NTS Fall 2015 Abstract

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Sep 03

Kiran Kedlaya

On the algebraicity of (generalized) power series

A remarkable theorem of Christol from 1979 gives a criterion for detecting whether a power series over a finite field of characteristic p represents an algebraic function: this happens if and only if the coefficient of the n-th power of the series variable can be extracted from the base-p expansion of n using a finite automaton. We will describe a result that extends this result in two directions: we allow an arbitrary field of characteristic p, and we allow "generalized power series" in the sense of Hahn-Mal'cev-Neumann. In particular, this gives a concrete description of an algebraic closure of a rational function field in characteristic p (and corrects a mistake in my previous attempt to give this description some 15 years ago).

Sep 10

Sean Rostami

Fixers of Stable Functionals

The epipelagic representations of Reeder-Yu, a generalization of the "simple supercuspidals" of Gross-Reeder, are certain low-depth supercuspidal representations of reductive algebraic groups G. Given a "stable functional" f, which is a suitably 'generic' linear functional on a vector space coming from a Moy-Prasad filtration for G, one can create such a representation. It is known that the representations created in this way are compactly induced from the fixer in G of f and it is important to identify explicitly all the elements that belong to this fixer. This work is in-progress.

Sep 17

David Zureick-Brown

Tropical geometry and uniformity of rational points

Let X be a curve of genus g over a number field F of degree d = [F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known to follow from the Bomberi-Lang conjecture. We prove a special case of this conjecture - we give an explicit uniform bound when X has Mordell-Weil rank r ≤ g-3. This generalizes recent work of Stoll on uniform bounds on hyperelliptic curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel-Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate. Our methods combine Chabauty-Coleman's p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs. This is joint work with Joe Rabinoff and Eric Katz.

Sep 22

Joseph Gunther

Embedding Curves in Surfaces and Stabilization of Hypersurface Singularity Counts

We'll present two new applications of Poonen's closed point sieve over finite fields. The first is that the obvious local obstruction to embedding a curve in a smooth surface is the only global obstruction. The second is a proof of a recent conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities. Note: The day of this seminar is Tuesday.

Sep 24

Brandon Alberts

The Moments Version of Cohen-Lenstra Heuristics for Nonabelian Groups

Cohen-Lenstra heuristics posit the distribution of unramified abelian extensions of quadratic fields. A natural question to ask would be how to get an analogous heuristic for nonabelian groups. In this talk I take and extend on recent work in the area of unramified extensions of imaginary quadratic fields and bring it all together under one Cohen-Lenstra style heuristic.

Oct 08

Ana Caraiani

On vanishing of torsion in the cohomology of Shimura varieties

I will discuss joint work in progress with Peter Scholze showing that torsion in the cohomology of certain compact unitary Shimura varieties occurs in the middle degree, under a genericity assumption on the corresponding Galois representation.

Oct 15

Valentin Blomer

Arithmetic, geometry and analysis of a senary cubic form

We establish an asymptotic formula (with power saving error term) for the number of rational points of bounded height for a certain cubic fourfold, thereby proving a strong form of Manin's conjecture for this algebraic variety by techniques of analytic number theory.

Oct 22

Brian Cook

Configurations in dense subsets of Euclidean spaces

A result of Katznelson and Weiss states that given a suitably dense (measurable) subset of the Euclidean plane realizes every sufficiently large distance, that is, for every prescribed (sufficiently large) real number the set contains two elements whose distance is this number. The analogue of this statement for finding three equally spaced points on a line, i.e. for finding three term arithmetic progressions, in a given set is false, and in fact false in every dimension. In this talk we revisit the case of three term progressions when the standard Euclidean metric is replaced by other metrics.

Oct 29

Aaron Levin

Integral points and orbits in the projective plane

We will discuss the problem of classifying the behavior of integral points on affine subsets of the projective plane. As an application, we will examine the problem of classifying endomorphisms of the projective plane with an orbit containing a Zariski dense set of integral points (with respect to some plane curve). This is joint work with Yu Yasufuku.

Nov 12

Vlad Matei

A geometric perspective on Landau's theorem over function fields

We revisit the recent result of Lior-Bary-Soroker. It deals with a function field analogue of Landau's classical result about the asymptotic density of numbers which are sums of two integer squares. The results obtained are just in the large characteristic and large degree regime. We obtain a characterization as qn goes to infinity, which is the desired analogue of the result over the integers. We take a geometric perspective in computing the number of polynomials of degree n which are split in the extension Fq[T1/2] / Fq[T] and we obtain a geometric explanation for the "mysterious" binomial coefficients appearing in his asymptotic.

Nov 19

Tonghai Yang

A generating function of arithmetic divisors in a unitary Shimura variety: modularity and application

In the original Gross-Zagier formula and Zhang's extension to Shimura curves, the modularity of the generating function is a very important step, where are the Heeger divisors and is the rational canonical divisor of degree 1 (associated to Hodge bundle). In the proof, they actually use arithmetic version in calculation: which is also modular. In this talk, we define a generalization of this arithmetic generating function to unitary Shimura variety of type (n, 1) and prove that it is modular. It has application to Colmez conjecture and Gross-Zagier type formula.

Dec 03

Daniel Hast

Finiteness of integral points on curves via the Chabauty-Coleman-Kim method

Decades before Faltings proved the Mordell conjecture, Chabauty was able to prove the conjecture for some special higher genus curves by studying how the curve embeds in the p-adic points of its Jacobian. This method was refined by Coleman to give explicit bounds on the number of rational points of such curves. Recently, Minhyong Kim developed a nonabelian version of the method that makes use of the whole unipotent fundamental group of the curve, not just its abelianization, leading to a new proof of Siegel's theorem. We will survey these results and some related conjectures of Kim that hint at the possibility of a p-adic proof of the Mordell conjecture. This talk will be expository in nature.

Dec 10

Brandon Levin

The weight part of Serre's conjecture

Serre's modularity conjecture (now a Theorem due to Khare-Wintenberger and Kisin) states that every odd irreducible two dimensional mod p representation of the absolute Galois group of Q comes from a modular form. I will begin with an overview of the Serre's original conjecture on modular forms focusing on the weight part of the conjecture. Herzig gave a generalization of the conjecture for n-dimensional Galois representations which predicts the modularity of so-called shadow weights. After briefly describing Herzig's conjecture, I will discuss joint work with D. Le, B. Le Hung, and S. Morra where we prove instances of this conjecture in dimension three.

Dec 17

Nathan Kaplan

Rational Point Counts for Varieties over a Fixed Finite Field

We will address several statistical questions about families of curves and surfaces over a fixed finite field. For example, what is the average number of rational points on an elliptic curve over Fq containing a rational 5-torsion point? How many cubic surfaces in P3(Fq) have exactly q2+7q+1 Fq-points? One major idea will be to use techniques from coding theory, specifically the MacWilliams theorem and its generalizations, to answer questions about distributions of rational point counts. No prior familiarity with coding theory will be assumed. Part of this is joint work with Ian Petrow.