems a worthwhile exercise to deduce Theorem 5.1.2 directly from the transformation law for 0. We need the following lemma. Lemma 5.3.1. Consider k = F where the negative real axis is chosen as the branch cut for the square root function. Proof: This lemma can be found in [MW93, p. 668] in the case r = 1. The more general result is obtained from the r = 1 case in [MW99, pp. 307-308]. A Recall that our transformation law says 0(B) = ley + ^ where .4, B, K, and ui are as defined as in section 4.4. To obtain the Inversion Formula, we choose A = ( -1 0 ) • '" case we have 2 11/2 U) = ^ IOC neOoo/(^)o«/ 2\ 57 Consider the case where y = 0. Note that in this case we may choose y = in our transformation formula. We have a; = 1 and \cy + = \y\]J^ = Now suppose J/ # 0. Then we have \cy + It remains to show that Ifil" ieOoo/{f)OooE However, this result follows immediately from Lemma 5.3.1. 5.3.1 A Comparison of Transformation Laws Now that we have shown directly that our "inversion formula" for 0 coincides with Theorem 5.1.2 for 0mw, we want to compare our full transformation law for 0 with Theorem 5.1.3 for Omw- Since we have shown that 0 and 0mw are equal as functions, we already know that these two transformation laws must be the same. However, we would like to derive this result directly. We will need the following lemma. Lemma 5.3.2. Let c, d TTFp[T]. Then ueT7p[T]/d(,T?p[T]) \[(W}i \ y / ueTFp[r]/c(TFp[r])^ ^ ^ where (d) ~ —m and v is the coefficient ofT^ in the oc-adic expansion of —2- Proof: Note that ^ u6TFp(r]MTFp[T]) 58 By our "inversion formula" for 0, which we know agrees with Theorem 5.1.2 for ^Afvvi we have and the result follows immediately. A In order to derive Merrill and Walling's transformation law directly from the transformation law we gave for 0 in the last chapter, we need to show that + = (5.1) y/z' Note that the left hand side comes from our transformation law and the right hand side come from Merrill and Walling's transformation law. There are four cases to consider. Case 1: Suppose c = 0 and y = 0. Note that c = 0 implies d € for all v which in turn implies that [rfjoo = 1- Since c = 0. we have a; = 1 and the left hand side of (5.1) is \d\]J^ = 1. In order to compute the right hand side we note that ^ ~ ~ ^ ~ I^u€7Tp[r]/(/(TFp[ri) ~ Case 2: Suppose c = 0 and y # 0. We have |d|oo = 1 by the same argument as in case 1. .Again oj = 1 and the left hand side of (5.1) is equal to 1. Now consider the right hand side. We have ^(d) = 1 just as in case 1. Finallv, = |rfU = 1- Case 3: Suppose c ^ 0 and y = 0. We now have c ^ 0 so the left hand side is \ ue/-^o/7f-io We have CX^2 -1/2 7,-1/2 \v^oo / 59 Thus the left haxid side is \uef-'o/yf-'o / Now we may split up the sum to obtain -1 \d\oo\x\:,'( ^ yW€Ooo/7ooOoo / ( E \"6n.#oo/»~'c«/'rv/»-'o„ Woo We can evaluate the sum over OoohooOoo- -1 -1 (J?"') ^uGC?OO/7ooC?OO y ,ueOoo/^Ooo -I ( cx' -1 1/2 uriM where u is the coefficient of T in the oo-adic expansion of ^ and Uoo (^) = —m + 2v^{x). Note that m is even if and only if Voo (^) is even. Now note that T¥p[r]/c{T¥p[T]) = Ilu^iooWe may therefore write the left hand side of (5.1) as -1 \cU' (l)' E N'-(R-)) ,ueTFp[r]/c(TFp[r]) Vv^too f)' 60 For the right hand side we have = jdloo- Thus equation (5.1) is equivalent to E VugTFp[T]/c(rFp[T]) (5.2) uG7Tp[T]/ If we apply Lemma 5.3.2 to equation (5.2), then we have \ 1 VueTFp[T]/c(rFp(r]) ueTFp(T]/c(TFp(r]) (5.3) where i.d) ~ —m' and i/ is the coefficient of T^' in the cx5-adic expansion of -f. Note that = (p) {f) ^{"f) " equation (5.3) becomes |C|oc = T. ^ tieTFp[T]/c(rFp[T]) (rO • which is a true statement since d,,2 G k for all u and ip is trivial on k. Case 4: Suppose c^Q and y # 0. The left hand side of equation (5.1) is \ ue/-io/7/-'c» ^ ^ j We have ... .^2 -1/2 71-"^ = n = \cy + d\]J^^\x cy + d \t)^00 OO 61 Thus the left hand side of (5.1) is given by / -I \cy + dloo|2:|00 \"€OOC/^0=C -1 II ^oc ,ueTFp[r]/c7Tp[T] -1 cy + d 1/2 = |cy+ (i|oo|x| -1 OO cx-' (p) y( P ) ) H ^oo ^u6TFp[T]/crFp[r] (-H -1 |cy+ (2^ ,u€rFp[T]/crFp[r] where = —e + 2uoo(a:) and 77 is the coefficient of T ) in the oo-adic expansion of Note that e is even if and only if is even. For the right hand side we have 11/2 d[cy+d) / _/ \ ** / \'' / / 1 \ 6—a = \d\]ff\cy +d\]i'('L) M^) \p J \p J y \ p/ where Vodv) = —a, (d{c^+d)) ~ coefficient of T°- in the oo-adic expansion of y, and r}" is the coefficient of in the oo-adic expansion of 5(^^p5) • 62 Thus equation (5.1) is equivalent to = \IW^ E y ^ ^ u€TFp[T]/ A short computation reveals that (;)'(?)'(^)' Jill slifT' where v and m are as in Ccise 3. Thus, (5.4) is equivalent to (5.2) which we showed to be a valid equation in cjise 3. We conclude that the transformation law for 6\fw given by Merrill and Walling is equivalent to our transformation law for 0. 63 Chapter 6 REPRESENTATIONS OF ELEMENTS AS SUMS OF SQUARES 6.1 Fourier Series and Restricted Representation Numbers in the Genus One Case 6.1.1 General Formulas for Fourier Coefficients As before, k will be a one variable algebraic function field of characteristic p > 0. Let F be the field of constants of k. Recall from section 4.3 that our specialized theta function can be written as (6.1) We will use the notation 9{x, y) for this theta function. We want to fix x and consider this theta function as a function of y. Again we let S be a finite set of places of k. We fix x as follows. For all u ^ 5, let = I. For all u € 5, choose a nonnegative integer and then choose x„ such that v{x) = riv Note that there are only finitely many ^ € (A: n Y\vis fv^^v) such that i?(^) > —v{x) for all V G S. Thus, for fixed x, the sum in equation (6.1) is in fact a finite sum. Recall the following invariance properties of 6. 1 (a) Let ^ E k. Then d(x, y) is left invariant under rfc(A) for A = 0 d{x,y) = 0(x,y+ ^). (b) Let ^ G k. Then d{x,y) is left invariant under rfc(/l) for A = Thus 6{x,y) = 0(^x,^2y). 64 (c) Let u e . Then 0{x,y) is right invariant under for K ( 0 =e{ux,y). (d) Let u € O. Then 6{x,y) is right invariant under r/-ic»(A') for K = (iO- Thus d{x, y) = 0(i, y+ x^u). By property (a) 9{x, y) is periodic in y with periods in k. Since 9 is periodic in y, 6^' must also be periodic in y. Thus 9^ has a Fourier expansion given by 9^^(x,y) = C2/(0,x) + ^ C2i{^,x)xp{^y) ?e/t* [Weill, p. 19], Let F be the field of constants of k and set (F| = 9 = p'" where p is a prime. .\ote that F = k n O. Let ^ e F'^ and ^ G k. By property (b) we have ci{^,x) = Ci{^3^, /3~^x) where we use Ci to denote the Fourier coefficients of 9{x,y). By property (c), Ci(,f, x) depends only on the divisor of x, so in fact ci(^, x) = x). The Fourier coefficients of 9^'(x,y) of course have the same property. Note that C2i(^,x) = 0 unless ^ e A: fl flugs^ —2v(x) for all V e S. Thus for a fixed x, 9^'{x, y) has a finite Fourier expansion. We would Hke to derive expressions for the Fourier coefficients of 0^'(x, y). A slight rearrangement of the Fourier expansion of gives C2/(0, x) = 9^'{x, y) - (6.2) Recall the Haar measure, druy, defined in section 4.1 for all places v. We define a Haar measure on A via dmy. Let dy be the Haar measure induced by dm^ on k\ A. We now integrate both sides of equation (6.2) over k\A. Since the sum over k^ has only a finite number of nonzero terms, we may freely distribute the 65 integral over the sum. Doing so gives f C2i{0,x)dy= f 0^(x,y)dy-y2 f x)'0(^7/)dy. JK\X JK\A. •'Ar\A Moving the Fourier coefl&cients outside of the integrals, we obtain C2I{0,x) F dy= F 0^^{x,y)dy - C2I{^,x) F rp{^y)dy. J k\A. J k\K f 6fc* Note that / ip{^y)dy = 0 for all ^ e • Recall that 6 is right invariant under Jk\A Tf-io(K) for /i' = ^ Q 1 ) ^ SL^iP). Thus, 0'^'{x,y + x^tj) = 0^^{x,y) for all Tj E O. Let dy denote the measure on A: \ k-jx^O induced by dy. Then C2j(0,x) F dy= F E'^\x,y)dy. Jk\A/x'^0 Jk\A/x^O Since k\A/x'^0 is a finite set, we have \k\P^/x''0\cn{Q,x)= YL ^"(^'2/)- (6-3) yek\A/x^O Choose a place w G S. By strong approximation the map kuilxl^O^ —)• k\A/x^O is surjective. Given a divisor D we define C{D) by C{D) = {/ G fc : (/) + £>> 0}. Note that C{D) is an F-vector space and let 1{D) be its dimension. Theorem 6.1.1 (Riemann-Roch). Let k be an algebraic function field of genus g. Let K denote the canonical class. If D is a divisor, then 1{D) -1{K -D)= deg (£>) + !