P-PJ)IC FROPH~T!ES OF MODULA~ SCHEMJ~S AND MODUI~ FORMS Nicholas M. Kat% International Summer School on Modular Functions ANTWERP 1972 70 Ka-2 TABLE OF CONTENTS Introduction 73 Chapter i" Moduli schemes and the q-expansion principle 77 i.i Modular forms of level i 1.2 Modular forms of level n i.~ Modular forms on Fo(P) l.h The modular schemes M and M n n 1.5 The invertible sheaf ~ on ~, and modular forms holomorphic at 1.6 The q-expansion principle 1.7 Base-change for modular forms of level n > 3 1.3 Base-change for modular forms of level i and 2 1.9 Modular forms of level i and 2: q-expansion principle i .I0 Modular schemes of level i and 2 i.ii Hecke operators i .12 Applications to polynomial q-expansions; the strong q-expansion principle 1.13 review of the modular scheme associated to Fo(P)__ Chapter 2: p-adlc modular forms 97 2.0 The Hasse invariant as a modular form; its q-expansion 2.1 Deligne's congruence A ~ Ep_ I mod p 2.2 p-adic modular forms with growth conditions 2. 3 Determination of M(R0,r,n,k) when p is nilpotent in R 0 2.4 Determination of S(R0,r,n,k) when p is nilpotent in R 0 2.5 Determination of S(Ro,r,n,k) in the limit 2.6 Determination of a "basis" of S(Ro,r,n,k) in the limit 2.7 Banach norm and q-expansion for r = i 2.9 Bases for level i and 2 2. 9 Interpretation via formal schemes Chapter 3: Existence of the canonical subgroup: applications i12 3-1 The existence theorem : statement 3.2 First principal corollary 3"3 Second principal corollary 3.4 Construction of the canononicsl subgroup in the case r = i 3-5 Hint for general r 3.6 Lemmas on the formal group 71 Ka- 3 3.7 Construction of the canonical subgroup as a subscheme of the formal group 3.8 The canonical subscheme is a subgroup 3-9 Conclusion of the proof of 3-1 3.10 Finiteness properties of the Frobenius endomorphism of p-adic modular functions 3.11 Applications to the congruences of Atkin - the U operator 3.12 p-adic Hecke operators 3-13 Interpretation of Atkin's congruences on j Chapter 4: p-adic representations and congruences for modular forms 142 4.1 p-adic representations and locally free sheaves 4.2 ApplicAtions to modular schemes 4.3 Igusa's theorem 4.4 Applications to congruences between modular forms ~ la Serre 4.5 Applications to Serre~s "modular forms of weight M" Appendix i: Motivations 158 AI. i Lattices and elliptic curves g la Weierstrass ; the Tate curve A1.2 Modular forms and De Rham cohomology A1.3 The Gauss-Manin connection, and the function P: computations AI.4 The Gauss-Manin connection and Serre's 8 operator Ai.5 Numerical Formulae Appendix 2 : Frobenius 175 A2.1 Relation of the De Rham and p-adic modular Frobenii A2.2 Calculation at A2. 3 The "canonic~l directlon in A2.4 P as a p-adic modular function of weight two Appendix 3: Hecke Polynomials, coherent cohomology, and U 181 A3.1 The Fredholmdeterminant of U A3.2 Relation to mod p 4tale cohomology and to coherent cohomology A3.3 Relation to the Cartier operator Ka-4 72 List of Notations i.o %/s 1.1 Tate(q), ~can' S(R0'I'k) 1.2 nE,(~n;S(R0,n,k ) 1.3 rO(O) 1.4 Mn, % 1.9 S(K,n,k) i .ii T~ 2.0 A 2 .i Ep.I,E k 2.2 M(R0r,n~k), S(Ro,r,n,k ) 2.6 B(n,k,j), B(R0,n,k,j), Brigid(R0,r,n,k) 2.8 P, Pl(projectors ) 2.9 Mn(R0,r), Mn(R0 ,r ) 3 .i H,Y 3.3 m 3.4 F,V 3.11 tr (p, U 4.1 Wn(k), ~, S n m m ,11- 4.4 G X , R_amanujan's series P AI.I AI.2 AI.3 V, Weierstrass's A~.4 e,~ A2.~ F(~) A2.2 ~ can ' qcan 73 Ka- 5 (In l) Introduction This expose represents an attempt to understand some of the recent work of Atkln~ Swinnerton-Dyer, and Serre on the congruence properties of the q-expansion coefficients of modular forms from the point of view Of the theory of moduli of elliptic curves, aS developed abstractly by Igusa and recently reconsidered by Dellgne. In this optic, a modular form of weight k and level ®k n becomes a section of a certain line bundle ~ on the modular variety M - n which "classifies" elliptic curves with level n structure (the level n structure is introduced for purely technical reasons). The modular variety M n is a smooth curve over Z[1/n], whose "physical appearance" is the same whether we view it over C (where it becomes ~(n) copies of the quotient