Reduction modulo p of cuspidal representations and weights in Serre's conjecture

Abstract

Let O be the ring of integers of a p-adic field and p its maximal ideal. We compute the Jordan-Holder decomposition of the reduction modulo p of the cuspidal representations of GL2(O/p) for e ≥ 1. We also provide an alternative formulation of Serre’s conjecture for Hilbert modular forms. 1. Cuspidal representations and weights 1.1. Cuspidal representations. Let K/Qp be a local field, where p is a prime, and let O be the ring of integers and p its maximal ideal. Let Re = O/pe. In particular, R1 = O/p is the residue field; let q = pf be its cardinality. Let K be the unramified quadratic extension of K, and let O and p be its ring of integers and maximal ideal. The cuspidal complex representations of GL2(Re) are well known (see for instance [PS]) in the case e = 1 and have been constructed for general e, under various names, by several authors; see, for instance, [Shi], [Ger], [How], [Car], [BK], and [Hil]. Aubert, Onn, and Prasad proved ([AOP], Theorem B; note that the notions of cuspidal and strongly cuspidal representations coincide for GL2 by Theorem A) that they are parametrized by Gal(K/K)-orbits of strongly primitive characters ξ : (O/pe)∗ → C∗. A strongly primitive character of (O/p)∗ is one that does not factor through the norm map N : O/p → O/p. See [AOP], 5.2, for the definition of strongly primitive characters for general e. We denote by Θe(ξ) the cuspidal representation of GL2(Re) corresponding to ξ. Fix an isomorphism C ' Qp, and from now on we view ξ and Θe(ξ) as p-adic representations. In this note we compute the Jordan-Holder constituents of Θe(ξ), the reduction mod p of Θe(ξ), and use the notions introduced to reformulate the Serre-type conjecture for Hilbert modular forms of [Sch]. See the last section for some remarks about motivation. The author is very grateful to the referee for comments that improved the exposition, and particularly for an observation that considerably simplified the computations in section 2. 1.2. Brauer characters. Let Θe(ξ) be a cuspidal representation of GL2(Re). The Jordan-Holder constituents of Θe(ξ) are determined by its Brauer character, hence by the values of the character of Θe(ξ) at p-regular conjugacy classes. The p-regular conjugacy classes of GL2(Re) are sent by the natural surjection π : GL2(Re) → GL2(R1) to p-regular conjugacy classes of GL2(R1). Moreover, Date: May 24, 2008. 1