Harmonic Maass forms associated to real quadratic field

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.