Modular forms (mod \(p\)) and Galois representations. (English) Zbl 0978.11018

Gross, Benedict H.

Int. Math. Res. Not. 1998, No. 16, 865-875 (1998).

This article is a continuation of the author’s paper [Isr. J. Math. 113, 61-93 (1999; Zbl 0965.11020)]. Its aim is to review the theory of modular forms (mod \(p\)) on reductive groups \(G\) over \(\mathbb{Q}\) as described in the author’s paper [loc. cit., §9]. In section 1 he formulates the basic conjecture concerning the existence of Galois representations \(\rho\) associated to simple submodules \(N\) of modular forms and proves that the kernel of \(\rho\) is uniquely determined by \(N\). After discussing local Galois representations in section 3 the author deals with ordinary modular forms (mod \(p\)) in section 4 predicting the restriction of \(\rho\) to the decomposition group at \(p\) in case the reductive group \(G\) is simply connected and the simple module \(N\) is ordinary at \(p\). Sections 2 and 5 present an example.

Reviewer: N.Klingen (Köln)

MSC:

11F33	Congruences for modular and \(p\)-adic modular forms
11F80	Galois representations

Keywords:

modular forms; Galois representations; Hecke algebra; reductive algebraic group; arithmetic subgroup

Citations:

Zbl 0965.11020