Abstract: where G is the Galois group GaI ( I ) /Q) and F is a finite field of characteristic I > 3. Suppose that p is modular of level N, i.e., that it arises from a weight-2 newform of level dividing N and trivial "Nebentypus character." Then p is an odd representation: the matrix p(c) (where c is a complex conjugation in G) has eigenvalues + 1, 1. Since + 1 and 1 are distinct in F, p is absolutely irreducible and has a model over every subfield of F containing the set trace(p ). We assume that F has been chosen so that it is generated by this set. Assume that p is a prime which exactly divides N (we write p I[ N), and restrict p to a decomposition group of G for the prime p. View the restriction as a representation p p of Gal(t)p/Qp). We say that p is finite at p ([34] or [35], w if there is a finite flat F-vector space scheme H over Zp for which the action of Gal( t )p/Qp) on the F-vector space H(1)p) gives pp. ( I f / ~ p,/9 is finite at p if and only if pp is unramified.) It is clear that p is finite at p whenever p is already modular of level N/p. In w of [34] (cf. [35]), Serre conjectured the converse, i.e., that if p is finite, then p is modular of level N/p. Soon thereafter, Mazur 1-20] proved this conjecture in the case

Abstract: Introduction 1. Modular forms 2. Invariant means on L (Sn) 3. Ramanujan graphs 4. Bounds for Fourier coefficients of 1/2-integral weight Bibliogrpahy Index.

Abstract: Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq) to the space of classical holomorphic Siegel modular forms of weight (p +q)/2 and genusn (resp. holomorphic hermitian modular forms of weightp +q and genusn). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of η in terms of periods of η over certain totally geodesic cycles—generalizing Shintani’s solution [21] of a conjecture of Shimura. We then choose η to be the Poincare dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in theholomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher’s Theorem we may now conclude that all such coefficients are zero.

Abstract: It is well-known (see Fueter [8] and Aigner [1]) that Fermat’s equation X+Y 3 = 1 has no solutions in a quadratic field K = Q( √ D) ( for |D| ≡ 1 (mod 3)), provided the class number of K is not divisible by 3. This subject was further investigated by Frey [6], who gave estimates for the 3-rank of the class group of K in terms of 3-descent on a curve Y 2 = X +D. In this paper we consider curves ED : DY 2 = 4X − 27 ,which are quadratic twists of the Fermat’s curve X +Y 3 = 1. We give a precise formula for the rank of the Selmer group corresponding to the complex multiplication √ −3 : ED −→ E−3D in terms of the 3-rank of the class group of Q( √ D) resp. Q( √ −3D). This result may be considered as a quantitative version of [1] and [8]. We discuss also Birch and Swinnerton-Dyer’s conjecture for curves ED. According to a theorem of Waldspurger [22], [23], natural rational factor of L(ED/Q, 1) may be expressed in terms of coefficients of certain modular forms of weight 3/2. We identify these forms explicitely and verify (mod 3)-part of Birch and SwinnertonDyer’s conjecture for ED in most cases (for D of density 5/6 ).

Abstract: In this paper we give a negative answer to a question of W. Harvey about the arithmeticity of Teichmuller modular forms and we prove a number of results about the nonarithmeticity of subnormal subgroups of modular groups. In the Appendix we announce and discuss a theorem according to which the appropriately defined rank of Teichmuller modular groups is equal to 1.

Abstract: Let ρ(z) denote the Poincare metric on C – {0,1}. It is known that where λ(τ) is the elliptic modular function We define the function We obtain the following mequalitics: , where 0≦ x ≦ 1 by using the above inequalities and Poincare metric. we get a new version of Schottky' theorem.

Abstract: My purpose is to give a succinct and readable account of recent developments in the theory of modular integrals (with associated rational period functions) and the Mellin transforms of these. As I have already given one such exposition [13] which emphasizes the rational period functions on the modular group T(l) at the expense of the modular integrals (and their Mellin transforms), the present article will redress the balance, dealing mainly with the latter and putting aside discussion of rational period functions per se, whenever possible. Unavoidably there is a good bit of overlap between the present note and [13], to which it should be regarded as supplementary. Proofs are - or are to be - given elsewhere [4,10,11,12]. All of the results presented here can be generalized to accommodate (reasonably) general multiplier systems, but for the sake of clarity we restrict attention to multiplier system identically one.