- g. Note that deg {K) = 2g — 2. Let j be the smallest integer greater than 1 ( \ 2g + 2nu • degv I. Choose £> = — 52--es 2n„ - v+j-w. Then deg(D) > deg to I v#u; \ Vjtw ) 66 2g which implies 1{K — D) — 0. By Riemann-Roch, 1{D) > ^ + 1 > 1- Thus if we choose an integer a such that a • degtw < —2g — 2n^ • degu, V^Wv€S then there exists ^ E k such that w{^) = a and ^ 6 xlO„ for aX\ v ^ w. We let TT^ be a uniformiser and conclude that for a • deg lu < — I 2^ — 1 + 2n^ • deg v ), \ ) the map 4> : t^Z^wI^IjOw k\ kjx^O is surjective. Thus equation (6.3) takes the form \k\h./x'^0\c2i{Q,x)=i ^ 9^'{x,y)\ . (6.4) \yen°0^/xl0u, / Let Fu, denote the residue field at w. Note that |Fu;| = Using this notation, we have \kerd>\ = ^ (6 5) ' |fc\A/x2C>| \k\A/xW\' ^ ' If we apply equation (6.5) to equation (6.4), then we arrive at the expression ^2/(0'^) = IF uX^)-a) \ 51 ^''(^,2/) ) • (6.6) ' "" \ye^Ou,/xlOu, J By a calculation almost identical to the one just completed, we have C2i{^.x) = ( X) e'''{x,y)ip{-e,y)\ . (6.7) \y€-rZO^/xlO^ J 6.1.2 Fourier Coefficients and Representation Numbers Now that we have expressions for the Fourier coefficients of 6^^(x, y), we would like to relate these Fourier coefficients to representation numbers. From the relation 67 between 0 and B defined in section 4.4, it is clear that, for a fixed x, 0^' has a finite Fourier expansion. If we consider the Fourier expansion of © we see that the Fourier coefficient of V'(Cy) is given by 1 if ^ = 0 2 if 0 ^^ € x~^f~'^0 and ^ is a square in k 0 otherwise We will use the notation C2ii^,x) for the Fourier coefficients of 0^'. Recall from the introduction the definition of the restricted representation num bers. We defined r2/(f, n) to be the number of representations of ^ as a sum of 21 squares, where the summands, are such that for all i we have e for all V ^ S and u(^,) > —n^ for all v E S. In general, C2i{^, x) gives the number of representations of ^ as sum of 21 squares, where for all i, v{^i) > —for u e S and € f~^0^ for v ^ S. In the case where k is of genus 1 we may choose a holomorphic differential, u, on k such that the divisor of / is the zero divisor. In this case we see that r2i{^,n) = C2i{^,x). We will restrict ourselves to the case where k is of genus 1. We now want to consider the relationship between Fourier coefficients of 9 and Fourier coefficients of 0. We have 0(x,y) = |x| ^^^d{x,y) (6.8) 0='(x,y) = \x\-'d^'{x,y) (6.9) C2/(^,X) = |X|~'C2<(^,X). (6.10) Thus, to evaluate the restricted representation numbers in the case where k is of genus 1, it will suffice to evaluate the Fourier coefficients C2i{^,x). It is worth noting that if m„ > n„ for all v, then r2/(^,m) > r2/(^, n). This relation is immediate from the definition of the restricted representation numbers. If we consider the special case where S = {u;} and let m = and n = n„. 68 then the corresponding relation on Fourier coefficients is given by C2/(^, > TT^). This relation is somewhat crude, but it will be useful in some cases later. 6.2 Summary of Results .A. number of results will be proved regarding the representation numbers in the subsequent sections of this chapter. Most of these results will require some tedious calculation. So that the reader may better keep track of the results and how they fit together, the results to be shown will be summarized in this section. We want to consider representation numbers in the following setting. Let k be a one-variable algebraic function field of genus 1 and characteristic p > 2. Let F be the field of constants of k and set |F| = q = p^. Since k is of genus 1, we may choose a holomorphic differential, tu, on k such that div(f) is the zero divisor. We choose a degree 1 place w oi k and set 5 = {u;}. We will use R to denote the set k n Y[v:^w^v- Let Fu, denote the residue field at w and note that |Fu;| = q since w is of degree 1. Fix a uniformiser, for w and let n be a nonnegative integer. Then let x e be given by = 1 for aXl v ^ w and Finally, since k is of genus 1 and S contains a single place, w, we may choose a = — 1, where a is as in section 6.1. Except where stated otherwise, we will also make the additional assumption that i-e., either —1 is a square in F or / is even. We will use the above notation and cissumptions for the remainder of this paper. By applying our version of the Inversion Formula to the expressions for C2/(0, x) and C2i{^,x) computed in section 6.1, we obtain recurrence relations for these Fourier coefficients. Note that the recurrence is on the valuation of x. The re 69 currence relations are given by C2z(0,0 + (g - 1) {q^~^ + (9 - 1)?'"^) - ~ l)c2i(l,Jri) + (9^ - 1) ^^C2l(l.oj +9^02/(1, + 9^02/(1,7r;j)) . and for ^ ^ F 52c2,(e+5.i) = 9»<'-''-^' /3eF 2n-l 53 (9 - l)9~°C2i(0,x7r-") (o=-ti;({) / C2/(;3,x7r-'')g-^"+^ l ^ 1/;^ ("^y~ ) • y€(Ou,/5r2,'—"Ow)" / Using these recurrence relations, we are able to obtain upper and lower bounds on certain sums of Fourier coefficients. Before obtaining these bounds, we note for comparison that the average value of C2/(^, tt^) over all ^ G il such that w(^) > —2n is Thus, the average value of a sum of q Fourier coefficients would be gn(/-2)-t-i 'pjjg bounds are as follows. Theorem 6.5.1. Suppose we are in the setting outlined at the beginning of this section. LetLn>5. Then \C2l (0,7r;i) + (g - l)c2i(l,0 - 70 Theorem 6.5.2. Suppose we are in the setting outlined at the beginning of this section. Let ^ € Hv^ ^ ~2, and suppose I > m + 5 and n > max{Tn + 5,9}. Then I + ^, i) - ,"('-^>+11 < 6,"0-2)+".+2-', 0e¥ These bounds show that the size of these sums of Fourier coefficients is on the order of the size of their average value. Our next goal is to obtain information regarding the size of the individual Fourier coefficients. We have the following proposition. Proposition 6.5.4. Again assume the setting outlined at the beginning of this section. Let ^ E . Then C2ii^,x) < C2/(0, or). Theorem 6.5.6. Assume the general setup given at the beginning of this section. Suppose n > o and I >9. Then C2i(0,x) < -t- .Vote that this theorem implies that all the Fourier coefficients are bounded above by Combining this observation with the bounds in the two theorems above, we obtain the following corollaries. Corollary 6.5.7. Assume the setup at the beginning of this section. Let m = —w{^), n > max{9, m + 5}, and I > max{9,m + 5}. Then • |c2/(0,x) - < 2g"('-2)+8-'. .|C2/(1, x) - ^(2g "('-2)+8-i ^ 2q<^-2)+A-l _ qnU-2)+3-l ^ 71 • For —2n < —m < —2, |C2i(^,x) - < 3g'»('-2)+m+2-/ _J_ 2^n(J-2)+9-/ _ 2qn{l-2)+a-l Corollary 6.5.8. Assume the setup at the beginning of this section. Let m = —w{C) o.nd n > max{9,m + 5}. Then as I —¥ oo - C2;(0,X) 1 C2I(KX) V 1 • ,n(l-2) • ^ 1 for -2n <-m< -2. Corollary 6.5.9. Assume the setup at the beginning of this section. Let m = —w{^) and I > max{9, m + 5}. Then as n —> oo .C2,(0,I)=0(7""-«). • C2i{^.x) — /or —2n < —m < —2. Note that these corollaries show that the Fourier coefficients are asymptotic to their average value as Z —> oo and asymptotic to a constant times their average value as n —> oc . Recall our assumption that k is not of characteristic 2 and consider an elliptic curve, E, over F with function field k. Choose a place w of degree 1. We may write the equation of E in the form = h{X) where h{X) is a cubic polynomial in X and such that w is the place at infinity. We will now assume that w is the place at infinity of E, but we no longer require the assumption that =1- We have the following proposition. Proposition 6.6.1. Assume the setting outlined at the beginning of this section. 