##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## Aaron Pollack

#### UCSD

## A Cohen-Zagier modular form on $G_2$

##### Abstract:

I will report on joint work with Spencer Leslie where we define an analogue of the Cohen-Zagier Eisenstein series to the exceptional group $G_2$. Recall that the Cohen-Zagier Eisenstein series is a weight $3/2$ modular form whose Fourier coefficients see the class numbers of imaginary quadratic fields. We define a particular modular form of weight $1/2$ on $G_2$, and prove that its Fourier coefficients see (certain torsors for) the 2-torsion in the narrow class groups of totally real cubic fields. In particular:

1) we define a notion of modular forms of half-integral weight on certain exceptional groups,

2) we prove that these modular forms have a nice theory of Fourier coefficients, and

3) we partially compute the Fourier coefficients of a particular nice example on $G_2$.

### February 17, 2022

### 2:00 PM

Pre-talk at 1:20 PM

APM 6402 and Zoom;

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

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