# Geometry of eigenvarieties for definite unitary groups

Lynnelle Ye

## Institution:

Stanford University

## Time:

Thursday, February 13, 2020 - 3:00pm to 3:50pm

## Location:

RH 306

We will discuss questions about the geometry of Chenevier's eigenvarieties for automorphic forms on definite unitary groups. For example, we will give bounds on the eigenvalues of the $U_p$ Hecke operator that appear in these eigenvarieties. These bounds generalize ones of Liu-Wan-Xiao for rank 2, which they used to prove the Coleman-Mazur-Buzzard-Kilford conjecture in that setting, to all ranks. If time permits, we will discuss possible avenues for recovering additional information not obtainable from these bounds and coming closer to fully generalizing Liu-Wan-Xiao's results.

# Taylor-Wiles-Kisin patching and mod l multiplicities in Shimura curves

Jeff Manning

UCLA

## Time:

Thursday, October 31, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

In the early 1990s Ribet observed that the classical mod l multiplicity one results for modular curves, which are a consequence of the q-expansion principle, fail to generalize to Shimura curves. Specifically he found examples of Galois representations which occur with multiplicity 2 in the mod l cohomology of a Shimura curve with discriminant pq and level 1.

I will describe a new approach to proving multiplicity statements for Shimura curves, using the Taylor-Wiles-Kisin patching method (which was shown by Diamond to give an alternate proof of multiplicity one in certain cases), as well as specific computations of local Galois deformation rings done by Shotton. This allows us to re-interpret and generalize Ribet's result. I will prove a mod l "multiplicity 2^k" statement in the minimal level case, where k is a number depending only on local Galois theoretic data. This proof also yields additional information the Hecke module structure of the cohomology of a Shimura curve, which among other things has applications to the study of congruence modules.

# Elliptic Curves of Prime Conductor

Aly Deines

## Institution:

Center for Communications Research

## Time:

Thursday, November 7, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R
The torsion order elliptic curves over $\mathbb{Q}$ with prime conductor have been well studied. In particular, we know that for an elliptic curve $E/\mathbb{Q}$ with conductor $p$ a prime, if $p > 37$, then $E$ has either no torsion, or is a Neumann-Setzer curve and has torsion order 2. In this talk we examine similar behavior for elliptic curves of prime conductor defined over totally real number fields.

# The tame Belyi theorem in positive characteristic

Kiran Kedlaya

UC San Diego

## Time:

Thursday, October 24, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

If an algebraic curve over a field of characteristic 0 admits a finite
map to the projective line ramified only over three points, then it must
be definable over some number field. This fact has a famous converse due
to Belyi: any curve over a number field admits such a finite map over
its field of definition.

Similarly, if an algebraic curve over a field of characteristic p>0
admits a finite *tamely ramified* map to the projective line ramified
only over three points, then it must be definable over some finite
field. We prove the converse: any curve over a finite field admits such
a finite map over its field of definition.

A construction of Saidi shows that this reduces to the existence of a
single tamely ramified map. This is easy to establish over an infinite
field of odd characteristic, and only slightly harder (using
Poonen-style probabilistic techniques) over a finite field of odd
characteristic. To handle the case of a finite field of characteristic
2, we use a construction of Sugiyama-Yasuda that they used to establish
existence of tame morphisms over an algebraically closed field of
characteristic 2.

Joint work with Daniel Litt (Georgia) and Jakub Witaszek (Michigan).

# Expanding polynomials for sets with additive or multiplicative structure

Cosmin Pohoata

Caltech

## Time:

Thursday, December 5, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

Abstract: Given an arbitrary set of real numbers A and a two-variate polynomial f with real coefficients, a remarkable theorem of Elekes and R\'onyai from 2000 states that the size |f(A,A)| of the image of f on the cartesian product A x A grows asymptotically faster than |A|, unless f exhibits additive or multiplicative structure. Finding the best quantitative bounds for this intriguing phenomenon (and for variants of it) has generated a lot of interest over the years due to its intimate connection with the sum-product problem from additive combinatorics. In this talk, we will quickly review some of the results in this area, and then discuss some new bounds for |f(A,A)| when the set A has few sums or few products. If time permits, will also discuss some new results over finite fields.

