Number/Representation Theory

Classical holomorphic modular forms on $\mathrm{SL}{}_2$ play a central role in modern number theory, and their analogues on many higher-rank groups have been extensively studied. However, many groups—including all but two exceptional Lie groups—do not admit holomorphic modular forms, motivating the search for a replacement. Quaternionic modular forms (QMFs) give a promising alternative. Gan–Gross–Savin studied QMFs on the split exceptional group $G{}_2$, and more recently, a theory of Fourier expansions has been developed by Aaron Pollack for quaternionic exceptional groups and by Hilado–McGlade–Yan for unitary groups $U(2,n)$. In this talk, I will present what is known and what is still unknown about QMFs, including dimension formulas and the arithmeticity(rationality/algebraicity) of cusp forms and certain Eisenstein series.