72 but do not assume = 1- Suppose that w is the place at infinity. If w{x) < 3, then for all^ G R, C2i{^,x) is independent of the choice of elliptic curve = f{X). Since for n < 3 the Fourier coefficients do not depend on the choice of elliptic curve, we can compute exact formulas for C2i{0,x) and C2/(l,x) in these cases. Formulas for n = 0 and 1 £ire computed in a slightly more general setting in the section on recurrence relations. They are given by C2f(0,1) = + (q- j , C2/(l, 1) = , C2/(0,7ru,) = (~^) ' C2z(l,7r„) = • Formulas in the cases where n = 2 and 3 are C7,(0,ni) = +^ + (zi]" q \ p J q^ C2i(l,7r2) C2z(0, ) = q-^ + q^{q^ - 1) + (q^ - l)(q^'~^ + q - 1)J , and C2i(l. TT^,) = ^ + q'(q - 1) - + 9 - 1) j . 6.3 Exponential Sums In this section we will prove numerous results on exponential sums. These results will serve as tools to simplify recurrence relations and prove results about Fourier 73 coefficients. Proposition 6.3.1. [Lan70, pp. 85-87] Let a € F* for p an odd prime. Then The next proposition follows from the Hasse-Davenport relations. Proposition 6.3.2. [IR72, pp. 147-151] Let q = p^ for p an odd prime and let F, denote the finite field with q elements. Let a € F* and let N and Tr denote the norm and trace respectively from F, doxvn to Fp. Then (EE— Now we prove some results on character sums. The following lemma is central to the proofs of many of these results. Lemma 6.3.3. Let x = for some positive integer n and let ^ with w{0) > —2n. Let m be an integer such that m < 2n. Then 2^ - I Q ifw{/3) < -m • y67r-C>„/7r2"Ou, ^ Proof: Suppose w{P) > -m. Then for all y G /3y € which implies ^wiPy) = 1. Counting the number of terms in the sum, we have ^^n-m Now suppose w{P) = —b < —m. Let the tu-adic expansion of y be given by 74 Then i^wi0y) = ^ ^wi^^Vi-^C] Giu /f 2/* . = 51 ^u; [/3 ^ y.-Tri, J m = H V'u; y.^") (Pyb-iT^w ^) t^fc-i \ i^b—l / yj-iSl yi€F, = 0 because /?y6_i7r^ ^ goes through a complete set of residues as yb-i varies over Fu,. A Proposition 6.3.4. Let 0 E with w{0) > —2n and n > 1. Then 53 ^wi^y) w(y)<0 53 ^wi0y) - 53 ^'miPy) y^Tr'^O^/Trl^'^Ou, ye-^u,0,u/T^^"0^ f "" 1) if t^iP) > 1 -9,2n-l if — I < w{P) < 1 0 ifw{0) < -1. Proof: The first equality is clear. The evaluation of the sums follows immediately from Lemma 6.3.3. A Proposition 6.3.5. Let 0 E with w{P) > —2n + 2 and n > 1. Then r g2n ifw{P) > 1 53 M0y) = < —ifw{l3)= 0 —1. ve'ru,'Ouj/»2,"-^Ou.-xu/^w 0 ifw{/3) < w{y)=l Proof: The result again follows immediately from Lemma 6.3.3. 75 Proposition 6.3.6. Let ^ ^ ku., with w{^) > —2n. Then 23 M^vK) y6(0u,/7r5,"~'0„)* M^yK) - ^ M^vK) 2n—a/ y€5ru,Ott,/ir2>" "Ou, {q- 1)^2" " ^ > -a _^n-a-l if w{^) = —a — 1 { 0 othcTwise. Proof: The first equality is clear. To obtain the second equality, apply Lemma 6.3.3 to the right hand side of the first equality keeping in mind that for 2n — a = 1 we have = 1- /"^Mj Out Proposition 6.3.7. Let € k-u, with w(^) > —2n, w{0) > 2a — 2n, and 2n — a > \. Then (9 - 1)?^" " ^ if w(P) > a and w(^) > —a _^2n-a-l if w{^) > —a and w{P) = a — I or w{^) > a and w{^) = —a — 1 if w{P) < a — I and w(0) ^ w(^) + 2a or w{P) > a and w{^) < —a — 1. Proof: To obtain the first equality, apply the change of variable y i—> i. Now we evaluate the sum. If w{^) > —a, then y€(Ou,/ri?~°Ou,)'' ye(Ou,/5r2,""''Ou,)'' 76 Note that ye(C7u,/7r2,"-''0„)'' = ^ '^w {-PyK") - y€0w/x2,"~"0„ yeir„Ov,/T^~'^Ov (6.11) and apply Lemma 6.3.3 to the right hand side. Note that for 2n — a = 1 the lemma does not apply to the rightmost sum in (6.11). However, in this case we have E i'ui 1 result follows. y e JTu, 0„,/n-J,"""C>„ Suppose ty(0 < ^(/3) < a — 1, and 1^(0 + 2a w{0). Then w{—0Tr~'^) and ii;(—/37r~°) < —1. Without loss of generality, assume that w{—0-K~°-) < u;(—^TT^,) and set w{—0Tr~'^) = —b. Given y G (Cu,/7r2"-<'C»^)'', let y = ^ that both yo and j/q are nonzero. Note that (Z)i>6_i T/,'7r^) G O^. Thus ip^ does not de pend on y[ for z > 6—1. Since the y,' for z < 6— 1 depend only on the for z < 6— 1, we can isolate the sum Y. (6.12) j/6_l 6fa7 Since w{ — 3~~°-) = —b, —goesthrough acomplete set ofresidues £is y6_i runs through Fu,. We conclude that the sum in (6.12) is zero. Now suppose w{^) > a. Then y€(ou/ti""°ou,)'' ye{0^/i^l''-''0^Y .Apply Lemma 6.3.3 to the right hand side and the result follows. A Remark: The sum 53 ~ is called a Kloosterman ye(Ou,/ir5,"-''Ou,)'' sum. 77 Lemma 6.3.8. Letn he a positive integer. Suppose^ e with —2n < w(0 < -2. Then :/w;(0 = -2 51 H - I _ l\q2n-2 if w{^) < —2. ye(Ou,/T2,"-'Ou,)* ^ - ' V Proof: 51 51 -ey7r„) 53 M-^y-^w) 53 J /,2ii-lo ^x /?CFX \ y / - 53 ^wi-^y-^w) ye(o„/ir2,"-'c?«,)'< if u;(^) = —2 -{ —{q — if iy(0 < ~"2. A Lemma 6.3.9. Let P be a homogeneous polynomial of degree 2 in the variables Cj for i = 1, ...,n with coefficients in F. Let P € F'^ and let Tr denote the trace from F dovm to Fp. Then E •E g27riTr(^P)/p _ ^ E-E ^2^iTr[P)lp CI€F CnGF CIGF CnGF Proof; VVe proceed by induction on n. Suppose n = 1. Then P = Lcf for £ € F. If Z, = 0. then the result is obvious. The case L # 0 follows immediately from Proposition 6.3.2. .\o\v suppose the result holds for n = m. Suppose P is a homogeneous poly nomial of degree 2 in c, for i = 1,.... m + 1. Then PP = /3(Lc^^i + Mcm+i + N) where L, M, and N do not depend on {3 or Cm+i. Moreover, L is independent of the Ci. M is a linear homogeneous polynomial in ci,..., Cm, and N is a. homogeneous polynomial of degree 2 in Ci, 78 E-• Eg27nT"r(^P)/p _ ^ g25riTr(^(LcJ,^i+AFCM+i+N'))/p CIGF Cm+l6F CIGF Cm+l6F =^ ^^2TiTr{0N)lp ^ ^2^iTT(0{Lcl,^,+Mcm+i))lp Cl€F Cm6F Cm+l€F Case 1: L = 0. Apply the change of variable Cm-k-i Cm+i0~^ to obtain ^ . . . ^ ^2^iTr(0N)/p ^ g25nTr{Afc„+,)/p CIGF CmGF Cm+l€F ={ 0 if M 0 Now we consider the sum E-E^2niTr(N)/p ^ ^ g2iriTr(Lc^^^+Mc,n+i)/p ci^F Cm€F Cm+i€F and suppose L = 0. Then we have ^ . . . ^ g27nTr(yV)/p ^ g2:riTr(A/c„+i )/p ci€F CmCF Cm+ieF _ r 0 if M 7^ 0 ~ I if'W=0 and the induction hypothesis implies the result. Csise 2: L ^ 0. We complete the square to obtain ^ ^ ^2niTrm-^ +N))/p ^ g2,riTr{;9I-(c„+i -^)2)/P CI6F CMGF CM+I€F Recall that |F| = p*" and let N denote the norm from F down to Fp. Apply the change of variable Cm+\ ^ c^+i — ^ and we have ^ . . . ^ ^2niTr{0(-^+W))/p ^ g27r.Tr(^tc^+.)/p ClGi? CttiGF Cm+ lGF ^ ^ E2XTTR(/3(-^+,V))/P ^ ^2^iTr{L Now consider the sum ^ ^ ^2iriTr{N)/p ^ g2^Tr(Z,<4^,+Afc„+i)/p ciGF Cm6F Cm+i€F We complete the square and apply the change of variable 0^+1 c^+i — ^ to obtain ^ ^ g2,r.Tr{-:^+/V)/p ^ g2,r.Tr(Lc^^,)/p ciSF Cm£F Cm+j€F and once again the induction hypothesis implies the result. A 6.4 Recurrence Relations Recall the setting from the beginning of section 6.2, but do not assume = 1- Given a prime, p, we will use Fp to denote the finite field with p elements. Before working toward recurrence relations, it is useful to evaluate (6.6) and (6.7) directly in the few trivial cases where it is possible, namely when w{x) = 0 or 1. These cases will become important later as cases in which our recurrence relations will not be valid and as base cases for our recurrence relations. We will use Proposition 6.3.1 and Proposition 6.3.2 in these calculations. C2,(0.1) = i Z «"(!.!') /ow g yGyenz^Ou,/Ou,^U; O-uj/Ow / When Res^{yuj) = 0, we have I ) = 9^'- When ReSu,{yuj) # 0, we have \f6F / 80 (E^W^)) =,'(^) .Thus, .«eF c2/(0,1) = - 1 g^' + (g - l)g' ( A similax computation gives C2i{l. 1). C2/(l,l) = - ^ e^^{l,y)ip{-y) q y€-nZ^ Ouf lO^ vejr:;'Ou,/c»u, / Since our function fields are of genus 1 and u; is a place of degree 1, there are no global functions with a single pole at the place, w, and no other poles. Thus when w{x) — 1, only constants can be represented as sums of squares and the number of representations is the same as in the case w{x) = 0. Using the relationship between representations numbers and Fourier coefficients given in (6.10) and the resulting bound explained at the end of section 6.1.2, we find that C2z(0, TTu,) = ^C2/(0,1) and Q C2l{l,T^w) = -7C2Z(1, 1). 