Abstract: This paper is concerned with computing the L-functions of the title in terms of modular forms. Because one can produce zeros of these L-functions, they seem to provide interesting test cases for various conjectures relating L-functions to cycles. In the first three sections we state the theorem, review some consequences, and give a brief sketch of the proof. This proof is based on recent results of Katz and Mazur, so in sections 4 through 6 we establish notation by reviewing the relevant parts of their work. In sections 7 and 8 we prove an Eichler-Shimura style result on a Hecke operator. The proof proper occupies sections 9 through 12. The paper [11] provides an introduction to the universal curves over Igusa curves.

Abstract: In this paper, we will deal with a very practical problem The problem is that of finding the lowest level of a given modular form which is known to have some level For forms on principal (homogeneous) congruence subgroups, the answer is the greatest common divisor of the widths at all cusps [6, p 82] This is often not an easy calculation to make unless the transformation formulas are already known However, frequently another congruence group appears: namely the Γ0 type group Here, the calculation of finding the minimal level turns out to be much easier and involves just the inversion formula Since it is often the case that if one non-trivial transformation is known for the form in question, it is the inversion formula, this will be very convenient This is the subject of Theorem 1

Abstract: In this paper we demonstrate the meromorphic continuation and functional equation for Koecher-Maas-series attached to modular forms of quaternions. The Koecher-Maas-series associated with modular forms in the Maas space, which are simultaneous eigenforms under all Hecke operators, possess Euler product expansions.

Abstract: In this paper we follow our investigation about the construction of alternating differential forms on the Siegel's modular variety. Let A~ represent the quotient of the Siegel's upper half-space Hg by the full integral symplectic group Sp(29, Z) = Fg. It is a well known fact that Ag is a quasi projective algebraic variety. We shall denote by ,4g its Satake's compactification. It is a normal projective, but singular variety. We shall denote by A0 ~ the regular part of Ag and by/T 0 a desingularization of Ag; without loss of generality we can assume Ao ~ contained in Ao" For any complex manifold M we shall denote by Or(M) the sections of the sheaf of alternating holomorphic differential forms on M of degree v. In [3] it has been shown that for g > 2 the injection

Abstract: — Let / b e an elliptic modular form level of N. We present a criterion for the integrality of/at primes not dividing N. The result is in terms of the values at CM points of the forms obtained applying to / the iterates of the MaaB differential operators.

Abstract: Cohomology of arithmetic groups, automorphic forms and L-functions.- Limit multiplicities in L 2(??G).- Generalized modular symbols.- On Yoshida's theta lift.- Some results on the Eisenstein cohomology of arithmetic subgroups of GL n .- Period invariants of Hilbert modular forms, I: Trilinear differential operators and L-functions.- An effective finiteness theorem for ball lattices.- Unitary representations with nonzero multiplicities in L2(??G).- Signature des varietes modulaires de Hilbert et representations diedrales.- The Riemann-Hodge period relation for Hilbert modular forms of weight 2.- Modular symbols and the Steinberg representation.- Lefschetz numbers for arithmetic groups.- Boundary contributions to Lefschetz numbers for arithmetic groups I.- Embedding of Flensted-Jensen modules in L 2(??G) in the noncompact case.

Abstract: §12.0. In the last three chapters we developed a representation theory of arbitrary Kac–Moody algebras. From now on we turn to the special case of affine algebras. We show that the denominator identity (10.4.4) for affine algebras is nothing else but the celebrated Macdon aid identities. Historically this was the first application of the representation theory of Kac–Moody algebras. The basic idea is very simple: one gets an interesting identity by computing the character of an integrable representation in two different ways and equating the results. In particular, Macdonald identities are deduced via the trivial representation. Furthermore, we show that specializations (11.11.5) of the denominator identity turn into identities for q -series of modular forms, the simplest ones being Macdonald identities for the powers of the Dedekind η-function. We study the structure of the weight system of an integrable highest-weight module over an affine algebra in more detail. This allows us to write its character in a different form to obtain the important theta function identity. This identity involves classical theta functions and certain modular forms called string functions, which are, essentially, generating functions for multiplicities of weights in “strings.” Furthermore, we consider branching functions, which are a generalization of string functions when instead of the Cartan subalgebra a general reductive subalgebra is considered. Finally, we introduce one of the most powerful tools of conformal field theory, the Sugawara construction and the coset construction, in relation to the study of general branching rules.