##### Department of Mathematics,

University of California San Diego

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### Math 209: Number Theory Seminar

## Aaron Pollack

#### UC San Diego

## Arithmeticity of quaternionic modular forms on G_2

##### Abstract:

Arithmeticity of quaternionic modular forms on G_2 Abstract: Quaternionic modular forms (QMFs) on the split exceptional group G_2 are a special class of automorphic functions on this group, whose origin goes back to work of Gross-Wallach and Gan-Gross-Savin. While the group G_2 does not possess any holomorphic modular forms, the quaternionic modular forms seem to be able to be a good substitute. In particular, QMFs on G_2 possess a semi-classical Fourier expansion and Fourier coefficients, just like holomorphic modular forms on Shimura varieties. I will explain the proof that the cuspidal QMFs of even weight at least 6 admit an arithmetic structure: there is a basis of the space of all such cusp forms, for which every Fourier coefficient of every element of this basis lies in the cyclotomic extension of Q.

### October 5, 2023

### 2:00 PM

APM 6402 and Zoom

See https://www.math.ucsd.edu/~nts

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