of the upper half plane by the principal congruence subgroup l~n) of SL(2,Z)) or over the algebraic closure of Z/p~ ~ (by "reduction modulo p") for primes p not dividing n. This very fact rules out the possibility of obtaining p-adic properties of modular forms simply by studying the geometry of M n ® ~pE and its llne bundles ®k w ; we can only obtain the reductions modulo p of identical relations which hold over C . The key is instead to isolate the finite set of points of M n ® ~/pZ corresponding to supersingular elliptic curves in characteristic p, those whose Hasse invarlant vanishes. One then considers various "rigid-analytic" open subsets of M ® ~ defined by removing p-adic discs of various radii around n p the supersingular points in characteristic p. This makes sense because the Hasse invariant is the reduction modulo p of a true modular form (namely Ep_l) over %, so we can define a rigid analytic open subset of M n ® Z P by taking only those p-adic elliptic curves on which Ep_ 1 has p-adic absolute value greater than some ¢ > 0. We may then define various sorts of truly p-adic modular forms as functions of elliptic curves on which IEp_ll > ~, or equivalent- ly as sections of the line bundles ~k restricted to the above-constructed Ka- 6 74 (In 2) rigid analytic open sets of M ® ~ [The role of the choice of s is to n p specify the rate of growth of the coefficients of the Laurent series development around the "missing" supersingular points]. The most important tool in the study of these p-adic modular forms is the endomorphism they undergo by a "canonical lifting of the Frobenius endomorphism" from characteristic p. This endomorphism comes about as follows. Any elliptic curve on which IEp.ll > E for suitable ~ carries a "canonical subgroup" of order p, whose reduction modulo p is the Kernel of Frobenius. The "canonical lifting" above is the endomorphism obtained by dividing the universal elliptic curve by its canonical subgroup (over the rigid open set of M ®~ where it exists). n p This endomorphism is related closely to Atkin's work. His operator U is simply (~i times) the trace of the canonical lifting of Frobenius, and certain of his results on the q-expansion of the function j may be interpreted as statements about the spectral theory of the operator U. The relation to the work of Swinnerton-Dyer and Serre is more subtle, and depends on the fact that the data of the action of the "canonical lifting of Frobenius" on _-i over the rigid open set tEp°ll ~ i is equivalent to the knowledge of the representation of the fundamental group of the open set of M ® Z~ where the Hasse invariant is invertible on the p-adic Tate module n Tp (which for a non-supersingular curve in characteristic p is a free Z -module of rank one). Thanks to Igusa, we know that this representation is P as non-trivial as possibl% ~nd this fact, interpreted in terms of the action ®k of the canonical Frobenius on the ~ , leads to certain of the congruences of Swinnerton-Dyer and Serre. In the first chapter, we review without proof certain aspects of the moduli of elliptic curves, and deduce various forms of the "q-expansion principle." This chapter owes much (probably its very existence) to discussions with Deligne. It is not"p-adic"~ and may be read more or less independently 75 Ka-7 (In 3) of the rest of the paper. The gecond chapter develops at length various "p-adic" notions of modular form~ in the spirit described above. A large part of it (r ~ 1 ) was included with an eye to Dwork-style applications to Atkin's work, and may be omitted by the reader interested only in Swinnerton-Dyer and $erre style congruences. The idea of working 8t such "p-adic modular forms" is due entirely to $erre, who in his 1972 College de France course stressed their importance. The third chapter develops the theory of the "canonical subgroup." This theory is due entirely to Lubin, who has unfortunately not published it except for a tiny hint [33]. The second half of the chapter interprets certain congruences of Atkin in terms of p-adic Banach spaces, the spectrum of the operator U, etc. The possibility of this interpretation is due to Dwork.