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Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ) into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

In this paper, we extend previous results to prove that generalized modular forms with rational Fourier expansions whose divisors are supported only at the cusps and certain other points in the upper half plane are actually classical modular forms. We discuss possible limitations to this extension and pose questions about possible zeroes for modular forms of prime level., Comment: 12 pages

The behaviour of Hecke polynomials modulophas been the subject of some studies. In this paper we show that ifpis a prime, the set of integersNsuch that the Hecke polynomialsTN,χℓ,kfor all primes ℓ, all weightsk≥ 2 and all characters χ taking values in {±1} splits completely modulophas density 0, unconditionally forp= 2 and under the Cohen–Lenstra heuristics forp≥ 3. The method of proof is based on the construction of suitable dihedral modular forms.

In this note we will show how to compute U p acting on spaces of overconvergent p-adic modular forms when X 0 ( p ) has genus 1. We first give a construction of Banach bases for spaces of overconvergent p-adic modular forms, and then give an algorithm to approximate both the characteristic power series of the U p operator and eigenvectors of finite slope for U p , and present some explicit examples. We will also relate this to the conjectures of Clay on the slopes of overconvergent modular forms.

In this paper, we will use techniques of Conrey, Farmer and Wallace to find spaces of modular forms Sk(Γ0(N)) where all of the eigenspaces have Hecke eigenvalues defined over Fp, and give a heuristic that these are all such spaces.

Nous demontrons que les pentes de l'operateur U 5 agissant sur 5-adique formes modulaires surconvergentes de poids k avec caractere de Dirichlet primitif x de conducteur 25 sont {1 4.[8i 5]: i ∈ N} ou {1 4.[8i+4 5]: i ∈ N}. Nous prouvons aussi que l'espace de forms parabolique de poids k et caractere Χ a une base des formes propres pour les operateurs de Hecke Τ p et U 5 definie sur Q 5 (4√5, √3).

We prove that near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of “evenly-spaced” annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

In this article we report on extensive calculations concerning the Gorenstein defect for Hecke algebras of spaces of modular forms of prime weight p at maximal ideals of residue characteristic p such that the attached mod p Galois representation is unramified at p and the Frobenius at p acts by scalars. The results lead us to the ask the question whether the Gorenstein defect and the multplicity of the attached Galois representation are always equal to 2. We review the literature on the failure of the Gorenstein property and multiplicity one, discuss in some detail a very important practical improvement of the modular symbols algorithm over finite fields and include precise statements on the relationship between the Gorenstein defect and the multiplicity of Galois representations. Appendix A: Manual of Magma package HeckeAlgebra, Appendix B: Tables of Hecke algebras., Comment: 52 pages LaTeX, 2 appendices

Let $\tau$ be the primitive Dirichlet character of conductor 4, let $\chi$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $\tau$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $\chi \cdot \tau^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space of classical cusp forms of weight $k$ and character either $\tau$ or $\chi\cdot\tau^k$ split completely over $\qtwo$.

This paper answers a question of Gross and others, by exhibiting specific examples of Hecke algebras where mod 2 multiplicity one fails for some modular forms, and the associated Hecke algebras are not Gorenstein. It shows that the methods of Wiles, Gross, Edixhoven, Buzzard, Mazur and others have an upper bound to their applicability., This replaces the May 2001 version; it has been rewritten, and the exposition has been improved, thanks to the help of William Stein and Kevin Buzzard