9 Now we move on to computing recurrence relations. In order to manipulate 81 the formulas for our Fourier coefficients into recurrence relations we will need the Inversion Formula listed below. It is a special case of Theorem 4.4.4. Notation: In the theorem below 9^ (xy~^, is used to denote the value of 9^{a,b) where a„ = 1 for zill v ^ ty and and 6„ = 0 for all t; ^ ti; and We will continue to use this notation for the remainder of the paper. Theorem 6.4.1 (Inversion Formula). Let 9 be the theta function in Theorem 4-4-4 and assume the setting given at the beginning of section 6.2. Let y € A such that i/u = 0 for all V ^ w and w{y) = a. Then Proof: Choose A = e SL2{k) and let u; be as in Theorem 4.4.4. We need only show that f . However, this is follows immediately from Lemma 5.3.1. A 6.4.1 The case ^ = 0 Recall that C2z(0,x) = ^ 9'^\x,y) (6.13) yG'r~'Oui/5r5,"Ou, / \ q2n+\ ^ 02'(x,y)+ ^ 0'^\x,y) .(6.14) \ u'(y) Now consider just the second sum in (6.14). Let w{y) = a and apply the 82 Inversion Formula to obtain y€5r2;C«>/*®"C7«, = w;'+ E 0#v6'r5,Ou,/ir2,"O„ ^ ^ ^ Since = |xy~^|{j, for w{xy~^) < 0, we have yeir2 0u,/7r2"0„ E.Jy)'"'" (•'-•-;) »6*u,Ow/»d, Oiu ^(y) Let y € ir^OuilTr^Oui with valuation a. Then y € ^ ~Zi'^ {0,^I-k^~°-0-uiY^ . Since we chose a > 2, we have —a < —1. The iw-adic expansion of — ^ is given by 5Z^_a where 6, E for all z. We have assumed that /c is of genus 1 and w is of degree 1. Thus, we see from section 6.1 that the map (f) : —>• k\A./xy~^0 is surjective. Using the left and right invariance properties of 6, we have that 6 {^xy~^, ~y ~ ^ (^2/ ^ ~y) ^ E k and for ^ € xy~^0. We conclude that 6 ^xy~\ —only depends on bi for — l Thus, 53 =1^1^'+~ (~) yex20„,/jr2nO^ o=2 VP/ Yi «"{<-', z) ^zeirur'Oa,/jr2>" ^Ou, ^ - E (^)°"|^.-'IL o#»e«-s+*c>„/,r2»Ou, = kl;;;'+ ~ (~^) '^2/(o,x7r-")^ = 1^:1;:'+ ~ 1)9^""°"' ('^) ^2/(0, XTT-")^ *(.L>(T)") = 1^:1;:;'+ ~ C2/(0,a:7r-")^ • 2n(/-l) _ „(n+l)(/-l) + (g - l)g-"('-2)-i I ^ ^ ^ V p ; = i^lu,'+ ^^(9- 1)9^" " ^ (~^) '^2/(0,x7r^°) (n-l)W +(? -1),",n+,-2 j (v)^ "-'(f) - (6.15) 84 We now consider the first sum in (6.14). Recall that R = ^ CJU; /'FU/* w(y) ^O^/w'^O'uWti;/ »€«-u-OII>/*2j"OW uj(y)<0 u,(y)=l u;{y)<0 u;(z)=-l \ c2ii0,x)^w{Py) B 6 ir^, 'Oil; / irj™ Ou, sen (v) + J (T)" ,E, C2i{0,xtz )I1}^{^Z) BeR w(z)=-l Im'u; \w(.p)>-2n* r a\'*s^ r% Ail the sums are finite, so we may freely switch the order of summation. Doing so gives Tl 0^^{x,y)= ^ C2ii0,x) ^ ip{3y) y€T;'Ou./TH,"Ou; ro\t"^ o V6»;;'Ou,/»E>''0<" w(y) + (~) H C2li^,X7r^') d€R ^Ou, w{fi)>—2n u;(z)=-l (6.16) For all /5 € /? with w{P) > —2n, we want to evaluate the above sums over y and 2. First note that since k is genus 1 and ttr is a place of degree 1, there are no /? G /? with w{^) -- —1. We assume that n > 1 and apply Proposition 6.3.4 and Proposition 6.3.5 to 85 (6.16) to obtain y€*'u;^Ou,*/*'Ou-/ir2>' 2/* O w w(!/)<0 = XI ^C2iW,x)- ^ ^C2i(^,X7r^^)'j ^eF* V VP/ / +(G - 1)(9^" + 9^"~^)C2/(0, X) + (l ~ 1)9^"~^C2/(0, XTT^^). (6.17) Combining (6.15) and (6.17), we have <7^"+^C2i(0, x) = |x|-' + ^^(g - 1)9^""""^ C2i(0,X7r-°)^ _^(n+l)r/ /g(n-l)('-l) + (9 - 1)9 n+<-2 (T) /3€F*J€F* \ \P / / -r (9 - 1) (9^" + 9^""') C2/(0, x) + (9 - l)9^"~^C2/(0,x7r~^). Note that both sides have terms involving C2/(0, x). Rearranging terms and doing 86 some minor simplifications, we have (fi+i)r/ I (6.18) As noted in the computation above, we assumed that w(x) > 2. Thus our recur rence relation is valid only in this case. We must also assume that I > 2 to insure that the expression above is defined. We can simplify (6.18) slightly, but we must first prove a property of the Fourier coefficients of 0^^{x,y). Proposition 6.4.2. Let ^ . Then 0{x, 0y) = ±d{x, y). Proof: Note that we may assume !„ = 1 and y„ = 0 for all v ^ w. Let {V^} be a basis for the F-vector space C{{x)), defined in section 6.1, and let ^ G iC((z)) be given by ^ = Ylj ^j^j- Let Tr denote the trace from F down to Fp. We have 0{x,py) = (ecax)) — ^ ^27riTr(0Re3w{y^^'^))/p ^€£((1)) = E Cj€F We observe that ReSw{y(^j CjVj)'^uj) is a homogeneous polynomial of degree 2 in 87 the Cj. By Lemma 6.3.9 we have e{x,0y) = C,€F = ±6{x, y) A Corollary 6.4.3. Let ^ . Then C2ii^P,x) = C2i(^,x) for all 0 E • We can now simplify (6.18) slightly by applying Corollary 6.4.3. We obtain ^C2/(0,x) = |xU {n+l)W I - (9 - 1) ^9^" ^C2i{l,x) + 9^" ^C2/(l,X7r^^)j . (6.19) The recurrence relation (6.19) is our first main expression for C2/(0, x). Note that this expression is very similar to the recurrence relation obtained in the genus zero case by Merrill and Walling [MW93, p. 672], except for the "extra terms" involving C2i{l,x) and C2/(L, TT-^X). We would like to remove some of the recursion from (6.19). In other words, we want to write an expression for C2/(0, x) that does not involve any terms of the form C2/(0, X7r~"). We will employ a method using generating functions. Write /—1 \ ^^2/(0. K) - E*' -1)'"° (—) 1 \P / 88 where 1 / /—1\ ^ — 1 (f) - («- 1) ^<(^""'C2|(1,t;) + 9^""^C2,(l,7r;-')^^ (n-l)r/ X , X (n+l)W ( g{n-m-l) f = _ I =5'-Kt' + to - 1),'-"-' (:^) - (9 - I) ^C2,(l,0 + ica(l,<-'A (n-l)rl / 1 \ ("+!)'•' ^gC"~0('-l) - 1 I '-(f) -1 - (9 - 1) ^<^21(1, K) + (^) <'')) Now set / — 1 \ ' B(X)=1-^(5-1),-'( —) i' 1=1 \ P / °° / _ 1 \ "•' =9-Eta-1)9-(y) x' =9 9-1 89 where the last three equalities are meant only formally. Set D{x) = C2i(0,7r^)x* t=0 and consider the formal equality OO = D(i)B(i) - ca(0,1). R=1 Solving for D{x) we have D{x) = ^2^0^) +Er=o'4(n)x" B{x) We want to find expressions for the coefficients of D{x). In order to do so, we need a series expansion of We have -g((y)'-K^r)' -S(-i)(Tr' Now we can find an expression for the coefficients of D{x). Wie have C2/(0,<,)= (l-^) (^)"'c2<(0,l) + yl(n)+24(n-a) (l " i) 90 Substituting for A{n), A{n — a), and C2i(0,1) we obtain C2,(0, ^Z) = (i - i) (y) (f"-' + (9 - 1)?'-' (y) ) \pJ [ ,-,(^) _i / -(9- 1) ^C2,(l,jr;) + ^C2i(l,'rS~')j *S('-i)(y)" / , 1, , / \ (n—a—l)r/ \ (n-a+l)r/ / ^ ) — 1 + (, -1),'—-• (^) iii, -(9-1) ^cj,(l,7rr°)+ (6.20) Note that we have now removed all the terms of the form 02/(0,1-"") from our recurrence relation. We now restrict ourselves to the case where ( —^ ) = 1. Recall that ( —^ ) = \ P J \ P J 1 implies that either —1 is a square in F or Z is even. This assumption will be needed when we compute bounds and asymptotic formulas to insure that the last line of (6.20) is a negative number. We introduce it now to aid in placing (6.20) in a cleaner form. Using this extra assumption, we obtain 91 C2/(0,0 = ^1 - ^ + (q- 1)9' /_(n-l)(/-l) _ 1 \ + 9'+""-" + (9 -1)9'--' ) - (9 - 1) (^ca(l,ir^) + ^C2,(l,7rJ-')^ -(9 - 1) . This expression can be simplified by evcduating the terms in the series over a which do not involve any Fourier coefficients. These series are simply geometric series. Evaluating them gives C2i(0, ~Z) =,'"-2"*' + rr^ + (1 - i) (,='-• + (9 - 1)9'-') + (l^) (9"-'"-" - 1) + ( JLLI-) _ „'-2) („(/-2)(—l) _ n - (9 - l)C2j(1.0 - - £ (1 - J) ((' - i)c2,(i.Tr'') + . Note that the above expression is only defined for I > 3. Moreover, since we have already obtained exact formulas for all the Fourier coefficients in the case where n < 1, we may as well assume that n > 2. If we collect terms and make a few 92 minor simplifications, then we have C2/(0, O + iq-l) {q^-^ + (<7 - 1)?'"^) + (^7^) (,"<'-2)+3 _ ,n(,-2)+l _ ,,+. + ^,-1) (6.21) - (9 - 1) o + """V +C2i(l, + C2j(l, irj)^. This is the final form of our recurrence relation for C2/(0, x). 6.4.2 The Case ^ ^ 0 We now return to the setting outlined at the beginning of section 6.2 without the assumption that = 1- Recall that Within the sum, a will denote the valuation of y. We apply the Inversion Formula and obtain = larl"'+ ^ y)^u,(-^y) »6»u,'o/ir2no^ w(y)<0 . (6-22) + Oj^yCTu, Ou,/jr5," CJu, 93 We now simplify (6.22) by evaluating the theta functions when possible, i.e. when w{xy~^) < 0, and expanding 6"^ into its Fourier series in the other cases. This gives '?