New Varieties and Forms from Missouri

Automorphic Forms on Shimura Varieties

Hyperbolicity, automorphic forms and Siegel modular varieties

We study the hyperbolicity of compactifications of quotients of bounded symmetric domains by arithmetic groups. We prove that, up to an étale cover, they are Kobayashi hyperbolic modulo the boundary. Applying our techniques to Siegel modular varieties, we improve some former results of Nadel on the non-existence of certain level structures on abelian varieties over complex function fields.

On Toric Varieties and Modular Forms

Let M∗(l) = M∗(Γ1(l),C) be the ring of holomorphic modular forms on Γ1(l). In this talk we use the combinatorics of complete toric varieties to construct a subring T∗(l) ⊂ M∗(l), the subring of toric modular forms (§2). This is a natural subring, in the sense that it behaves nicely with respect to natural operations on M∗(l) (namely, Hecke operators, Fricke involution, and the theory of oldform...

Modular Forms and Calabi-yau Varieties

n=1 anq n be a holomorphic newform of weight k ≥ 2 relative to Γ1(N) acting on the upper half plane H. Suppose the coefficients an are all rational. When k = 2, a celebrated theorem of Shimura asserts that there corresponds an elliptic curve E over Q such that for all primes p N , ap = p + 1 − |E(Fp)|. Equivalently, there is, for every prime `, an `-adic representation ρ` of the absolute Galois...

Cayley Forms and Self-dual Varieties

Generalized Chow forms were introduced by Cayley for the case of 3-space; their zero set on the Grassmannian G(1, 3) is either the set Z of lines touching a given space curve (the case of an ‘honest’ Cayley form), or the set of lines tangent to a surface. Cayley gave some equations for F to be a generalized Cayley form, which should hold modulo the ideal generated by F and by the quadratic equa...