Abstract: Modular functions on a lattice (m(x)+m(y)=m(x∪y)+m(x∩y)) live on modular lattices in that they are induced by modular functions on a quotient modular lattice. Those which identify pairs of the distributive inequality live on distributive lattices in the same sense. The structure of all modular functions on a lattice of finite height is determined. The “distance function” derived by Kranz from a modular function is shown to satisfy the triangle inequality.

Abstract: Absact. The Hilbert modular function field over Q(V2) has generators satisfying modular equations when the arguments are multiplied by factors of norm two. These equations are found by machine use of Fourier series and are further used to show computationally that Weber's ring class field theory for rationals has an illustration of Hecke's type for Q(V2). This has bearing on Hilbert's twelfth problem, the generation of algebraic fields by transcendental functions.

Abstract: Let , where are integers, is some congruence subgroup of , and is a congruence-character of , be the space of all Siegel modular forms of genus , weight and character with respect to . In this paper, for a very broad class of congruence subgroups of , including all those previously investigated and practically all those groups encountered in applications, the author constructs a sufficiently large commutative ring of Hecke operators, acting on , a canonical decomposition (*)and a canonical inner product on . It is shown that the Hecke operators preserve the canonical decomposition (*) and that they are normal with respect to the canonical inner product .Bibliography: 17 titles.

Abstract: On Shimura's correspondence for modular forms of half-integral weight.- Period integrals of cohomology classes which are represented by Eisenstein series.- Wave front sets of representations of Lie groups.- On p-adic representations associated with ?p-extensions.- Dirichlet series for the group GL(n).- Crystalline cohomology, Dieudonne modules and Jacobi sums.- Estimates of coefficients of modular forms and generalized modular relations.- A remark on zeta functions of algebraic number fields.- Derivatives of L-series at s = 0.- Eisenstein series and the Riemann zeta function.- Eisenstein series and the Selberg trace formula I.

Abstract: One proves that the convolution of a modular form of semiintegral weight and genus one with the theta-series of an indefinite quadratic form of signature (3.2) is a Siegel modular form of genus two of an even weight. One establishes the relationship between the zeta-functions of these modular forms.

Abstract: Relations between representations of the Hecke rings of the symplectic and special linear groups on the space of Fourier coefficients of Siegel's modular forms of arbitrary genus are studied. The results are applied to the problem of finding relations between the Fourier coefficients of eigenfunctions of Hecke operators and the corresponding eigenvalues.

Abstract: Let F be a Siegel cusp modular form of genus two and of even weight k > 0. One proves that if the zeta-function ZF(s) of the form F has a pole at the point s=k, then the Fourier coefficients a(N) of this form depend only on the determinant and the divisor of the matrix N.

Abstract: The statement used in proving Theorem 2 of [1] needs explanation. This was pointed out to us by Professor S. Raghwan of Tata Institute of Fundamental Research, Bombay, and we gave the explanation of this in [2]. For the sake of completeness we give here the full proof of the theorem; filling the gap in the proof. We use the same notations and definitions as those of [1]. Also for simplicity of notation we write kmn, gmn and amn to mean km,n, gm,n and a respectively. We need the following lemma.

Abstract: We investigate the arithmetic character of the Fourier coefficients of a Hilbert-Siegel modular form which is an eigenfunction of all Hecke operators. We prove the analytic continuability and a functional equation for the corresponding zeta function.

Abstract: In this chapter we develop the tools needed to describe the subgroup of H1(X(Γ);ℚ/ℤ) corresponding to the cuspidal group C(Γ).

