Modular forms from Noether–Lefschetz theory

Analytic Theory of Modular Forms

Modular Forms and Applications in Number Theory

Modular forms are complex analytic objects, but they also have many intimate connections with number theory. This paper introduces some of the basic results on modular forms, and explores some of their uses in number theory.

Modular Forms and Algebraic K-theory

In this paper, which follows closely the talk given at the conference, I will sketch an example of a non-trivial element of K2 of a certain threefold, whose existence is related to the vanishing of an incomplete L-function of a modular form at s = 1. To explain how this fits into a general picture, we begin with a simple account, for the non-specialist, of some of the conjectures (mostly due to...

Eichler-shimura Theory for Mock Modular Forms

We use mock modular forms to compute generating functions for the critical values of modular L-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms. T...

Recovering modular forms from squares

The purpose of this appendix is to provide a proof of the fact that a holomorphic newform f of weight 2k, level N and trivial character, with Hecke eigenvalues {ap | (p, N) = 1}, is determined up to a quadratic twist, in fact on the nose if N is square-free, by the knowledge of ap for all primes p in a set of sufficiently large density. We will in fact prove a more general statement below, incl...