^"^'c2z(^,x) = |x|J + ^ I ^ C2i(/3,x)ii;^{py)ip^(-^y) v€«i;^Ow/iri"Ou- \^e/:((i')) u;(y)<0 + XI (") C2i{0,xy~^)7p^ [-0-^ 7p^{-^y) V6>r„0«,/»2'>0u, ^ P ^ ffeCUx^y-^)) ^ iu(y) If we divide up the sums over y by valuation and collect terms, then we have g^"'^^C2/(^,x) = 1^1;;;'+ C2i{p,x) ^ ^u.((,5-0y) 0ec{{x^)) ye y6(C»u,/r2,"""0«,)'' 2n—1 ^ . V arl Q=n4-1 ye(0^/nl''-''0u,)' There are three different character sums, i.e. sums involving in (6.23). We may evaluate these character sums in almost all cases using Propositions 6.3.4, 6.3.6, and 6.3.7. First consider the case where ^ and n > 2. We apply these propositions to our recurrence relation and use the fact that the terms in (6.23) involving Fourier 94 coefficients associated to give no contribution unless € F. This gives = g"' + - l)c2i{^,x) - g^""^C2i(0,x) n ^/_^\arl^ arl - 9,2n-l ' ^ca(^, I) + J] ( -^ I (9 - ix;-) ^erx a=l ^ P / - 9^" ^ (—) 53 C2iiP,xirJ) 2n—1 ^ ^ -v arl + Z (v) a=n+l ^ P / If we rearrange terms, evaluate the geometric series in the last term, and use the fact that for all ^ E , C2/(^, TT^) = C2/(L,TT^), then we have 9'""'c2i(0,x) = 9"' -(?-!) r9^"~'c2,(l,i) - 9^""^C2/(1,I7r;')j (n-l)r/ , X (n+l)r/f r /g("« in— III#—fOC I) — 1 I — 1 n+l-2 / -1 \ I ^ \ P / Note that the recurrence relation above does not give us any new information. It is exactly the same as the recurrence relation we obtained for C2i(0,x) in (6.19). Now suppose that n > 2 and < —2. As before we apply Propositions 6.3.4, 95 6.3.6, and 6.3.7 to (6.23) and obtain ^€F* {ii;(0+l)r/ -(T) g2"+«'(Oc2/(0,x7r-«)+i) arl + S (^) (g-l)9^" " ^C2/(0,X7r^°) , a=-u;(0 arf + C2/(^,X7r^") (t) l ye(Ow/i After rearranging terms and dividing by we have ^C2,(e + ^,:r)=9"('-2)+i 0eF {«;(0+l)ri (t) arl + £(T) {q - l)q "021(0, Xn^") .a=-w(^) arl + ^ (T) C2/(/5,X7r^°)9 \u;(^)=u;(0+2a ye(Ou,/^2r-"Ou,)'' / As in the ^ = 0 case, we will now make the additional assumption that =1. This assumption gives only a slight simplification, but it will prove useful in obtaining bounds and asymptotic formulas. Using our assumption, we 96 obtain 5€F 271-1 ^(Q- 1)9~°C2/(0, xtt'") (a=-u;(0 ( + C2/(/S,X7r-")g-2"+i l ye(0„/5r2,"-''C»„)'' This is the final form of our recurrence relation for C2i{^,x) for w(^) < —2. 6.5 Bounds and Asymptotic Formulas We continue to work in the setting outlined at the beginning of section 6.2. 6.5.1 The Case ^ = 0 First we recall the recurrence relation for C2i{0,x) valid for n > 2 and I > 3. It is given by C!,(0. -Z) =?"<'-"'+' + (9 - 1) (9''-' + (9 - 1)9'-") + (9""-"+'- 9""-"+' - 9'-"' +9'-') (6.24) -(9-1)1 ^<^2,(1.0 +^ 9" Q +C2/(1,<~^) + 02/(1, TT;^)) . Our goal in this section is to prove the following theorem. 97 Theorem 6.5.1. Suppose we are in the setting outlined at the beginning of sec tion 6.2. Let l,n>5. Then |C2/(0, O +{q- 1)C2/(1,0 - < 29"('-2)+4-i _ qn(l-2H3-l 4^n{i-2)+l-i We will actually obtain bounds slightly stronger than those stated in the theorem. The theorem will then be proved by obtaining a simpler, although slightly weaker form of these bounds. The simpler form of these bounds will prove useful later. We now obtain upper and lower bounds on C2/(0, tt^) + (? — The proof of the theorem using the bounds will be given at the end of the section. Considering the relationship between Fourier coefficients and representation numbers, it is clear that C2/(l,7rJj,) > 0 for all i > 0. We see in (6.24) that all of the Fourier coefficients on the right hand side have negative coeffi cients. If we throw away all the terms involving C2/(l,7rJ„) for 2 < z < n — 1 and plug in the formulas we obtained earlier for C2i{l, 1) and C2/(l,7r^), then we obtain the following upper bound on C2i(0, ttJJ;) + (9 — l)c2/(l, C2i(0, i)c2,(i.O + («-!) + {g- 1),'-=) + (7^) + 9'"') - (9'+ 9 +1)(9'- 1) (6.25) Obviously, the bound on C2/(0, +{q— 1)02/(1, TTJJ,) given in (6.25) is an upper bound for C2/(0,7r2) and for [q — l)c2/(l,7r^) as well. Now we return to (6.24) and plug in our upper bound for C2/(l,7r^) for i < n. Since the coefficients of C2/(l,7rJj,) are negative for all i < n, plugging in an upper bound for these coefficients gives 98 us a lower bound on C2i{0,Tr^) + (9 — l)c2/(l, namely C2z(0,O+(g - 1)C2/(1,0 > + {q-l) {q^-^ + (9 - + (7^) (9°"""*' - 9"""""" - V*'+9'-') - ((i^) " -(n - 2)q^'-^{q^ - 1)^ + q'~\in - 2)q'^ - (2n - 5)q^ + (n - 3))] + ((9-i)W-i)+ i)) ((,. + ,+1 _ (,. +1)(,. _ D), (6.26) Now that we have obtained upper and lower bounds on C2/(0,7r{^) + {q — l)c2;(l, ttJJ,), we can prove Theorem 6.5.1. Proof of Theorem 6.5.1: Suppose that I and n are both at least 5. Then a simpler form of our upper bound is given by C2/(0, TT^) + (q - 1)C2/(1, ttZ) < + ?' + ^ i < gn(/-2)+l ^ 2g"('-2)+3-' + 2<72'-1 - ^"('-2)+l-i < gn{l-2)+l 29"('-2)+3-i _ ^n(l-2)-l (6.27) Note that the last inequality is only true for n > 5. Since the upper bound required n > 5, we will assume n > 5 as we compute a simpler lower bound. C2/(0, +{q- 1)C2/(1, O > _ g2/-2 _ 2^'-l 4. ^n(t-2)4-3-i _ ^2 - 2g"('-2)+i-' _ 2qr"('-2)+-»-' _ _ 2)9"'"^® - (n - 3)9-2'+^ - (n + l)g'+2 > ^n(/-2)H-l _ 2g2/-2 ^n(l-2)+3-t _ ^^n{l-2)+l-l - 2g"('-2)+4-i > ^n(/-2)+l _ 2qn(l-2)+4-l ^n(l-2)+3-l _ ^^„(/-2)+ l-/ Note that the last two inequalities require I > 5. 99 Now we return to (6.27) and see that ' < 2q"^' '. Thus 2^n(J-2)+3-i _ qn{l-2)-l < 2g''a-2)+4-t _ ^n(l~2)+3-l ^ ^^„(i-2)+l-J and the result follows. A 6.5.2 The Case ^ ^ 0 .A.S in the previous section we first recall the recurrence relation in question valid for n > 2 and w{^) < —2. It is given by ^C2,(e + ,/5,2:)=9"('-2)+i del 2n-l - ^ {q- 1)9-"C2,(0, XTT"") \a=-tu(f) ( C2z(^,X7r-'')g-2"+^ l ye(o..,/5ri:—"Ou,)'' (6.28) The goal of this section is to prove the following result. Theorem 6.5.2. Suppose we are in the setting outlined at the beginning of sec tion 6.2. Let ^ G Ylv7^w^v with w{^) = —m < —2, and suppose I > m + 5 and n > max{m + 5, 9}. Then I + 0,^) — < Qq^(l-'2')+rn+2-l ^€F 100 Proof: We begin by working toward an upper bound. The first step is to set obvious negative terms in (6.28) to zero. Doing so gives 2n-l (a=-tu(f) / + ^ C2/(^,X7r-'')9-2"+i (5 29) l y&{0^ln^-''0u,Y Clearly, C2i{^, x) satisfies the same upper bound. We want to remove the dependence on Fourier coefficients and character sums from the upper bound. Before we can do this, we need some upper bounds on the Fourier coefficients and some results on character sums. In most cases we will not need the best possible bounds to obtain our result. Again we return to the recurrence relation, (6.24). We will need to assume that I and n are both at least 5. However, these assumptions are weaker than those made in Theorem 6.5.2. If we remove all terms with negative coefficients from the upper bound (6.27) in the proof of Theorem 6.5.1, then we have an upper bound on C2/(0, x) given by C2i(0,O + 2g"('-2)+3-' <2^"('-2)+I This upper bound is clearly also valid for C2/(l,7rJ^). Note that as n increases, the representation numbers cannot decrease. In other words we have C2z(0, tt^) > C2i(0,7r^~^). Using the relationship between the Fourier coefficients of 0 and those of 0 we have q^C2i(0, ttJJ) > C2/(0, 101 Suppose —w{^) = m. We are interested in finding an upper bound for 2n-l ^2 (Q ~ 1)9~"C2/(0, XTT"") for 2 < m < 2n — 1. Consider the case where m 2n-l XI (9 - l)'7~''c2/(0,X7r-") a=m n—6 a=m o=n—5 2n-l + (g - l)<7~°C2f(0, <~°) O=TH-1 Now we split up the middle sum again and use the upper bound we just computed for C2/(0, ttJJ,). This gives 2n-l - l)"? °C2/(0,X7r^") a—rn n—6 < Yl 2(9 - + ^(9 - a=m a=0 + (9 - 1)9^""C2/(0, 7r„) + (g - l)g""c2/(0,1) 2n-l + E (9 - 1)9-9'°-"" a=n 4-1 n—6 < 5^2((7- l)g-y"-a)('-2)+l +^2(9- l)^5-"+«('-l) + (9 - + 2q' + q)+ {q - a=n+l 102 Now we evaluate these sums to obtain 2n-l (9 - l)9~°C2/(0,x7r-'') a=m (gn(/-2)+l ^^(m-l)(l-0 _ ^(l-0(n-6)^j +^ (29«-^ + 27^'-® + 2q^-^ + 29''"^ + + 2g' + q) Using the fact that q^~^ — 1 > and doing some simplification, we have 2n-l ^(9 - 1)9-"C2,(0, XTT-") <29-('-2)—('-1)+3 ^ 3g8/-6-n ^ ^n(/-2)-i+3 a~m ^ ' <4^"('-2)+3-/ if we assume n > 9, I > 0, and m > 2. However, these assumptions are made in the statement of Theorem 6.5.2. We now have the tools we need to remove the dependence on Fourier coefficients and character sums from (6.29). Assume that I > —U!