Abstract: We modify and generalize proofs of Täte and Serre in order to show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to T0( N) mod /. We also summarize results for r,(/V). Using these results, we show that an arbitrary prime divides the discriminant of the classical Hecke ring to a power which grows linearly with k. In this way, we find a lower bound for the discriminant of the Hecke ring. After limiting ourselves to cusp forms, we also find an upper bound. Lastly we use the constructive nature of Täte and Serre's result to describe the structure and dimensions of the generalized eigenspaces for the Hecke operators mod /. Introduction. Distinct eigenforms for the Hecke operators in characteristic zero often have congruent ¿/-expansions modulo some prime. In fact, in the case of the full modular group, Atkin, Täte, and Serre proved that whereas there are infinitely many systems of eigenvalues for the Hecke operators in characteristic zero, there are only a finite number mod /. In particular, this implies that the Hecke operators are very nonsemisimple mod /. Täte and Serre's proof has been previously unpublished. In this paper, we modify the proofs of Täte and Serre and generalize some of their results to the case of modular forms on the congruence subgroups T0(N). In particular, we show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to T0(N) mod /. Implicit in our proof is a bound for the number of such systems of eigenvalues. Later in the paper, we suggest a better bound, although our proof is only complete when the level is small. We also summarize the situation for TX(N). Using the above-mentioned results, we obtain a lower bound for the discriminant of the Hecke ring in characteristic zero. In the process, we show that an arbitrary prime / divides the discriminant of the Hecke ring of weight Ac to a power which grows linearly with Ac. After limiting ourselves to cusp forms, we also use the Petersson conjecture to find an upper bound for the discriminant. Lastly, we use the results about the finite number of systems of eigenvalues to derive information about the structure and dimension of the generalized eigenspaces for the Hecke operators mod /. The author wishes to thank John Täte and David Kazhdan for their advice and concern, and especially Barry Mazur for his help and encouragement. Received by the editors July 18, 1979 and, in revised form, February 6, 1981. 1980 Mathematics Subject Classification. Primary 10D12, 10D23. © 1982 American Mathematical Society 0002-9947/81/0000-1034/$0S. 2 5 269 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 270 NAOMI JOCHNOWITZ 1. Background. Let A be a positive integer and let Ak(N) (resp. Sk(N)) be the space of all modular forms (resp. all cusp forms) of weight k for the congruence subgroup r^)=j(^)eSL2(Z)|cE0(mod^)). We refer to Ak(N) (resp. Sk(N)) as the space of modular (resp. cusp) forms of level N and weight k. By the «/-expansion of a modular form /, we mean the ¿/-expansion of/ at infinity. We recall the following formula which gives the action of the Hecke operator T (p a prime not dividing A) on the ¿/-expansion of an element of Ak(N): TP^nQ\"^lanpq\"+pk-^anq\"P. More generally, if aai is any integer relatively prime to N, we have Tm:lanq«»^dk-x2anm/dqnd. n d\\m n For fixed weight Ac, each operator Tm can be expressed as a polynomial in the operators Tp. In this paper we deal exclusively with Hecke operators Tm where m is relatively prime to the level. Fix a prime / not dividing N. Let Mk(N) be the subset of Ak(N) consisting of all forms whose ¿/-coefficients at infinity are rational and /-integral. Definition 1.1. We define the space of modular forms mod / of weight Ac and level N, Mk( N ), to be the F^vector space {/= 2änq\"\\f= 2a„q\" E Mk(N)} C F,[[q]], where the symbol än denotes the reduction of an mod /. Thus, according to this definition, a modular form mod / is identified with its ¿/-expansion. Note that Mk(N) « Mk(N)/lMk(N) *. Mk(N) ® F,. Moreover, using the fact that Mk(N) contains a basis for Ak(N), one easily shows that dim^M^A) = dimc^A). For the remainder of this paper, we assume that / s* 5 unless otherwise specified. We define the space of all modular forms mod / of level N, M( N ), to be the subalgebra of F,[[q]] which