Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

Yongquan Hu

doi:10.4171/jems/1078

Abstract

We prove a new upper bound for the dimension of the space of cohomological automorphic forms of fixed level and growing parallel weight on $\mathrm{GL}_2$ over a number field which is not totally real, improving the one obtained by Marshall. The main tool of the proof is the mod $p$ representation theory of $\mathrm{GL}_2(\mathbb{Q}_p)$ as started by Barthel-Livne and Breuil, and developed by Paskunas.

Highlights

DimC Sd(Kf ) ∼ C · ∆(d) for some constant C independent of d
To compare our result with the previous ones, let us restrict to the case when F is imaginary quadratic
In [12], Finis, Grunewald and Tirao proved the bounds d d2 ln d using base change and the trace formula respectively

Summary

Non-commutative Iwasawa algebras

Let G be a p-adic analytic group of dimension d and G0 be an open compact subgroup of G. To see that F ⊗R P has finite length, we may assume B is of type (III), in which case E is a free R-module of rank 4. To prove the injectivity of (3.5), it suffices to prove that HomcKont(Mη∨ , σ∨) is infinite dimensional This is already established in the proof of Proposition 3.11, together with Remark 3.12 and Lemma 3.8 if σ ∈ {Sym0F2, Symp−1F2} up to twist. Let Mη∨ and Eη∨ be as in §3.4; recall that Eη∨ is isomorphic to F[[x, y]] and is identified with Eab (the maximal abelian quotient of E) by the discussion before [27, Lem. 9.2]. [27, Cor. 10.43] states this for i = 1 but the proof works for all i ≥ 1 This implies that TorEi (F, Pπα∨ ) is isomorphic to a finite direct sum of 1∨G since e1(1G, 1G) = 0. This proves that R′ is one of the rings considered there

Key computation

Main results

Findings

Application