##### Department of Mathematics,

University of California San Diego

****************************

### Math 209: Number Theory Seminar

## Finley McGlade

#### UCSD

## A Level 1 Maass Spezialschar for Modular Forms on $\mathrm{SO}_8$

##### Abstract:

The classical Spezialschar is the subspace of the space of holomorphic modular forms on $\mathrm{Sp}_4(\mathbb{Z})$ whose Fourier coefficients satisfy a particular system of linear equations. An equivalent characterization of the Spezialschar can be obtained by combining work of Maass, Andrianov, and Zagier, whose work identifies the Spezialschar in terms of a theta-lift from $\widetilde{\mathrm{SL}_2}$. Inspired by work of Gan-Gross-Savin, Weissman and Pollack have developed a theory of modular forms on the split adjoint group of type D_4. In this setting we describe an analogue of the classical Spezialschar, in which Fourier coefficients are used to characterize those modular forms which arise as theta lifts from holomorphic forms on $\mathrm{Sp}_4(\mathbb{Z})$.

### November 30, 2023

### 2:00 PM

APM 7218 and Zoom; see https://www.math.ucsd.edu/~nts

****************************