Slopes of overconvergent 2 - adic modular forms . Kevin Buzzard

Let p be a prime, and let N be a positive integer coprime to p. Let Mk(Γ1(N);Qp) denote the weight k modular forms of level Γ1(N) defined overQp. In recent years, work of Coleman and others (for example [5],[6],[7],[8],[9]) has shown that a very profitable way of studying this finite-dimensional Qp-vector space is to choose a small positive rational number r and then to embed Mk(Γ1(N);Qp) into a (typically infinite-dimensional) p-adic Banach space Mk(Γ1(N);Qp; p ) of p-overconvergent p-adic modular forms, that is, sections of ω on the affinoid subdomain of X1(N) obtained by removing certain open discs of radius p −r above each supersingular point in characteristic p (at least if N ≥ 5; see the appendix for how to deal with the cases N ≤ 4). The space Mk(Γ1(N);Qp; p ), for 0 < r < p/(p+ 1), comes equipped with canonical continuous Hecke operators, and one of them, namely the operator U := Up, has the property of being compact; in particular U has a spectral theory. Coleman exploited this theory in [5] to prove weak versions of conjectures of Gouvêa and Mazur on families of modular forms. One of us (K.B.) has made, in many cases, considerably more precise conjectures ([3]) than those of Gouvêa and Mazur, predicting the slopes of U , that is, the valuations of all the non-zero eigenvalues of U . These conjectures are very explicit, and display a hitherto unexpected regularity. However, they have the disadvantage of being rather inelegant. See also the forthcoming PhD thesis [13] of Graham Herrick, who has, perhaps, more conceptual conjectures about these slopes. We present here a very concrete conjecture in the case N = 1 and p = 2, which presumably agrees with the conjectures in [3] but which has the advantage of being much easier to understand and compute. Let Sk := Sk(Γ0(1),Q) denote the level 1 cusp forms of weight k. If F (X) is a polynomial with rational coefficients then by its 2-adic Newton polygon we mean its Newton polygon when considered as a polynomial with 2-adic coefficients.