{^) +5 and n > max{—w{^)+ 5.9} as in the statement of Theorem 6.5.2. Consider the case where u;(^) = —2. 103 We have ^C2/(^ + /9,x) 0&F <^n(i-2)+l + ^^(7 - l)9"''C2i(0,X7r-°)^ + C2i(/3,X7r-^)g-2n+l I ^ I /sepx \ye(Ou,/x2,""'Ou,)'' V ^ ^ J <^n(/-2)+l ^ ^^„(/-2)+3-i + C2/(l,X7r-^)g-2"+^ [ ^ ^ j yGCO^/ir^T'^Ou,)" \ ^ <^n(/-2)+l _j_ ^^„(f-2)+3-/ _,_ 2^("-l){'-2)+lg-2Ti+1^2n-2 -gn{l-2)+\ _j_ _^^„(/_2)+3-i _,_ 2^("-l)(i-2) <;^n(Z-2)+l ^ g^„(/_2)+3-< The case w{^) = —3 is particularly simple because there are no 0 E R with w{/3) = — 1 and so the last sum in (6.29) is empty. We have C2/(^ + 0,X) < 9"('-2)+ 1 4- [ ^ (9 - l)9""c2i(0, XTT"") j FL€F \ 0=3 / < ^n{l-2)+l ^^„(i_2)+3-i Now we proceed by induction. Suppose w{^) = —m < —3 and that for all ^ with w{^) > —m we have C2l{^,x) < 9"('-2)+1 + Qgri(t-2)-w{^)+2-l We will show that for w{^) = —m, C2,(E + /9, 2:) < 9"('-2)+1 + 69"('-2)+M+2-Z^ 0eF which in turn implies the same bound for C2i(^,x). Let b be the integer part of y. Then +P,X) 0eY 2n-l + ^ C2/(^,x7r-'')9-2"+M ^ H <^n(/-2)+l ^ 4^n(i-2)+3-i - E C2l(,6,XTT^'')q-a\„-2n+l l + C2/(l,X7r^'')9 2n+l j ^ _ ^yTT^ n \i/6(Ou,/>r2r-''Ou,)'' ^ ^ /y <^n('-2)-fl + 4gn(i-2)+3-i ^ J- ^^_a^^-2n+l _ i)^2n-a-l l 4- C2z(l, 3:7r-'')g-2n+l(^ _ jj2^2n-6-l <^n(/-2) + l ^^„(/-2)+3-/ + ^ g-2"+l(g _ l)2^2n-a-l^m-2a-l^^(n-a){/-2)+l l + 7g("-«)('-2)+m-2a+2-i^ _|_^_2„+l^^ _ j^2^2n-6-l ^29("-''){'-2)+l ) ^^n(;-2)+l ^^^„(/-2)+3-i ^ ^m+l-3a ^^(n-a)(/-2)+l _j_ jg(n-o)(/-2)+m-2a+2- l _^na-2) + l ^ 4^n(/-2)+3-/ __]; ^^„(/_2)+m+2 + ^ (^7g'»('-2)+2m+3-/ j ^ ^^(n-bW-2)+Z-b <^n(/-2)4-l _|_ ^^n(i-2)+3-/ ^n(l-2)+m+2-l _|_ y^n(i-2)+2m+l-2i <-^n(/-2)+l ^ 2g"('-2)+m+2-/ 105 Now we can use our upper bound to obtain a lower bound just as we did for C2/(0,x). We eliminate positive terms from (6.28) and we obtain 5€F + X) C2/(;5,x7r^")g j ^ w ^vK Consider the case where w{^) = —2. + ^,x) > -9 ^C2/(0,x7r^^) + ^ <^2i{^.XTZ^'^)q ^"+1 5eF ^€F* /^u V /y > _29-i9("-^)('-2)+i +g-ic2,(Lx7r-^) > ^n(/-2)+l _ 2gn(/-2)+2-i Again the case ly(^) = —3 is easier than the other cases. We have J^C2/(^ + /?,x) > - q-'^C2i{Q,xiT-'^) <3€F > ^"('-2)+l _ 2g-2g(n-2)(/-2)+l _ ^7I(/-2)+1 _ 2gn(i-2)+3-2< .\'o\v suppose zi;(^) = —m < —3. Again we let b be the integer part of y. We have + P,X) 0€F + 5^ C2i(y9,X7r-'')g-2"+W ^ f-yS-TT"" - ^yTT^ H >^"('-2) + L _ 2GL-'N^(N-'N+L)('-2)+L —a\„—2n-i-l X] C2i{0,X7r^'')q 53 l ^-2n+l^m-2a-l^^ _ jj2^2n-a-l(g(n-a)(/-2)+l L - ^ ^"•^^C2i(^,X7r^'') 3eF^ tfe(Ou,/5ri''-»C»„)>' >^"('-2) + L _ 2G("-"»+L)('-2)+2-M _ ^ ^M-3A+L ^^(N-A){F-2)+L ^^(N-A)(/-2)+M-2A+2-/^ L _J ^7^N(/-2)+2M+3-/ _ ^(-Z-3)(6-L)^^ _ 2^(N-6)(/-2)+3-6 >^N(Z-2) + L _ ^^(N-6)(Z-2)+3-6 _ ^n{l-2)+m+2-l _ J^N(Z-2)+2M+L-2/ 2) + L 2^"('~2)+M+2—/ '^^N(I—2)+M—4—Z >^N('-2)-F-L _ 3^N(Z-2)+M+2-/ The upper and lower bounds we have obtained imply the theorem. 107 6.5.3 Individual Fourier Coefficients So far all of the bounds have been on sums of Fourier coefficients. In this section we want to use our results on sums of Fourier coefficients to obtain bounds and asymptotic formulas for individual Fourier coefficients. Proposition 6.5.3. Given x = and y E ive have that 9'^^{x,y) is a rational number. Moreover, if = 1» then 6^{x,y) > 0. Proof: The relationship between the Fourier coefficients of y) and the repre sentation numbers tells us that the Fourier coefficients must be rational numbers. Recall the Fourier expansion of y). 0^(x,y) = C2/(0,x) + ^ C2i{^,x)ipi^y) W'e want to split up the sum over into a nested sum. The outer sum will be over principal divisors D, and the inner sum will be over elements of with a given divisor. We have 6^{x,y) = C2i(0,x) + ^ ^ C2i(^,x)'0(^y) D div{,0=D For each principal divisor D choose such that div{^D) = D. Then all other ^ e k^ with div(^) = D are of the form for /? e F''. Thus, by Corollary 6.4.3 we have C2/(0, x) + E E C2l{^,x)fpi^y) = C2l{0,x) -I- C2l{^D,^) ViP^DV) div{0=D (6.31) Since V ih(Bfny) - / ^ ~ ^ ^e5„(^Dy) = 0 2^ viP^oy) - I if ResA^oy) # 0, 108 the expression for 9^^{x,y) we wrote in (6.31) is a rational number. Now suppose that ( ^ 1 =1- Then either —1 is a square in F or Z is even. If — 1 is a square in F, then using the fact that ip{a) = ip(—a) we have d(x, y) = ci (0, x) + ^ ci (^, x)ipi^y) = ci(0,x) + Ci(^,x)i/;(-^y) Now apply the change of variable ^ and we have 0{x,y) = ci(0,x) 4- Ci{-^,x)ip{^y) eefc* = ci (0, x) + ^ Ci (^, x)tp{^y) = 0{x,y). .Note that Ci(—x) = ci(^, x) from the properties discussed at the end of sec tion 6.1.1 because —1 is a square in F. We conclude that 0{x,y) is a real valued function, which implies 6^^{x,y) > 0. If I is even, then let I' = ^- The computation at the beginning of this proof then implies that 0^''(x,i/) is rationed. Since 0^'(x,y) = (0^''(x, y))^, we conclude that 0^'(x, y) > 0. A Proposition 6.5.4. Again assume the setting outlined at the beginning of sec tion 6.2. Let ^ E . Then C2i{^,x) < C2/(0, x). Proof: C2i(0,x) = |c2z(0,x)i = We know from its relationship to representation numbers that C2/(^, r) is a positive 109 rational number. Thus, C2i{^,x) = |c2/(^,ar)| 1^ Yi d^'{x,y)7P{-^y)\ fx^Our < y&iru,^ Ow /i^Our ^ Y, \e^{x,y)\ y€ir,;'C7u,/x20ur C2z(0,x) A Combining Proposition 6.5.4 with Theorem 6.5.1 and equation (6.27) in the proof of Theorem 6.5.1, we obtain the following results. Proposition 6.5.5. Assume the general setting outlined at the beginning of sec tion 6.2. Let l,n> 5. Then • C2l{0.x) > _ 2g"('—2)+3—i _|_ gn{l—2)+2—l _ ^gn(l—2)-l • C2i(l,x) < Theorem 6.5.6. Assume the general setup given at the beginning of section 6.2. Suppose n > 0 and I >9. Then C2/(0,x) < + 9"('-2)+6-/ Proof: Since w is a place of degree 1, we have that for n > 2, £((x)) is an n- dimensional F-vector space. For j = 0,2,3, choose Vj e C{(x)) such that ^'(^}) = so that 1 is the coefficient of tt"-' in the itz-adic expansion of Vj. Then {VQ, V2, V3,..., is an F-basis for £((x)) and we may write ^ € iC((x)) in the 110 form Y2j where bj € F for all j. We have chosen 5 = {ly} and a holomorphic differential, w, so the definition of 9{x,y) in (6.1) can be written as e{x,y) = \x\^''^ ^ iecax)) 2n-l Let y = = IZj>o (^o # 0), and let Tr denote the trace from m= —1 F down to Fp. We may write 21 y€ir^^C7u,/5r2,"C»'uE ^ ' where F is a homogeneous polynomial in bQ,b^,...,bn with coefficients depending on the Um and e,. In fact since P was created by taking a residue and to is a place of degree 1, we know that each term of P is of the form a constcmt times Omeibab^ where Q may be equal to /3 and m + i — a — 0 = —1. Now we want to start with the sum over 6„ and factor out all the terms that do not depend on 6„. Then we want to do the same thing with the sum over 6„_i and continue down through the sum over 62- The result of this process gives us the following expression for C2:(0, x). YEIRZ^OW/'^LRO^ V FCOGF ^2MTR{M2BL+N2B:I)/P ^ ^2-NITR(M3BL+N3B3)/P . . . ^ ^277ITR(M„BL+N„B^)/P J 626? 