On the eigenvalues of Hecke operators

Proceedings of The National Academy of Sciences, 1982

E. Hecike initiated the application of representation theory to the study of cusp forms. He showed that, for p a prime congruent to 3 mod 4, the difference ofmultiplicities ofcertain conjugate representations of SL4(F) on cusp forms of degree 1, level p, and weight 22 is given by the class number h(-p) of the field Q(Vj;). We apply the holomorphic Lefschetz theorem to actions on the Igusa compactification ofthe Siegel moduli space of degree 2 to compute the values of characters of the representations of SpN(FV) on certain spaces of cusp forms of degree 2 and level p at parabolic elements ofthis group. Our results imply that here too, the difference in multiplicities of conjugate representations of Sp4(F,) is a multiple of h(-p).

Journal of the London Mathematical Society, 2003

Journal of Algebra, 1990

2016

Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482

2010

The behaviour of Hecke polynomials modulo p has been the subject of some study. In this note we show that, if p is a prime, the set of integers N such that the Hecke polynomials T N,χ ℓ,k for all primes ℓ, all weights k ≥ 2 and all characters χ taking values in {±1} splits completely modulo p has density 0, unconditionally for p = 2 and under the Cohen-Lenstra heuristics for p ≥ 3. The method of proof is based on the construction of suitable dihedral modular forms.

Inventiones Mathematicae, 1982

Publications mathématiques de l'IHÉS, 1967

Journal of Pure and Applied Algebra, 2005

We establish and comment on a surprising relationship between the behaviour modulo a prime p of the number s n (G) of index n subgroups in a group G, and that of the corresponding subgroup numbers for a subnormal subgroup of p-power index in G. One of the applications of this result presented here concerns the explicit determination modulo p of s n (G) in the case when G is the fundamental group of a finite graph of finite p-groups. As another application, we extend one of the main results of the second author's paper [16] concerning the p-patterns of free powers G * q of a finite group G with q a p-power to groups of the more general form H * G * q , where H is any finite p-group.

St. Petersburg Mathematical Journal, 2015

In this paper we study congruence properties of the representations Uα := U P SL(2,Z) χα of the projective modular group PSL(2, Z) induced from a family χα of characters for the Hecke congruence subgroup Γ 0 (4) basically introduced by A. Selberg. Interest in the representations Uα stems from their appearance in the transfer operator approach to Selberg's zeta function for this Fuchsian group and character χα. Hence the location of the nontrivial zeros of this function and therefore also the spectral properties of the corresponding automorphic Laplace-Beltrami operator ∆ Γ,χα are closely related to their congruence properties. Even if as expected these properties of Uα are easily shown to be equivalent to the ones well known for the characters χα, surprisingly, both the congruence and the noncongruence groups determined by their kernels are quite different: those determined by χα are character groups of type I of the group Γ 0 (4), whereas those determined by Uα are such character groups of Γ(4). Furthermore, contrary to infinitely many of the groups ker χα, whose noncongruence properties follow simply from Zograf's geometric method together with Selberg's lower bound for the lowest nonvanishing eigenvalue of the automorphic Laplacian, such arguments do not apply to the groups ker Uα, the reason being, that they can have arbitrary genus g ≥ 0, contrary to the groups ker χα, which all have genus g = 0.

Linear Algebra and its Applications, 2013

We prove a conjecture by W. Bergweiler and A. Eremenko on the traces of elements of modular group in this paper.