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

JournalsjemsVol. 23, No. 11pp. 3625–3678

Multiplicities of cohomological automorphic forms on $GL_{2}$ and mod $p$ representations of $GL_{2} (Q_{p})$

Yongquan Hu
Morningside Center of Mathematics, Beijing, China

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

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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 $GL_{2}$ over a number field which is not totally real, improving the one obtained in [Ann. of Math. (2) 175 (2012)]. The main tool of the proof is the mod $p$ representation theory of $GL_{2} (Q_{p})$ as started by Barthel–Livné and Breuil, and developed by Paškūnas.

Cite this article

Yongquan Hu, Multiplicities of cohomological automorphic forms on $GL_{2}$ and mod $p$ representations of $GL_{2} (Q_{p})$ . J. Eur. Math. Soc. 23 (2021), no. 11, pp. 3625–3678

DOI 10.4171/JEMS/1078