L 2 harmonic forms and finiteness of ends

Heegner Divisors, L-functions and Harmonic Weak Maass Forms

Recent works, mostly related to Ramanujan’s mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as “generating functions” for central values and derivatives of quadratic twists of weight 2 modular L-functions. To obtain these results, we cons...

Hyper - Kähler Calabi Metrics , L 2 Harmonic Forms , Resolved M 2 - branes , and AdS 4 / CFT 3 Correspondence

1 L 2 cohomology of Manifolds with flat ends

We give a topological interpretation of the space of L2-harmonic forms on Manifold with flat ends. It is an answer to an old question of J. Dodziuk. We also give a Chern-Gauss-Bonnet formula for the L2-Euler characteristic of some of these Manifolds. These results are applications of general theorems on complete Riemannian Manifold whose Gauss-Bonnet operator is nonparabolic at infinity. Résumé...

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....

Binary Forms , Equiangularpolygons and Harmonic

Consider binary forms F having complex coeecients and discriminant D F 6 = 0. In a sequence of previous papers, the rst author studied the area A F of the planar region jF(x;y)j 1 deened by forms F of this type in connection with the enumeration of integer lattice points. In particular, the rst author showed that the GL 2 (R)-invariant quantity jD F j 1=n(n?1) A F is uniformly bounded when the ...