is the sum of the Mk(N). This is not a direct sum since it can be shown that Mk(N) C Mk+/_X(N) for all weights k. Conversely, Mk(N) and Mk,(N) have a nontrivial intersection if and only if k = Ac'mod / — 1 ([13] and [15] for level one, [7] for arbitrary level). Definition 1.2. Given / E Mk( N ), define the filtration of /, w( f ), to be the quantity inf{j\\f E M^N)}. Definition 1.3. Wk = (Mk(N) ® F,)/(Mk_l+x ® F,). In other words, Wk is the quotient space of forms of weight k with coefficients in F;, modulo the subvector space of all such forms of lower filtration. The Hecke operators of weight Ac stabilize the set Mk( N ) and thus act on the space Mk(N). The operator T¡ coincides mod / with Atkin's U¡ operator and is therefore License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SYSTEMS OF EIGENVALUES OF MODULAR FORMS 271 denoted by the letter U. Thus in terms of the ¿/-expansion of a modular form mod /, we have the following formula: U: 2>„ I + 1, U decreases the filtration. Proof. By Fact 1.4, w(f\\0'~]) *£ w(f) + I2 — I with equality if and only if w(f) = lmod/. Since/| UV = f-f\\8'~x, we have w(f\\ UV) = w(f\\0'~i). Fact 1.7 concludes the proof. Q.E.D. Corollary 1.10. If k > I + 1, U annihilates the vector space Wk. Corollary 1.11. Ifw(f) = 1 mod /, i7k?aj/| U ^ 0. Remark. The operators 0, U, and V are related by the following exact sequence: 0 -* M(N) ^M(N) ^M(N) ^M(N) -> 0. In particular, Kernel 0 = Image V and Image 0 = Kernel U. Proposition 1.12. If k is sufficiently large, the Hecke operators do not act semisimply on Mk(N). Proof. One easily sees that when Ac is sufficiently large, Mk(N) contains f\\ V for some nonzero form /in the image of 0. But/| Kis annihilated by U2 and not by U. Q.E.D. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 272 NAOMI JOCHNOWITZ In fact, one can show that the above conclusion is true whenever 2/2 if A> 1 or/> 13, 3/2-/ if N= land/ = 11,7, ,3/2 + 3/ if N = 1 and/= 5. 2. The finiteness of the number of systems of eigenvalues mod /. Fix a level A and a prime / such that (/, 6A) = 1. Definition 2.1. {Xp}(p N)=x.pptime is called a system of eigenvalues of level N mod / if there exists a nonzero form / E M( N ) <8> F, such that /1 T = X f for all primes p not dividing A. To simplify notation, we sometimes write {Xp} to denote such a system. The proof of the following theorem is a modification and generalization of a proof by Serre. The original proof was stated in the context of level one, but with a few adjustments extends to arbitrary level. Theorem 2.2. There are only a finite number of systems of eigenvalues of level N mod /. Proof. Let R be the subring of End F/ M( N ) generated by the Hecke operators Tp where p is a prime not dividing A. We use the term Hecke module to mean a module over the ring R. Since the Hecke operators induce an action on the quotient space introduced in Definition 1.3, we can view Wk as a Hecke module. Definition 2.3. Let a be an integer with 1 < a < / — 1 and define Wk[a] to be the Hecke module Wk <8> F, where F is a one-dimensional F, vector space on which T acts as p\". The spaces Wk[a] and Wk are clearly isomorphic as vector spaces. If {Xp} is a system of eigenvalues corresponding to Wk, then {p\"X } corresponds to W¿[a]. We refer to Wk[a] (resp. to {p\"Xp}) as a twist of Wk (resp. of {Xp}). Each space Wk has only a finite number of twists. The following theorem clearly suffices to prove Theorem 2.2. Theorem 2.4. Any quotient Wj is isomorphic as a Hecke module to a twist of some Wk with 0 =s Â: < 2/. Proof. We prove this theorem by showing that if j > 21, Wj is isomorphic as a Hecke module to a twist of some Wm with aai < j. Case l.;z 1 (mod/). Fact 1.4 implies that 0 induces a vector space embedding 0: Wk =» lVk + /+x if I does not divide Ac. Lemma 2.5. If k > I + 1 and I does not divide k, then the above embedding is a vector space isomorphism. Proof. It suffices to show that Wk and Wk+I+X have the same dimension as F, vector spaces. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SYSTEMS OF EIGENVALUES OF MODULAR FORMS 273 Using Theorem 2.23 of [12] one easily calculates Formula 2.6. Let Ac > / + 1, then dimF; wk = àim¥Mk(N) dim¥Mk_l+x(N) = dimc^A(A) dimcAk_l+x(N) = (l-l)(g-l)+(l-l)m/2 + B([k/3]-[(k-l+l)/3]) + C(