# TBA (Note Special Day)

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

Jan Vonk

IAS

Friday, February 7, 2020 - 3:00pm to 4:00pm

Huy Dang

University of Virginia

Thursday, January 9, 2020 - 3:00pm to 4:00pm

RH 306

Garo Sarajian

UCSB

Thursday, February 6, 2020 - 3:00pm to 4:00pm

RH 306

Lynnelle Ye

Stanford University

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

RH 306

Jeff Manning

UCLA

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

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.

Aly Deines

Center for Communications Research

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

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.

Kiran Kedlaya

UC San Diego

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

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).

Cosmin Pohoata

Caltech

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

RH 440R

Lynn Chua

UC Berkeley

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

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.

Kevin McGown

CSU Chico

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

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.