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.

We show that the slopes of the U5 operator acting on slopes of 5adic overconvergent modular forms of weight k with primitive Dirichlet character χ of conductor 25 are given by eitherdepending on k and χ.

Let p be a prime, and let S 2 ðG 0 ðpÞÞ be the space of cusp forms of level G 0 ðpÞ and weight 2. We prove that, for p 2 f431; 503; 2089g, there exists a non-Eisenstein maximal ideal m of the Hecke algebra of S 2 ðG 0 ðpÞÞ above 2, such that ðT q Þ m is not Gorenstein. # 2002 Elsevier Science (USA)

Abstract. Let τ be the primitive Dirichlet character of conductor 4, let χ be the primitive even Dirichlet character of conductor 8 and let k be an integer. We show that the U 2 operator acting on cuspidal overconvergent modular forms of weight 2k − 1 and character τ 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 χ · τ k has slopes in the arithmetic progression {1, 2, . . . , n, . . . }.We also 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 τ or χ · τ k split completely over Q 2 .

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.MSC Classification: 11F80 (primary), 11F33, 11F25 (secondary).