8/22 
Ronen Mukamel 
Rice University 
Teichmuller curves in positive characteristic 

Abstract: We will investigate the arithmetic nature of Teichmuller curves in the spirit of the celebrated theory developed for the classical modular curves. For arithmetic Teichmuller curves in genus two, we introduce the Weierstrass polynomial, an analogue of the classical modular polynomial associated to modular curves. We find that Weierstrass polynomials often have integer coefficients and surprising congruences with modular polynomials in positive characteristic.

8/29 
School Closed 
9/5 
John Calabrese 
Rice University 
Quotient Categories 

Abstract: It is a well known fact that two varieties are birational if and only if they have the same function field. An a priori unrelated theorem is due to Gabriel: two varieties are the same if and only if their categories of coherent sheaves are the same. I'll explain how these two theorems are actually special cases of a broader one, involving quotients of Coh(X). This is joint work with Roberto Pirisi (UBC).

9/12 
Nathan Kaplan 
UC Irvine 
Counting Lattices by Cotype 

Abstract: The zeta function of Z^d is a generating function that encodes the number of its sublattices of each index. This function can be expressed as a product of Riemann zeta functions and analytic properties of the Riemann zeta function then lead to an asymptotic formula for the number of sublattices of Z^d of index at most X.
Nguyen and Shparlinski have investigated more refined counting questions, giving an asymptotic formula for the number of cocyclic sublattices L of Z^d, those for which Z^d/L is cyclic. Building on work of Petrogradsky, we generalize this result, counting sublattices for which Z^d/L has at most m invariant factors. We will see connections to cokernels of random integer matrices and the CohenLenstra heuristics. Joint work with Gautam Chinta (CCNY) and Shaked Koplewitz (Yale).

9/19 
Arindam Roy 
Rice University 
Unnormalized differences of the zeros of the derivative of the completed Lfunction 

Abstract: We study the distribution of unnormalized diferences between imaginary parts of the zeros of the derivative of the Riemann ξ function. Such distributions are capable of identifying the exact location of every zero of the Riemann zeta function. In particular, we prove that the Riemann hypothesis for the Riemann zeta function is encoded in the distribution of zeros of derivative of completed Dirichlet Lfunction. We also show that the differences tend
to avoid the imaginary part of low lying zeros of the Riemann zeta function.

9/26 
Zheng Zhang 
Texas A & M 
On the period maps for certain Horikawa surfaces and for cubic pairs. 

Abstract: It is an interesting problem to attach moduli meanings to locally symmetric domains via period maps. Besides the classical cases like polarized abelian varieties and lattice polarized K3 surfaces, such examples include quartic curves (by Kondo), cubic surfaces and cubic threefolds (by Allcock, Carlson and Toledo), and some CalabiYau varieties (by Borcea, Voisin, and van Geemen). In the talk we will discuss two examples along these lines: (1) certain surfaces of general type with p_g=2 and K^2=1; (2) pairs consisting of a cubic threefold and a hyperplane section. This is joint work with R. Laza and G. Pearlstein.

10/3 
Izzet Coskun 
University of Illinois at Chicago 
Birational Geometry of moduli spaces of sheaves on surfaces and the BrillNoether Problem 

Abstract: In this talk, I will relate sharp Bogomolov inequalities on a surface to the ample cone of the moduli space of sheaves via Bridgeland stability. I will discuss weak BrillNoether theorems on rational surfaces. As a consequence, I will describe the classification of Chern characters on the plane and on Hirzebruch surfaces such that the general bundle of the moduli space is globally generated. This is joint work with Jack Huizenga.

10/10 
Midterm Recess 
10/17 
Beth Malmskog 
Colorado College 
Solving Sunit equations in Sage and Applications to Algebraic Curves. 

Abstract: Many finiteness and enumerative problems in number theory rely on the finiteness/enumeration of the set of solutions to the equation x+y=1 over the group of Sunits in a number field, where Sis a finite set of primes. In 1995, Nigel Smart solved certain Sunit equations to enumerate all genus 2 curves defined over the rationals with good reduction away from p=2. Smart's work build on that of of Baker, de Weger, Evertse, Yu, and many others. In 2016, following Smart's methods, Malmskog and Rasmussen found all Picard curves over Q with good reduction away from p=3, and Angelos Koutsianas described methods for enumerating, and in some cases explicitly describes, all elliptic curves defined over a number field with good reduction outside S. Both projects required Sage implementation of special cases of Smart's general method. In January 2017, Alejandra Alvarado, Angelos Koutsianas, Beth Malmskog, Christopher Rasmussen, Christelle Vincent, and Mckenzie West combined these implementations and created new functions to solve the equation x+y=1 over the Sunits of a general number field K for any finite set S of primes in K. The code is available on SageTrac and is under review for inclusion in future releases of Sage. This talk will give an overview of motivating problems and applications, the methods involved, and next steps to advance the theory and/or to improve this implementation.

