We establish an asymptotic formula with a power savings in the error term for traces of CM values of a family of Maass-Poincaré series which contains the modular j-function as a special case. By work of Borcherds [B], Zagier [Za], and Bringmann and Ono [BO], these traces are Fourier coefficients of half-integral weight weakly holomorphic modular forms and Maass forms.

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a "determinant" map from this moduli surface to (Z/NZ)*; its fibres are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modul...

In his striking 1995 paper, Borcherds [2] found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant −d evaluated at the modular j-function. Among a number of powerful generalizations of Borcherds’ work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the e...

For a prime p we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space V of modular forms of weight 2 and level p2. For p ≡ 3 mod 4 we define a special Hecke stable subspace V0 of V which contains the space of modular forms with CM by the ring of integers of Q( √−p) and we describe the calculation of the corresponding Brandt matrices.

For a prime p we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space V of modular forms of weight 2 and level p2. For p ≡ 3 mod 4 we define a special Hecke stable subspace V0 of V which contains the space of modular forms with CM by the ring of integers of Q( √−p) and we describe the calculation of the corresponding Brandt matrices.

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....

Generalizing work of Gross–Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to quadratic spaces over an arbitrary totally real base field. As a special case, we obtain an explicit formula for the n...