NTS Spring 2014/Abstracts

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January 23

Majid Hadian-Jazi (UIC)
Title: On a motivic method in Diophantine geometry

Abstract: By studying the variation of motivic path torsors associated to a variety, we show how certain nondensity assertions in Diophantine geometry can be translated to problems concerning K-groups. Then we use some vanishing theorems to obtain concrete results.

January 30

Alexander Fish (University of Sydney, Australia)
Title: Ergodic Plunnecke inequalities with applications to sumsets of infinite sets in countable abelian groups

Abstract: By use of recent ideas of Petridis, we extend Plunnecke inequalities for sumsets of finite sets in abelian groups to the setting of measure-preserving systems. The main difference in the new setting is that instead of a finite set of translates we have an analogous inequality for a countable set of translates. As an application, by use of Furstenberg correspondence principle, we obtain new Plunnecke type inequalities for lower and upper Banach density in countable abelian groups. Joint work with Michael Bjorklund, Chalmers.

February 13

John Voight (Dartmouth)
Title: Numerical calculation of three-point branched covers of the projective line

Abstract: We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups. This is joint work with Michael Klug, Michael Musty, and Sam Schiavone.

February 20

Nir Avni (Northwestern)
Title: Representation zeta functions

Abstract: I will talk about connections between the following: 1) Asymptotic representation theory of an arithmetic lattice G(Z). More precisely, the question of how many n-dimensional representations does G(Z) have. 2) The distribution of a random commutator in the p-adic analytic group G(Zp). 3) The complex geometry of the moduli spaces of G-local systems on a Riemann surface, and, more precisely, the structure of its singularities.

February 27

Jennifer Park (MIT)
Title: Effective Chabauty for symmetric power of curves

Abstract: While we know by Faltings' theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of Sym^d(X) that are not parametrized by a projective space or a coset of an abelian variety, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.

March 11

Yueke Hu (Madison)
Title: Local integrals of triple product L-function and subconvexity bound

Abstract: Venkatesh proposed a strategy to prove the subconvexity bound in the level aspect for triple product L-function. With the integral representation of triple product L-function, if one can get an upper bound for the global integral and a lower bound for the local integrals, then one can get an upper bound for the L-function, which turns out to be a subconvexity bound. Such a subconvexity bound was obtained essentially for representations of square free level. I will talk about how to generalize this result to the case with higher ramifications as well as joint ramifications.

April 10

Kartik Prasanna (Michigan)
Title: Algebraic cycles and Rankin-Selberg L-functions

Abstract: I will give a survey of a circle of results relating L-functions and algebraic cycles, starting with the Gross-Zagier formula and its various generalizations. This will lead naturally to a certain case of the Bloch-Beilinson conjecture which is closely related to Gross-Zagier but where one does not have a construction of the expected cycles. Finally, I will hint at a plausible construction of cycles in this "missing" case, which is joint work with A. Ichino, and explain what one can likely prove about them.

April 17

Davide Reduzzi (Chicago)
Title: Galois representations and torsion in the coherent cohomology of

Hilbert modular varieties

Abstract: Let F be a totally real number field, p a prime number (unramified in F), and M the Hilbert modular variety for F of some level prime to p, and defined over a finite field of characteristic p. I will explain how exploiting the geometry of M, and in particular the stratification induced by the partial Hasse invariants, one can attach Galois representations to Hecke eigen-classes occurring in the coherent cohomology of M. This is a joint work with Matthew Emerton and Liang Xiao.

April 24

Arul Shankar (Harvard)
Title: The average 5-Selmer rank of elliptic curves

Abstract: We use geometry-of-numbers techniques to show that the average size of the 5-Selmer group of elliptic curves is equal to 6. From this, we deduce an upper bound on the average rank of elliptic curves. Then, by constructing families of elliptic curves with equidistributed root number we show that the average rank is less than 1. This is joint work with Manjul Bhargava.

May 8

Melanie Matchett Wood (UW-Madison)
Title: Jacobians of Random Graphs and Cohen Lenstra heuristics

Abstract: We will consider the question of the distribution of the Jacobians of random curves over finite fields. Over a finite field, given a curve, we can associate to it the (finite) group of degree 0 line bundles on the curve. This is the function field analog of the class group of a number field. We will discuss the relationship to the Cohen Lenstra heuristics for the distribution of class groups. If the curve is reducible, a natural quotient of the Jacobian is the group of components, and we will focus on this aspect. We are thus led to study Jacobians of random graphs, which go by many names (including the sandpile group and the critical group) as they have arisen in a wide variety of contexts. We discuss new work proving a conjecture of Payne that Jacobians of random graphs satisfy a modified Cohen-Lenstra type heuristic.

Organizer contact information

Robert Harron

Sean Rostami

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