10/24 
Abbey Bourdon 
Wake Forest University 
Degrees of CM Points on X_1(N) 

Abstract: The noncuspidal points of X_1(N) correspond to isomorphism classes of pairs (E,P), where E is an elliptic curve and P is a point on E of order N. If E has complex multiplication by an order in an imaginary quadratic field K, we say (E,P) is a KCM point. In this talk, I will give a compete classification of the degrees of KCM points on X_1(N)_{/K}, where K is any imaginary quadratic field. This is joint work with Pete L. Clark.

10/31 
Ari Shnidman 
Boston College 
Ranks of quadratic twists of abelian varieties 

Abstract: In 1977, Mazur showed that the "Eisenstein quotient" of J_0(p) has rank 0 (and so finitely many rational points). We show that for many primes p, there is a further quotient of J_0(p) such that a positive proportion of quadratic twists also have rank 0. This is a special case of a general result concerning abelian varieties with real multiplication. The proof uses recent our work with Manjul Bhargava, Zev Klagbsrun, and Robert Lemke Oliver, on the average size of the Selmer group of a 3isogeny in any quadratic twist family.

11/7 
Andrew Obus 
University of Virginia 
Reduction of dynatomic curves: The good, the bad, and the irreducible 

Abstract: The dynatomic modular curves parameterize oneparameter families of dynamical systems on P^1 along with periodic points (or orbits). These are analogous to the standard modular curves parameterizing elliptic curves with torsion points (or subgroups). For the family x^2 + c of quadratic dynamical systems, the corresponding modular curves are smooth in characteristic zero. We give several results about when these curves have good/bad reduction to characteristic p, as well as when the reduction is irreducible. We will also explain some motivation from the uniform boundedness conjecture in arithmetic dynamics.

11/14 
Renzo Cavalieri 
Colorado State University 
Graph formulas for tautological cycles 

Abstract: The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed by suitably decorated graphs.
The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence, and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.

11/21 
Nivedita Bhaskhar 
UCLA 
Brauer pdimensions of complete discretely valued fields 

Abstract: The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. One can analogously define the Brauerpdimension of F for p, a prime, by restricting to algebras with period, a power of p. The 'periodindex' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields.
In this talk, we look at the periodindex question over complete discretely valued fields in the socalled 'bad characteristic' case. More specifically, let K be a complete discretely valued field of characteristic 0 with residue field k of characteristic p > 0 and prank n (= [k:k^p]). It was shown by Parimala and Suresh that the Brauer pdimension of K lies between n/2 and 2n. We will investigate the Brauer pdimension of K when n is small and find better bounds. For a general n, we will also construct a family of examples to show that the optimal upper bound for the Brauerpdimension of such fields cannot be less than n+1. These examples embolden us to conjecture that the Brauer pdimension of K lies between n and n+1.
The proof involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier. This is joint work with Bastian Haase.

11/28 
Leo Goldmakher 
Williams College 
The unreasonable effectiveness of the PolyaVinogradov inequality 

Abstract: The PolyaVinogradov inequality, an upper bound on character sums proved a century ago, is essentially optimal. Unfortunately, it's also not so useful in applications, since it's nontrivial only on long sums (while in practice one usually needs estimates on sums which are as short as possible). The best tool we have to handle shorter sums is the Burgess bound, discovered in 1957; this is generally considered to supersede PolyaVinogradov, both because its proof is "deeper" (building on results from algebraic geometry) and because it is more applicable.
In this talk I will introduce and motivate both of these bounds, and then describe the unexpected result (joint with Elijah Fromm, Williams '17) that even a tiny improvement of the (allegedly weaker) PolyaVinogradov inequality would imply a major improvement of the (supposedly superior) Burgess bound. I'll also discuss a related connection between improving PolyaVinogradov and the classical problem of bounding the least quadratic nonresidue (joint with Jonathan Bober, University of Bristol).
