Title & Abstract

Welcome & Opening

[Speaker] Frank Thorne

[Title] An Overview of Number Field Counting.

[abstract] I'll give an overview of the question of counting number fields of bounded discriminant. In the first talk, I'll give a general overview -- what is known, what is believed, and what techniques are available?

[Speaker] Frank Thorne

[Title] An Overview of Number Field Counting.

[abstract] In the second talk, I'll give a more detailed overview of what can be said by analytic methods. Works by Delone-Faddeev, Davenport-Heilbronn, Bhargava, and others translate some of these questions into lattice point counting questions, which can be addressed by a variety of tools familiar in analytic number theory: geometry of numbers, Fourier analysis, zeta functions. I'll give a few examples of what can be proved this way.

[Speaker] Asif Zaman

[Title] An approximate form of Artin's holomorphy conjecture and nonvanishing of

Artin L-functions.

[abstract] Let $k$ be a number field and $G$ be a finite group, and let $\mathfrak{F}_{k}^{G}$ be a family of number fields $K$ such that $K/k$ is normal with Galois group isomorphic to $G$.  Together with  Robert Lemke Oliver and Jesse Thorner, we prove for many families that for almost all $K \in \mathfrak{F}_k^G$, all of the $L$-functions associated to Artin representations whose kernel does not contain a fixed normal subgroup are holomorphic and non-vanishing in a wide region. 

These results have several arithmetic applications. For example, we prove a strong effective prime ideal theorem that holds for almost all fields in several natural large degree families, including the family of degree $n$ $S_n$-extensions for any $n \geq 2$ and the family of prime degree $p$ extensions (with any Galois structure) for any prime $p \geq 2$. I will discuss this result, describe the main ideas of the proof, and share some applications to bounds on $\ell$-torsion subgroups of class groups, to the extremal order of class numbers, and to the subconvexity problem for Dedekind zeta functions.

[Speaker] Alexandre Perozim de Faveri

[Title] Simple zeros of GL(2) L-functions.

[abstract] I will discuss my work on simple zeros of automorphic L-functions of degree 2. The main result is that the completed L-function of a primitive holomorphic form f of arbitrary weight and level has Omega(T^z) simple zeros with imaginary part in [-T, T], for any z < 2/27. This provides the first power bound in this problem for f of non-trivial level, where the previous best bound was Omega(log log log T). The proof uses a method of Conrey-Ghosh combined with ideas of Booker and Booker-Milinovich-Ng, in addition to a new ingredient coming from zero-density estimates for twists of f. I will explain the basic method, the obstructions that arise when f has non-trivial level, and how to unconditionally get around such obstructions to obtain a power bound. This argument gives a curious connection between the quality of zero-density estimates for a certain family and the number of simple zeros for a single element of that family.

Break

[Speaker] Hae-Sang Sun

[Title] Distribution of modular inverses via dynamical approach.

[abstract] Let $R(n,x)$ be the number of integers modulo $n$ such that both the integer and its modular inverse mod $n$ are less than $x$. A folklore conjecture is that $R(p,p^{1/2+\epsilon})>0$ for a prime $p$. It is a well-known result that $R(p,p^{3/4+\epsilon})\gg p^{1/2+\epsilon}$ for primes $p$. An open problem is to improve the exponent $3/4$. In the talk, I will introduce how to study the average version of the problem via transfer operator method for the dynamics of continued fractions. This is research in progress.

[Speaker] Anders Södergren

[Title] Non-vanishing at the central point of the Dedekind zeta functions of non-Galois cubic fields.

[abstract] It is believed that for every S_n-number field, i.e. every degree n extension of the rational numbers whose normal closure has Galois group S_n, the Dedekind zeta function is non-vanishing at the central point. In the case n = 2 Soundararajan established, in spectacular work improving on earlier work of Jutila, the non-vanishing of the Dedekind zeta function for at least 87.5% of the fields in certain families of quadratic fields. In this talk, I will present recent joint work with Arul Shankar and Nicolas Templier, in which we study the case n=3. In particular, I will discuss some of the main ideas in our proof that the Dedekind zeta functions of infinitely many S3-fields have non-vanishing central value.

