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|The sup-norm problem for automorphic forms over function fields and geometry|
The sup-norm problem is a purely analytic question about automorphic forms, which asks for bounds on their largest value (when viewed as a function on a modular curve or similar space). We describe a new approach to this problem in the function field setting, which we carry through to provide new bounds for forms in GL_2 stronger than what can be proved for the analogous question about classical modular forms. This approach proceeds by viewing the automorphic form as a geometric object, following Drinfeld. It should be possible to prove bounds in greater generality by this approach in the future.
|CM values of modular functions and factorization|
The theory of complex multiplication tells us that the values of the j-invariant at CM points are algebraic integers. The norm of the difference of two such values has nice and explicit factorization, which was the subject of the seminal work of Gross and Zagier on singular moduli in the 1980s. In this talk, we will recall this classical result, review some recent factorization formula for other modular functions, and report some progress on a conjecture of Yui and Zagier. This is joint work with Tonghai Yang.
|Proportion of ordinary curves in some families|
An abelian variety in characteristic [math]p[/math] is said to be ordinary if its [math]p[/math] torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the [math]p[/math] -torsion of an abelian variety. I will talk about some families which do not obey these heuristics, namely Jacobians of Artin-Schreier and superelliptic curves, and discuss the structure of the respective moduli spaces that make it so.