Higher Siegel theta lifts on Lorentzian lattices, harmonic Maass forms, and Eichler–Selberg type relations

Theta-lifts of Maass Waveforms

Let O be an arbitrary order in an indeenite quaternion division algebra over Q. If O 1 is the group of elements in O with norm equal to 1, and H the complex upper half-plane, then X O := O 1 nH is a compact Riemann surface. Furthermore, let ? 0 (d) SL 2 (Z) be the Hecke congruence group of level d. Then X d := ? 0 (d)nH is a non-compact Riemann surface with nite volume. Let be the hyperbolic La...

Mock Theta Functions, Ranks, and Maass Forms

Algebraicity of Harmonic Maass Forms

In 1947 D. H. Lehmer conjectured that Ramanujan’s tau-function never vanishes. In the 1980s, B. Gross and D. Zagier proved a deep formula expressing the central derivative of suitable Hasse-Weil L-functions in terms of the Neron-Tate height of a Heegner point. This expository article describes recent work (with J. H. Bruinier and R. Rhoades) which reformulates both topics in terms of the algebr...

Coefficients of Harmonic Maass Forms

Harmonic Maass forms have recently been related to many different topics in number theory: Ramanujan’s mock theta functions, Dyson’s rank generating functions, Borcherds products, and central values and derivatives of quadratic twists of modular L-functions. Motivated by these connections, we obtain exact formulas for the coefficients of harmonic Maass forms of non-positive weight, and we obtai...

Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms

This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and q-series, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular L-functions, generalized Gross-Zagier formulae, to name a few....