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

Yongquan Hu Morningside Center of Mathematics, Academy of Mathematics and Systems Science, University of the Chinese Academy of Sciences

Number Theory mathscidoc:1802.24001

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.
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@inproceedings{yongquanmultiplicities,
  title={Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$},
  author={Yongquan Hu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180208110840331107914},
}
Yongquan Hu. Multiplicities of cohomological automorphic forms on $\mathrm{GL}_2$ and mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180208110840331107914.
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