Counting Discrete, Level$1$, Quaternionic Automorphic Representations on $G_2$
Abstract
Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm{GL}_2$. Here, we use "hyperendoscopy" techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$, eventually getting an analog of the EichlerSelberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard, and Taïbi to compute dimensions of spaces of level$1$ quaternionic representations. On the way, we prove a JacquetLanglandsstyle result describing them in terms of classical modular forms and automorphic representations on the compactatinfinity form $G_2^c$. The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being "regular". A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series. We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discreteatinfinity automorphic representations on arbitrary reductive groups instead of just classical ones.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.09313
 arXiv:
 arXiv:2106.09313
 Bibcode:
 2021arXiv210609313D
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory;
 11F55 (primary) 11F70;
 11F72;
 11F75;
 20G41;
 22E50;
 22E55 (secondary)
 EPrint:
 28 pages, 1 figure, 1 table This version: corrected a typo in the formula for the exact weights of modular forms in the interpretation of corollary 8.2.1