# On the concrete security of the unique Shortest Vector Problem

Lynn Chua

UC Berkeley

## Time:

Thursday, November 14, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

We study experimentally the Hermite factor of BKZ2.0 on uSVP lattices, with the motivation of understanding the concrete security of LWE in the setting of homomorphic encryption. We run experiments by generating instances of LWE in small dimensions, where we consider secrets sampled from binary, ternary or discrete Gaussian distributions. We convert each LWE instance into a uSVP instance and run the BKZ2.0 algorithm to find an approximation to the shortest vector. When the attack is successful, we can deduce a bound on the Hermite factor achieved for the given blocksize. This allows us to give concrete values for the Hermite factor of the lattice generated for the uSVP instance. We compare the values of the Hermite factors we find for these lattices with estimates from the literature and find that the Hermite factor may be smaller than expected for blocksizes 30, 35, 40, 45. Our work also demonstrates that the experimental and estimated values of the Hermite factor trend differently as we increase the dimension of the lattice, highlighting the importance of a better theoretical understanding of the performance of BKZ2.0 on uSVP lattices.

# An explicit upper bound on the least primitive root

Kevin McGown

CSU Chico

## Time:

Thursday, October 10, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R
Let $p$ be an odd prime. A classical problem in analytic number theory is to give an upper bound on the least primitive root modulo $p$, denoted by $g(p)$. In the 1960s Burgess proved that for any $\varepsilon>0$ one has $g(p)\ll p^{1/4+\varepsilon}$ for sufficiently large $p$. This was a consequence of his landmark character sum inequality, and this result remains the state of the art. However, in applications, explicit estimates are often required, and one needs more than an implicit constant that depends on $\varepsilon$. Recently, Trudgian and the speaker have given an explicit upper bound on $g(p)$ that improves (by a small power of log factor) on what one can obtain using any existing version of the Burgess inequality. In particular, we show that $g(p)<2r\,2^{r\omega(p-1)}p^{\frac{1}{4}+\frac{1}{4r}}$ for $p>10^{56}$, where $r\geq 2$ is an integer parameter. $\$ In 1952 Grosswald showed that if $g(p)<\sqrt{p}-2$, then the principal congruence subgroup $\Gamma(p)$ for can be generated by the matrix $[1,p;0,1]$ and $p(p-1)(p+1)/12$ other hyperbolic matrices. He conjectured that $g(p)<\sqrt{p}-2$ for $p>409$. Our method allows us to show that Grosswald's conjecture holds unconditionally for $p> 10^{56}$, improving on previous results.

# Holomorphy Conjectures on Certain L-functions

Liyang Yang

Caltech

## Time:

Thursday, November 21, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R
In this talk, we will mainly discuss two basic conjectures on entireness of certain Artin $L$-functions from Galois representation and certain Langlands $L$-functions from automorphic representation. Although the background and definitions are quite different, we will show these $L$-functions are closely related to each other by a generalized Jacquet-Zagier trace formula. Some applications will be provided.

# Applications of Cayley Digraphs to Waring's Problem and Sum-Product Formulas

## Speaker:

Yesim Demiroglu Karabulut

Harvey Mudd

## Time:

Thursday, September 26, 2019 - 3:00pm to 4:00pm

## Location:

RH 306
In this talk, we first present some elementary new proofs (using Cayley digraphs and spectral graph theory) for Waring's problem over finite fields, and explain how in the process of re-proving these results, we obtain an original result that provides an analogue of Sárközy's theorem in the finite field setting (showing that any subset $E$ of a finite field $\Bbb F_q$ for which $|E| > \frac{qk}{\sqrt{q - 1}}$ must contain at least two distinct elements whose difference is a $k^{\text{th}}$ power). Once we have our results for finite fields, we apply some classical mathematics to extend our Waring's problem results to the context of general (not necessarily commutative) finite rings. In the second half of our talk, we present sum-product formulas related to matrix rings over finite fields, which can again be proven using Cayley digraphs and spectral graph theory in an efficient way.

# Moments of cubic L-functions over function fields

Alexandra Florea

## Institution:

Columbia University

## Time:

Thursday, October 17, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R
I will focus on the mean value of $L$-functions associated to cubic characters over $\mathbb{F}_q[t]$ when $q \equiv 1 \pmod 3$. I will explain how to obtain an asymptotic formula which relies on obtaining cancellation in averages of cubic Gauss sums over functions fields. I will also talk about the corresponding non-Kummer case when $q \equiv 2 \pmod 3$ and I will explain why this setting is somewhat easier to handle than the Kummer case, which allows us to prove some better results. This is joint work with Chantal David and Matilde Lalin.