NTS ABSTRACTSpring2019

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Jan 23

Yunqing Tang

Reductions of abelian surfaces over global function fields

For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the product of two elliptic curves. This result can be viewed as a generalization of a theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.

Jan 24

Hassan-Mao-Smith--Zhu

The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

Abstract: Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.

Jan 31

Kyle Pratt

Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions

Abstract: I will discuss recent work, joint with Bui, Robles, and Zaharescu, on a moment problem for Dirichlet $L$-functions. By way of motivation I will spend some time discussing the Lindel\"of Hypothesis, and work of Bettin, Chandee, and Radziwi\l\l. The talk will be accessible, as I will give lots of background information and will not dwell on technicalities.

Feb 7

Shamgar Gurevich

Harmonic Analysis on $GL_n$ over finite fields

Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: $$trace (\rho(g))/dim (\rho),$$ for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (TAMU).

Feb 14

Tonghai Yang

The Lambda invariant and its CM values

Abstract: The Lambda invariant which parametrizes elliptic curves with two torsions (X_0(2)) has some interesting properties, some similar to that of the j-invariants, and some not. For example, $\lambda(\frac{d+\sqrt d}2)$ is a unit sometime. In this talk, I will briefly describe some of the properties. This is joint work with Hongbo Yin and Peng Yu.

Feb 28

Brian Lawrence

Diophantine problems and a p-adic period map.

Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theory. Joint with Akshay Venkatesh.

March 7

Masoud Zargar

Sections of quadrics over the affine line

Abstract: Abstract: Suppose we have a quadratic form Q(x) in d\geq 4 variables over F_q[t] and f(t) is a polynomial over F_q. We consider the affine variety X given by the equation Q(x)=f(t) as a family of varieties over the affine line A^1_{F_q}. Given finitely many closed points in distinct fibers of this family, we ask when there exists a section passing through these points. We study this problem using the circle method over F_q((1/t)). Time permitting, I will mention connections to Lubotzky-Phillips-Sarnak (LPS) Ramanujan graphs. Joint with Naser T. Sardari

March 14

Elena Mantovan

p-adic automorphic forms, differential operators and Galois representations

A strategy pioneered by Serre and Katz in the 1970s yields a construction of p-adic families of modular forms via the study of Serre's weight-raising differential operator Theta. This construction is a key ingredient in Deligne-Serre's theorem associating Galois representations to modular forms of weight 1, and in the study of the weight part of Serre's conjecture. In this talk I will discuss recent progress towards generalizing this theory to automorphic forms on unitary and symplectic Shimura varieites. In particular, I will introduce certain p-adic analogues of Maass-Shimura weight-raising differential operators, and discuss their action on p-adic automorphic forms, and on the associated mod p Galois representations. In contrast with Serre's classical approach where q-expansions play a prominent role, our approach is geometric in nature and is inspired by earlier work of Katz and Gross. This talk is based joint work with Eishen, and also with Fintzen--Varma, and with Flander--Ghitza--McAndrew.

March 28

Adebisi Agboola

Relative K-groups and rings of integers

Abstract: Suppose that F is a number field and G is a finite group. I shall discuss a conjecture in relative algebraic K-theory (in essence, a conjectural Hasse principle applied to certain relative algebraic K-groups) that implies an affirmative answer to both the inverse Galois problem for F and G and to an analogous problem concerning the Galois module structure of rings of integers in tame extensions of F. It also implies the weak Malle conjecture on counting tame G-extensions of F according to discriminant. The K-theoretic conjecture can be proved in many cases (subject to mild technical conditions), e.g. when G is of odd order, giving a partial analogue of a classical theorem of Shafarevich in this setting. While this approach does not, as yet, resolve any new cases of the inverse Galois problem, it does yield substantial new results concerning both the Galois module structure of rings of integers and the weak Malle conjecture.

April 4

Wei-Lun Tsai

Hecke L-functions and $\ell$ torsion in class groups

Abstract: The canonical Hecke characters in the sense of Rohrlich form a set of algebraic Hecke characters with important arithmetic properties. In this talk, we will explain how one can prove quantitative nonvanishing results for the central values of their corresponding L-functions using methods of an arithmetic statistical flavor. In particular, the methods used rely crucially on recent work of Ellenberg, Pierce, and Wood concerning bounds for $\ell$-torsion in class groups of number fields. This is joint work with Byoung Du Kim and Riad Masri.

April 11

Taylor Mcadam

Almost-prime times in horospherical flows

Abstract: Equidistribution results play an important role in dynamical systems and their applications in number theory. Often in such applications it is desirable for equidistribution to be effective (i.e. the rate of convergence is known). In this talk I will discuss some of the history of effective equidistribution results in homogeneous dynamics and give an effective result for horospherical flows on the space of lattices. I will then describe an application to studying the distribution of almost-prime times in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture.

April 18

Ila Varma

Malle's Conjecture for octic $D_4$-fields.

Abstract: We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results.

April 25

Michael Bush

Interactions between group theory and number theory

Abstract: I'll survey some of the ways in which group theory has helped us understand extensions of number fields with restricted ramification and why one might care about such things. Some of Nigel's contributions will be highlighted. A good portion of the talk should be accessible to those other than number theorists.

April 25

Rafe Jones

Eventually stable polynomials and arboreal Galois representations

Abstract: Call a polynomial defined over a field K eventually stable if its nth iterate has a uniformly bounded number of irreducible factors (over K) as n grows. I’ll discuss some far-reaching conjectures on eventual stability, and recent work on various special cases. I’ll also describe some natural connections between eventual stability and arboreal Galois representations, which Nigel Boston introduced in the early 2000s.

April 25 NTS

Jen Berg

Rational points on conic bundles over elliptic curves with positive rank

Abstract: Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses the Brauer group, Br X, to preclude the existence of rational points on a variety X. In this talk, we'll explore the arithmetic of conic bundles over elliptic curves of positive rank over a number field k. We'll discuss the insufficiency of the known obstructions to explain the failures of the Hasse principle for such varieties over a number field. We'll further consider questions on the distribution of the rational points of X with respect to the image of X(k) inside of the rational points of the elliptic curve E. In the process, we'll discuss results on a local-to-global principle for torsion points on elliptic curves over Q. This is joint work in progress with Masahiro Nakahara.

April 25

Judy Walker

Derangements of Finite Groups

Abstract: In the early 1990’s, Nigel Boston taught an innovative graduate-level group theory course at the University of Illinois that focused on derangements (fixed-point-free elements) of transitive permutation groups. The course culminated in the writing of a 7-authored paper that appeared in Communications in Algebra in 1993. This paper contained a conjecture that was eventually proven by Fulman and Guralnick, with that result appearing in the Transactions of the American Mathematical Society just last year.

May 2

Melanie Matchett Wood

Unramified extensions of random global fields

Abstract: For any finite group Gamma, I will give a "non-abelian-Cohen-Martinet Conjecture," i.e. a conjectural distribution on the "good part" of the Galois group of the maximal unramified extension of a global field K, as K varies over all Galois Gamma extensions of the rationals or rational function field over a finite field. I will explain the motivation for this conjecture based on what we know about these maximal unramified extensions (very little), and how we prove, in the function field case, as the size of the finite field goes to infinity, that the moments of the Galois groups of these maximal unramified extensions match out conjecture. This talk covers work in progress with Yuan Liu and David Zureick-Brown

May 9

David Zureick-Brown

Arithmetic of stacks

Abstract: I'll discuss several diophantine problems that naturally lead one to study algebraic stacks, and discuss a few results.