# Difference between revisions of "NTS ABSTRACTFall2019"

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## Sep 5

 Will Sawin 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.

## Sep 12

 Yingkun Li 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.

## Sep 19

 Soumya Sankar Proportion of ordinary curves in some families An abelian variety in characteristic $\displaystyle{ p }$ is said to be ordinary if its $\displaystyle{ p }$ torsion is as large as possible. In 2012, Cais, Ellenberg and Zureick-Brown made some conjectures about the distribution of the size of the $\displaystyle{ p }$ -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.

## Oct 3

 Patrick Allen On the modularity of elliptic curves over imaginary quadratic fields Wiles's proof of the modularity of semistable elliptic curves over the rationals uses the Langlands-Tunnell theorem as a starting point. In order to feed this into a modularity lifting theorem, one needs to use congruences between modular forms of weight one and modular forms of higher weight. Similar congruences are not known over imaginary quadratic fields and Wiles's strategy runs into problems right from the start. We circumvent this congruence problem and show that mod 3 Galois representations over imaginary quadratic fields arise from automorphic forms that are the analog of higher weight modular forms. Our argument relies on a 2-adic automorphy lifting theorem over CM fields together with a "2-3 switch." As an application, we deduce that a positive proportion of elliptic curves over imaginary quadratic fields are modular. This is joint work in progress with Chandrashekhar Khare and Jack Thorne.

## Oct 10

 Borys Kadets Sectional monodromy groups of projective curves Let K be a field. Fix a projective curve X \subset \mathbb{P}^r_K of degree d. A general hyperplane H \in \mathbb{P}^{r*} intersects X in d points; the monodromy of X \bigcap H as H varies is a subgroup G_X of S_d known as the sectional monodromy group of X. When K=\mathbb{C} (or in fact for \mathrm{char} K = 0), the equality G_X=S_d was shown by Castelnuovo; this large monodromy fact is important in studying the degree-genus problem for projective curves. I will talk about the behaviour of sectional monodromy groups in positive characteristic. I will show that for a large class of curves the inclusion G_X \supset A_d holds. On the other hand, for a seemingly simple family of curves X_{m,n} given by the equation x^n=y^mz^{n-m} in \mathbb{P}^2 I will completely characterize the possibilities for G_{X_{n,m}}; the list of possibilities includes linear groups \mathrm{AGL}_n(q), \mathrm{PGL}_2(q) as well as some sporadic simple groups.