A Cohen-Zagier modular form on G_2

Abstract: I will report on joint work-in-progress 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 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.

number theory

Audience: researchers in the topic

UCLA Number Theory Seminar