Classically, definite quadratic forms give rise to theta series,

which are holomorphic modular forms. In 1926, Hecke attached weight one

holomorphic theta series to indefinite quadratic forms of signature (1,

1). This ingenious construction reminds one of the Rankin-Selberg

unfolding method, which appeared 10 years later. In 2003, Bruinier and

Funke introduced the notion of harmonic Maass forms, which have poles at

the cusps and map to classical holomorphic modular forms under a suitable

differential operator. In this talk, we will construct harmonic Maass

forms of weight one that map to Hecke's indefinite theta series, and use

their Fourier coefficients to give a closed expression for the Petersson

norm of $\eta^2$. This is a joint work with Pierre Charollois.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |