Number Theory Seminar

Classical holomorphic modular forms are number-theoretic objects that have been intensely studied. The split exceptional group G_2 does not support a theory of holomorphic modular forms, but it does possess so-called quaternionic modular forms. These are a special class of automorphic forms that appear to behave similarly to holomorphic modular forms. In the talk, I will describe a theory of modular forms of half-integral weight on G_2 and other exceptional groups. In particular, we prove the existence of a modular form of weight 1/2 on G_2 whose Fourier coefficients are related to the 2-torsion in the narrow class groups of totally real cubic fields. This is joint work with Spencer Leslie.