Asymptotic values of modular multiplicities for \operatorname{GL}_2

Abstract

We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of the group Res K/Qp GL 2 for K a finite extension of Q p . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

The Monomial Lattice in Modular Symmetric Power Representations

“…(1) r to ind Γ B (χ r 2 ), see, for example, [Roz14,Lemma 2.4]. We generalize this result as follows:…”

Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity

“…By Proposition 2.10 and Corollary 2.11 of [BDJ10], ρ is modular of some weight σ = ⊗ v|p σ v and central character ⊗ v|p (N kv /Fp ) n−1 . The results of [Roz12] show that σ is a Jordan-Holder constituent of the reduction of ⊗ τ Symm k−1 O 2 E for some sufficiently large k, where the tensor product is over embeddings τ : F → E for a sufficiently large number field E, viewed as contained both in C and in Q p . Another application of Proposition 2.10 of [BDJ10] shows that ρ is modular of parallel weight k and level prime to p. Moreover the presence of a good dihedral prime q allows us to use an indefinite quaternion algebra of discriminant dividing q and hence assume the open compact subgroup has level dividing n in the first application of Proposition 2.10 of [BDJ10].…”

Crystalline lifts of two-dimensional $\mathrm{mod} \: p$ automorphic Galois representations

“…There are two main ingredients to the proof of Theorem 1.1. The first of these is Proposition 2.1 below, which is a statement purely about mod p representations of GL 2 (F) where F is a finite field of characteristic p. The result can be deduced from the main result of [12], but we give instead a short self-contained proof that could be useful if one wishes to extract an explicit value of n 0 in the conclusion of Theorem 1.1.…”