636F 6„6F / Note that the Ma have no dependence on the bj. Moreover, Ma does not depend on Cm for m > 2a. Thus, Ma is of the form 020-1^0 +(other terms) and all terms of Ma are of the form a constant times 0^6, where m+i = 2ot—\. The Na are homogeneous Ill linear polynomials in the bj. Note that Na does not depend on bj for j > a or on Om for 772 > 2a — 1. Thus, Na = flza—2^q—iCo+fl2a—3^a—2®o~'"<^2a—3^q—iCi + (other terms) and all terms of Nq £ire of the form a constant times Omeibj where m+ i—j = a—1. We want to consider evaluating the inside sums, i.e. the sums over the bj, from right to left. Depending on the coefficients in the tu-adic expansion of y, there are 3 possibilities for the value of a sum of the form e2TiTr(Cj b]+Djbj)/p TSjQj-g we 6jeF use Cj and Dj because, if we completed the square in some of the sums further to the right, then we may or may not have Cj = Mj and Dj = Nj. However, as we will see below Cj will still be independent of the 6's and Dj will still be a homogeneous linear polynomial in the 6's. The possible values are as follows: 1. If Cj 7^ 0, then we can complete the square. We obtain a factor of Tjq^l'^e ^ ^' where 77 is a fourth root of unity such that (779'''^)^' = g' ^= 9'. We can factor 779^/^ completely out of the nested sum. The e ^ ' term moves to the left. However, since Cj is in dependent of the 6's and Dj is a homogeneous linear polynomial in the 6's, —^ is a homogeneous polynomial of degree 2 in the 6's and this term will not change the form of the sums to the left. 2. If Cj = Dj = 0, then we get a factor of q. 3. If Cj = 0 and Dj ^ 0, then we get zero. However, we will always have Dj = 0 if we set 6, = 0 for some number of i < j. After setting these bi = 0, our sum falls into the case above. The sums over the bi that are set to zero then contribute a factor of 1 as only the 6, = 0 term remains in the sum. As we proceed from right to left, let s be the number case 1 sums, t the number of case 2 sums, and u the total number of 6, that are set to zero. We need not 112 worry about case 3 sums as they simply cause some of the 6, to be set to zero and become case 2 sums. Using this notation we have s + t = n — u. Thus for a fixed y, the value of 6^'{x^y) is Note that since we are assuming that = 1, then we know from Proposition 6.5.3 that 6'^{x,y) > 0. If we now consider all y € then the possible nonzero values of d^'{x,y) are given by o < s + 2t < 2n. We proceed in two cases. The case 5 = 0 and t = n will be considered as an exceptional case later. We consider the other cases below. Fix a value, for some 0 < 5 + 2t < 2n — 1, of 0^^{x,y). We want to determine the number of Tr~^(D y € _2n/n"' value of y). The number of y giving this value is "u; a number between 0 and Thus the number of y giving the chosen value can be written as a polynomial in q with coefficients between 0 and (g — 1). Let m be the degree of this polynomial. Note that the polynomial is bounded above by We will use this fact later in the proof. Since m is the largest power of q in the polynomial expression for the number of y giving our fixed value of y), g(«+20'-"'-2n-i+m jg largest power of q in the contribution to C2/(0, x) from these y. From equation (6.27) in the proof of Theorem 6.5.1 we know that for n > 5 and Z > 5, we have C2/(0, x) < Recall that 0^'(x, y) > 0 for all y 6 Therefore all the contributions are positive and we must have (s + 2t)l — nl — 2n—l + m< n{l — 2) + 1. Rearranging terms gives m < 2nl — (s + 2t)l + 2. 113 This condition on m holds as long as n > 5 and I > 5. From the formula for C2/(0, x) written above, it is clear that the number of y which give the fixed value of 6{x, y), and therefore of 9^{x, y), does not depend on I. Thus we may choose 1 = 5 and we have m < lOn — 5(s + 2t) + 2 for n > 5. As m is independent of /, the above inequality must also be independent of I. We will now assume, as in the statement of the theorem, that I > 9. Recall that the number of y giving our fixed value of 6^^{x,y) was bounded by The contribution to C2/(0,a:) from these y is then bounded by q(s+2t)l-nl-2n-l+Tn+l ^ ^{i+2£)i-n/-2n+10n-5(s+2t)+2 _ ^(a+2t)(f-5)-n(i-8)+2 If we add up the bounds for the contributions for all the values of y) with 0 < s + 2t < 2n — 1, we see that the total contribution from these values is bounded above by _-n(/-8)+2 ^a(/-5}-n(/-8)+2 ^ ^ (g2n('-5) _ Q=0 ' ~ ^ < ^„(/-2)+6-/ Now we consider the exceptional case s = 0 and t = n. The value of 0-'(x, y) in this case is g"'. We want to know the largest possible number of y that give us this value of y). We will determine this number by beginning with a general y and determining conditions on the coefficients in the u;-adic expansion of y. In order to obtain a value of for d^^(x, y), we must have M, = Ni = 0 for all i. We will work from largest to smallest. M, depends only on Om for —1 < m < 2z — 1 and the only term involving aai-i is of the form eoa2i-i- To set M, = 0 we can solve 114 for 02,-1 and write it in terms of a,„ for — 1 < m < 2i — 2, or in the case where 2 = 0 simply set a_i =0. Now we want to set all the Ni equal to zero. Recall that Ni depends on Om and bj for —1 < m < 2i — 2 and j = 0, 2,3, ...,i — 1. Moreover, after setting all of the Mi =0, we see that Ni in fact only depends on for — 1 < m < 2i — 2 and m even or m = 1. Suppose z > 3 and consider all terms involving Note that one term is of the form 60021-26161-1 and there are no other terms involving 6,6i_i which depend on 021-2- Since we have assumed u = 0, we must have Ni = 0 independent of the bj. As Ni must be zero independent of the bj and cq # 0, the coefficient of 6,6i-i must be zero. Thus we may set this coefficient equal to zero and solve for a2i-2 in terms of a-m for —1 < m < 2i — 3 and m even or m = 1. For i = 3 we may impose an additional condition. There is a term of the form 60026360 and no other terms involving 6360 which depend on 02- Thus we may write 02 in terms of Ci and oq. Now consider N for i = 0, 2. Note that NQ = 0. For n = 2 there is no term involving 6261 since there is no coefficient 61. However, there is a term of the form 60016260 and there is no other term involving 6260 which depends on Oi. Thus we may write oi in terms of OQ. We see that we have now fixed 2n of the coefficients in the lu-adic expansion of y. In other words we have written all of the coefficients in terms of OQ. Thus the number of y for which 9^(x,y) = g"' is at most q. We get a contribution to C2i(0, x) from these y of at most • 9")^' = If we combine this contribution with the contribution from all the other cases computed earlier in this proof, then we conclude that C2/(0, x) < ^ Using the above Theorem, we have the following bounds and asymptotic formulas. 115 Corollary 6.5.7. Assume the setup at the beginning of section 6.2. Let m = —w{^), n > max{9,m + 5} and I > max{9, m + 5}. Then (a) |C2/(0, X) — < gn(l-2}+6-l (b) |C2/(1, ^ (gn(/-2)+6-i ^^n{l-2)+A-l _ ^n(/-2)+3-/ ^ 4^„(i-2)+l-i) (c) For —2n < —m < —2, |C2i(^,a:) - < 6gn(i-2)+m+3-/ ^ _ gn(l-2)+6-l Proof: (a) Apply the Theorem and Proposition 6.5.5. (b) Apply the Theorem to the lower bound obtained for C2/(0, x) + (g — l)c2/(l, x) in Theorem 6.5.1. This gives a lower bound on C2/(l,x). Combine this lower bound with the upper bound on C2/(l,x) computed in Proposition 6.5.5 to obtain the result. (c) By Proposition 6.5.4, C2i(^,x) < C2i(0,x). This fact combined with the Theo rem and Theorem 6.5.2 gives the result. A The Corollary below now follows immediately. Corollary 6.5.8. Assume the setup at the beginning of section 6.2. Let m -- and n > max{9,m + 5}. Then as I oo C2l(0,l) . 1 • ,n(l-2) ^ • ^1- ^ 1 for -2n <-m< -2. Q 116 Note that since we are assuming u; is a place of degree 1, there are elements in £((x^)) to be represented and a total of representations. Thus the average number of representations of an element is g2ni-2n corresponds to an average value of C2/(^, x) of We have thus shown that all the Fourier coefficients are asymptotic to the average value of the Fourier coefficients as Z —> oo. As n —oo we don't quite get asymptotic formulas, but rather asymptotic formulas up to a constant. The result is given by the following corollary. Corollary 6.5.9. Assume the setup at the beginning of section 6.2. Let m = —w(^) and I > max{9,m + 5}. Then as n oo • C2i{0,x) = • C2iil,x) = • C2i{^,x) = for —2n < —m < —2. 