# 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.

# The Weyl law for algebraic tori

Ian Petrow

ETH Zurich

## Time:

Wednesday, May 1, 2019 - 3:00pm to 4:00pm

## Location:

RH 440R

A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.

# Factorization of Hasse-Weil zeta functions of Dwork surfaces

Lian Duan

## Institution:

Univ. of Mass, Amherst

## Time:

Thursday, April 11, 2019 - 3:00pm to 4:00pm

## Location:

RH 306

As a classical example of K3 surfaces, the Dwork surface family is of interest in algebraic geometry and number theory. A lot of work has been done to understand the Hasse-Weil zeta functions of these surfaces. Recent works show that people can totally determine the algebraic part of the zeta function for a general Dwork surface. In this talk, we discuss how to use geometric method to find the explicit factorization of the algebraic part.

# Iwasawa theory for function fields

Bryden Cais

## Institution:

University of Arizona

## Time:

Thursday, June 6, 2019 - 3:00pm to 3:50pm

## Location:

RH 306

Let X_n be a Z_p-tower of smooth projective curves over a
perfect field $k$ of characteristic p that totally ramifies over a finite,
nonempty set of points of X_0 and is unramified elsewhere. In analogy
with the case of number fields, Mazur and Wiles studied the growth of
the p-parts of the class groups $Jac(X_n)[p^infty](\overline{k})$ as n-varies, and
proved that these naturally fit together to yield a module that is
finite and free over the Iwasawa algebra. We introduce a novel
perspective by proposing to study growth of the full p-divisible group
$G_n:=Jac(X_n)[p^infty]$, which may be thought of as the p-primary part of
the *motivic class group* $Jac(X_n)$. One has a canonical decomposition
$G_n = G_n^{et} \times G_n^{m} \times G_n^{ll}$ of $G$ into its etale, multiplicative, and
local-local components, as well as an equality $G_n(\overline{k}) = G_n^{et}(\overline{k})$.

Thus, the work of Mazur and Wiles captures the etale part of G_n, so
also (since Jacobians are principally polarized) the multiplicative
part: both of these p-divisible subgroups satisfy the expected
structural and control theorems in the limit. In contrast, the
local-local components G_n^{ll} are far more mysterious (they can not be
captured by $\overline{k}$-points), and indeed the tower they form has no analogue
in the number field setting. This talk will survey this circle of ideas,
and will present new results and conjectures on the behavior of the
local-local part of the tower G_n.

# ABC Triples in Families

Edray Goins

Pomona College

## Time:

Thursday, April 18, 2019 - 3:00pm to 4:00pm

## Location:

RH 306

Given three positive, relative prime integers A, B, and C such that the first two sum to the third i.e. A + B = C, it is rare to have the product of the primes p dividing them to be smaller than each of the three.  In 1985, David Masser and Joseph Osterlé made this precise by defining a "quality" q(P) for such a triple of integers P = (A,B,C); their celebrated "ABC Conjecture" asserts that it is rare for this quality q(P) to be greater than 1 -- even through there are infinitely many examples where this happens.  In 1987, Gerhard Frey offered an approach to understanding this conjecture by introducing elliptic curves.  In this talk, we introduce families of triples so that the Frey curve has nontrivial torsion subgroup, and explain how certain triples with large quality appear in these families.  We also discuss how these families contain infinitely many examples where the quality q(P) is greater than 1.  This joint work with Alex Barrios.

# Graphs of Hecke operators and Hall algebras

## Speaker:

Roberto Alvarenga

## Institution:

USP-Brazil, visiting UCI

## Time:

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

## Location:

RH 306

Abstract: Motivated by questions of Zagier, Lorscheid develops in his Ph. D thesis the theory of graphs of Hecke operators. These graphs encode the action of Hecke operators on automorphic forms over a global function field. In this talk, I intend to present this theory and connect it with the Hall algebras theory. This connection allow us to describe these graphs for an elliptic curve.