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 [math]\displaystyle{ p }[/math] is said to be ordinary if its [math]\displaystyle{ 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]\displaystyle{ 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.

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.

Oct 17

Yousheng Shi

Generalized special cycles and theta series

We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using the oscillator representation and the thesis of Greg Anderson, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups.

Oct 24

Simon Marshall

Counting cohomological automorphic forms on $GL_3$

I will give an overview of the limit multiplicity problem for automorphic representations. I will then talk about work of mine in this area that gives a new bound for the number of cohomological cusp forms on $GL_3$ of fixed weight and growing level. My proof uses $p$-adic techniques of Calegari and Emerton.

Nov 7

Asif Zaman

A zero density estimate for Dedekind zeta functions

Given a finite group $G$, I will discuss a zero density estimate for Dedekind zeta functions associated to Galois extensions over $\mathbb{Q}$ with Galois group $G$. The result does not assume any unproven progress towards the strong Artin conjecture. Building on a recent breakthrough of Pierce, Turnage-Butterbaugh, and Wood, this has applications to $\ell$-torsion in class groups and the Chebotarev density theorem on average. This is a joint work with Jesse Thorner.

Nov 14

Liyang Yang

Holomorphic Continuation of Certain $L$-functions via Trace Formula

In this talk, we will mainly discuss two basic conjectures on entireness of certain basic $L$-functions arising from Galois representations (algebra) and automorphic representations (analysis). Although the background and definitions are quite different, we will show that such $L$-functions are closely related by a generalized Jacquet-Zagier trace formula. As a consequence, we obtain holomorphy of adjoint $L$-functions for $\mathrm{GL}(n),$ where $n\leq 4.$ Some applications will be provided.

Nov 21

Tony Feng

Steenrod operations and the Artin-Tate pairing

In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.

Nov 26

Brandon Alberts

Counting Towers of Number Fields

Fix a number field $K$ and a finite transitive subgroup $G \le S_n$. Malle's conjecture proposes asymptotics for counting the number of $G$-extensions of number fields $F/K$ with discriminant bounded above by $X$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $T \triangleleft G$. Step one: for each $G/T$-extension $L/K$, first count the number of towers of fields $F/L/K$ with ${\mathrm Gal}(F/L) \cong T$ and ${\mathrm Gal}(F/K)\cong G$ with discriminant bounded above by $X$. Step two: sum over all choices for the $G/T$-extension $L/K$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $T$ is an abelian normal subgroup of $G$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $G$ has an abelian normal subgroup.

Dec 05

Benjamin Breen

On unit signatures and narrow class groups of odd abelian number fields: structure and heuristics

What is the probability that the ring of integers in a number field contains a unit of mixed signature? In this talk, we present Cohen-Lenstra style heuristics for unit signatures and narrow class groups of odd abelian number fields. In addition, we analyze the equation $x^3 - ax^2 + bx - 1 = 0$ to prove that there are infinitely many cyclic cubic number fields with no units of mixed signature. This is joint work with Noam Elkies, Ila Varma, and John Voight.