[Speaker] Daniel Fiorilli

[Title] Omega results for cubic field counts via the Katz-Sarnak philosophy.

[abstract] I will discuss recent joint work with P. Cho, Y. Lee and A. Södergren. Since the results of Davenport-Heilbronn, much work has been done to obtain a precise estimate for the number of cubic fields of discriminant at most X. This includes work of Belabas-Bhargava-Pomerance, Bhargava-Shankar-Tsimerman, Taniguchi-Thorne and Bhargava-Taniguchi-Thorne. In this talk I will present a negative result, which states that the GRH implies that the error term in this estimate cannot be too small. Our approach involves low-lying zeros of Dedekind zeta functions of cubic fields (first studied by Yang), and is strongly related to the Katz-Sarnak conjectures and the ratios conjecture of Conrey, Farmer and Zirnbauer.

[Speaker] Frank Thorne

[Title] An Overview of Number Field Counting.

[abstract] In the third talk, I'll focus more on algebraic methods. In many cases when the Galois group is restricted, it's possible to obtain quite strong results and I will say a bit about how. I'll also describe work with Robert Lemke Oliver improving on the most general known upper bounds for field counting.

[Speaker] Keunyoung Jeong

[Title] On an upper bound of the average analytic rank of a family of elliptic curves.

[abstract] The average of the rank of elliptic curves over rational numbers in a ``natural'' family is expected to be 1/2. For example, Goldfeld conjectured that the average of analytic ranks of the quadratic twist family of an elliptic curve over rational numbers is 1/2. In this talk, we will introduce a machinery which gives an upper bound of the average of analytic ranks of a family of elliptic curves. To run the machinery, we need to know the probability that an elliptic curve in the family has good/multiplicative/additive reduction (actually we need something more), and use trace formulas. Using the machinery on the set of all elliptic curves over rationals and the set of elliptic curves with a given torsion subgroup respectively, we can compute an upper bound of the n-th moment of the average. This is the first result on an upper bound of the average of the family of elliptic curves with a fixed torsion group, as far as we know. This is joint work with Peter J. Cho.

[Speaker] Yoonbok Lee

[Title] The number of zeros of linear combinations of $L$-functions near the critical line.

[abstract](This is a joint work with Youness Lamzouri.)

We investigate the zeros near the critical line of linear combinations of $L$-functions belonging to a large class, which conjecturally contains all $L$-functions arising from automorphic representations on $\textup{GL}(n)$. More precisely, if $L_1, \dots, L_J$ are distinct primitive $L$-functions with $J\ge 2$, and $b_j$ are any nonzero real numbers, we prove that the number of zeros of $F(s)=\sum_{j\leq J} b_j L_j(s)$ in the region $\Re(s)\geq 1/2+1/G(T)$ and $\Im(s)\in [T, 2T]$ is asymptotic to $K_0 T G(T)/\sqrt{\log G(T)}$ uniformly in the range $ \log \log T \leq G(T)\leq (\log T)^{\nu}$, where $K_0$ is a certain positive constant that depends on $J$ and the $L_j$'s. This establishes a generalization of a conjecture of Hejhal in this range. Moreover, the exponent $\nu$ verifies $\nu\asymp 1/J$ as $J$ grows.

[Speaker] Kohji Matsumoto

[Title] $M$-functions associated with cusp forms.

[abstract] The density function associated with the value-distribution of various $L$-functions is called the $M$-function. In this talk, I first describe the history and earlier results on the theory of $M$-functions. Then I report my recent work on $M$-functions associated with the value-distribution of symmetic square $L$-functions of holomorphic cusp forms.

closing