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 $g_2$.