6.6 Exact Formulas for Small Valuation Cases We have been working in function fields, k, of elliptic curves over finite fields, F, such that char{F) ^ 2. Since char{¥) ^ 2, every elliptic curve can be written in the form = f{X) where f{X) is a cubic polynomial in A". For curves of this form we have that the place at infinity is a place of degree 1. In this section we will work in the setting outlined at the beginning of section 6.2 except we will not assume that = 1- We will assume that w is the place at infinity. Note that w{X) = —2 and U!{Y) = —3. We want to consider the cases when w{x) = 2 or 3. We will use {1, X} as a basis for >C((7r^)) over F and {1, A', Y} as a FfATirVl basis for iC((7r^)) over F. Our function field, k, can be written as ^ . A ~ fi^j) 117 set of representatives for 0 €. R with w{0) > —6 is given by the set of polynomials in X and Y with total degree at most 2. Proposition 6.6.1. Assume the setting outlined at the beginning of section 6.2, but do not assume =1- Suppose w is the place at infinity, a place of degree 1. If w{x) < 3, then for all ^ E R, C2i{^,x) is independent of the choice of elliptic curve = f{X). Proof: This result was verified earlier for ^^(x) < 1. Assume w{x) = 3. Let ^ € R be given by ^ = di + d2X + d^Y + d^X"^ + d^XY + d^Y"^ with d,- € F. Note that if there are any other nonzero coefficients, then C2ii^,x) = 0. Let Pj e C{{x)). Then 0j = Qj + bjX + CjY for Cj, bj, cj € F and we have = aj + 2ajbjX + 2ajCjY + bjX'^ + 2bjCjXY + CjY"^. Note that for all Pj, pj is a polynomial in X and Y of total degree at most two, and thus is one of our set of representatives chosen above for elements of R with valuation at least —6. Thus there are no relations among the Uj, bj, and Cj, and for any elliptic curve Ylj = C is equivalent to the system of equations over F given by Ej o.j = di Ej = d4 2ajbj = d2 Ej = ds 2ayCj = da EiC^=^6- Since the number of solutions to the system of equations above is independent of the choice of elliptic curve, the representation numbers, and therefore the Fourier coefficients of the theta function, must be as well. If we set (fa, d^, de, and all Cj equal to zero, then the exact same argument proves the result for w{x) = 2. A We have now shown that to compute exact formulas for Fourier coefficients when w{x) = 2 or 3 we may choose any elliptic curve and use that curs'e to do the computations. VVe will now carry out these computations using Y"^ = X^ — X. 118 6.6.1 w{x) = 2 We want to evaluate C2/(0,X) = ^(x,y) |F^|Mr')+l) ^ e for the curve — X where w is the place at infinity and x = Since w is a place of degree 1, we have that |F,i;| = q = p''- By definition 1 21 \x\'^l^'^lo^{xOtp{y(^) / = kl' 53 \u;(f)>-2 21 -21 = Q 53 «€R \u;(0>-2 Now 2/ -21-5 C2i{0,x) = q 53 yGT^^OUR/TrJ^ C?u {6R \u){0>-2 / Let y = 2Ii=-i € = 60 + 62-^- Then we have Res^{y^^u) = —2(a_i6o + 0362 + 2016062)- Now consider just the inside sum in the expression for C2i(0, x). We have f ^ ^ ^ ^ ^2niTr{-2a-ibl-2a3bl-4aibob2)/p\ bo 62 _ (:I g27nTr(-2a_i6g)/p g2jrirr(-2a36|-4ai6o62)/pr 60 ^2 / 119 where as before Tr denotes the trace from F down to Fp. Now suppose 03 ^ 0. There are {q — \.)q'^ y with # 0. In this case we complete the square and obtain 21 g27rzTr((-2a_i+2^)6g)/p ^g2irtrr(-2a3(62 + 2i^)2)/p'\ K bo 6j / =qi y' ^^e2^'rr((-2a_i+2fi)6g)/p)• = I i~^) ~ = 0, (q- l)q^ such y [ if _ 2a_i + 2^ ^ 0, {q- l)^q^ such y. Now assume 03 = 0. There are {q — l)g^ y with as = 0 and oi ^ 0. In order to get any contribution in this case we must have 60 = 0 and we get a contribution of If 03 = 0 and ai = 0, then we have \ r q*' if a_i = 0. q"^ such y 27riTr(-2a_i6g)/p \ =J , s W 9^ ) I (f) q^{q - 1) such y If we now compile all of our computations into a final formula, we have C2,(0, 4) = [(y)"9"(9- 1)9'+ 9''(9 - 1)V W(i -1)9' + + (y) " -1) q \ p J q^ Now that we have a formula for C2/(0, tt^), we can use our recurrence relation 6.21 to obtain a formula for C2/(l, tt^). Doing so gives us rl\ c2i{i,K) = 9' M 9'•-(t) 120 6.6.2 u;(x)=3 Now we suppose w{x) = 3. We have C2/(0,x) = q ^ <€R \u,{0>-3 / Let y = o-iT^lu and let ^ = 6o + b2^ + ^3^- Then we have ReSyj{y^^uj) = —2(a_i6o + 03^2 + 20160^2 + 2026063 + 2046363 + 0563). Again we look at the inside sum in the expression for C2i(0,i). We have I ^ ^ ^ ^ g2jriTr(—2(a_i6o+<'3^2+2ai6o62+2a26o63+2a46j63+a563))/p j bo ^2 63 — I y " g2irirr(-2a-ifcg)/p ^ ^^^2itiTr(-2azb\-iaibotn)/p (6 32) \ bo &2 \ 21 y ^ g2n-iTr(—20563 —(4a26o+-4 63 / There are (9 — 1)?® y with cg ^ 0. In this case we can complete the square in the rightmost sum and obtain E g27riTr(—2a_i6o)/p ^ g2JriTr(—2036!—4aj 6062+^{<1260+202046062+0^62))/? (60 '2 \ g23riTr(-2as(63-^(a26o+(M62))^)/p 63 / We may now apply the change of variable 63 63 + 3^(0260 + 0462) and evaluate the rightmost sum. The result is ti / \ ^ y f g2:rirr((-2Q-t +^)6g)/p ^2:rirr((-2o3 + ^)fci+(^^^t^-4ai6o)&2)/p j (6.33) 121 fj2 Now suppose —03 -i—i ^ 0. Note that there are {q — l)^g® y with as ^ 0 and as Oa —03 H—- 7^ 0. We complete the square again and obtain as / rl 2TiTr ^(-2a_i + ^)6g- (3^) /P '•(t) J2e - ^ --'s 60 \ 21 62 Apply the change of variable 62 •—> 62 ~ ( —::2 ) ( — ai6o ) and we have as ( l-KiTr Q ^ e ^>0 \ Suppose the coefficient of is nonzero. There are now {q — 1)^9'^ y which satisfy all the assumptions. Via another change of variable, we obtain 9^' ^for these y. If the coefficient of ftg is zero then we have a contribution of g"*'. There are {q — 1)^?'' y that fit this case. a2 Now we return to equation (6.33) and consider the case where —03 H—^ = 0. as There are {q — l)g° y that fit this condition. To get any contribution we must force — ci J 60 =0. There are {q — 1)^9'' y for which f — ai ) ^0. In these ) \ <^5 / cases we must have 60 = 0 and we get a contribution of 9^' ^. There are 122 {q — y in the ( — ai ) =0 case. In this case we have \ 0.5 J ^Ziy' ^^e2"Tr((-2a_x + ^)6§)/pj 9^' = I ("^) ^ (9 - 1)9^ such y g-" if a_i + ^ # 0, (q - l)^q^ such y. Now we return to equation (6.32) and suppose 05 = 0. There are g® y which satisfy this assumption. To get any contribution we must force 02^0 +0.462 = 0. We have 4 cases. Case 1: ao = 04 = 0. There are 9^ y in this case. We have 21 I ^ ^ g27rirr(-2a-i6g)/p " ^2niTr(-2a3a\-4aibob2)fp | V 60 62 / = q (result from w{x) = 2 computation ) = + (, - + (^y (5^ - !),='«. Case 2: 02 = 0, 04 ^ 0. There are q'^{q — 1) y in this case. We must have 62 = 0. This gives us rl -.2/Jf (I X „2;r.rr(-2a_i6^)/pOrriT'r'f\2TTiTr(-2a-lbl)/p\ 1 __f J\ Q9" II ^ )1 if Q-i 0, (q - l^q^ such y 60 /I 9^' if a_i =0, (q — l)q^ such y Case 3: 02 ^0,04 = 0. There are (q — l)q'* y in this case. We must have 60 = 0. This gives us 2/ I ^2^iTr(-2a3b?)/p \ _ J 9^' I ^ ) if tta # 0, (q - 1)^9^ such y 9 ., . . ... if 03 =0, {q — l)q^ such y 123 Case 4: 02 #0,04 # 0. There are (q — \) y in this case. We must have do 62 = bo- This gives us (I4 ^21 ^27rtTr^-2a_i6g-2a3^5g-(-4ai^6g^/p^ <7^' (^) coefficient of 6q is nonzero , (q — l)^g^ such y {q'^'' otherwise , {q — 1)^9^ such y We now compile all our computations to obtain the following final formula. C2l(0,!r=) = (9 - 1)V + (? - 1)V?" + 9" j (9-1)- i;V + 2(9- +2(9 - + (9 - 1)'9V' + (9 - 1)^9=9^' (^)l = 9- (^9='-^ + M - 1) + (^) " + 9 - 1)] • .•\gain we can use our formula for C2/(0,7r^) along with (6.21) to obtain a formula for C2/(l, TT^). We obtain C2.(i, TJ)=i (^9''-^+,•(,-1) - +9 -1) V 124 REFERENCES [Coh47] E. Cohen. Sums of an even number of squares in GF[p", x\. Duke Journal of Mathematics, 14, 1947. [Coh48] E. Cohen. Sums of an odd number of squares in GF[p", x\. Duke Journal of Mathematics, 15, 1948. [Dic66] L. E. Dickson. History of the Theory of Numbers, volume 2. Chelsea Publishing Company, 1966. [EH91] G.VV. Effinger and D.R. Hayes. Additive number theory of polynomials over a finite field. In Oxford Math Monographs. 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