Kevin Buzzard — Page 2

arXiv2003

Slopes of overconvergent 2-adic modular forms

The slope of a p-adic overconvergent eigenform of weight k is the p-adic valuation of its U_p eigenvalue. We find the slope of all 2-adic finite slope overconvergent eigenforms of tame level 1 and weight 0. As a consequence we prove that any finite slope 2-adic overconvergent eigenform of tame level 1 and weight 0 has coefficients in Q_2. These results provide evidence towards conjectures of the first author that predict the slopes of classical p-adic modular forms under a mild ``Gamma_0(N)''-regular hypothesis on p.

math.NT↗

arXiv1999

Companion forms and weight one forms

In this paper we prove the following theorem. Let L/\Q_p be a finite extension with ring of integers O_L and maximal ideal lambda. Theorem 1. Suppose that p >= 5. Suppose also that ρ:G_\Q -> GL_2(O_L) is a continuous representation satisfying the following conditions. 1. ρramifies at only finitely many primes. 2. ρmod λis modular and absolutely irreducible. 3. ρis unramified at p and ρ(Frob_p) has eigenvalues αand βwith distinct reductions modulo λ. Then there exists a classical weight one eigenform f = \sum_{n=1}^\infty a_m(f) q^m and an embedding of \Q(a_m(f)) into L such that for almost all primes q, a_q(f)=tr(ρ(\Frob_q)). In particular ρhas finite image and for any embedding i of L in \C, the Artin L-function L(i o ρ, s) is entire.

math.NT↗