ICERM Fall 2015
Research Seminar

Thursday 4-5pm, 11th floor lecture room

September 17

David Harvey (University of New South Wales)
Counting points on hypersurfaces

I will explain some of the main ideas of my paper "Computing zeta functions of arithmetic schemes", which gives a suite of algorithms for computing zeta functions of varieties over finite fields or over the integers. These algorithms are completely general in the sense that they impose no smoothness or nondegeneracy conditions on the variety; yet as a function of p, their complexity matches the best known results for more restricted classes of varieties such as hyperelliptic curves.

September 24

Joe Silverman (Brown University)
Abelian Varieties, Canonical Heights, and Counting Rational Points

The number of rational points of height less than B on an abelian variety grows roughly like (log B)r, where r is the rank of the Mordell-Weil group. Neron originally developed the theory of canonical heights in order to prove an asymptotic growth rate C(log B)r. Neron's construction is local, while Tate gave an easier global construction. In this talk I will describe Weil's height machine, explain Tate's construction of canonical heights, give their basic properties, and survey some of the places that they appear in arithmetic geometry. As time permits, I will explain recent work (joint with Shu Kawaguchi) on semi-definite positivity of canonical heights associated to nef divisors on abelian varieties and how nef heights are used to prove a conjecture in arithemtic dynamics.

October 1


October 8

Holly Swisher (Oregon State University)
Hypergeometric functions over finite fields, I

Based on the developments of many people, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to the classical hypergeometric functions. We present a systematic approach for translating certain classical hypergeometric series transformations to the finite field setting via an explicit dictionary. Some specific examples will be discussed. Additional motivation and a Galois perspective will be discussed next week in part II by Ling Long. This is joint work with Jenny Fuselier, Ling Long, Ravi Ramakrishna, and Fang-Ting Tu.

October 15

Ling Long (Louisiana State University)
Hypergeometric functions over finite fields, II

In this talk, we will discuss the Galois perspective of hypergeometric functions over finite fields. In particular we will associate Galois representations to the classical 2F1 hypergeometric functions with rational parameters via the generalized Legendre curves. Then we will use the Galois perspective to translate several types of classical hypergeometric formulas to the finite field settings. This is a joint work with J. Fuselier, R. Ramakrishna, H. Swisher and F.-T. Tu.

October 22


October 29

Winnie Li (Pennsylvania State University)
Langlands L-functions and counting geodesic cycles

The zeta function of an algebraic variety counts points. The Selberg zeta function counts closed geodesic cycles on a compact Riemann surface. For combinatorial objects like graphs and complexes, their zeta functions count closed geodesic cycles. In this talk we shall explain how the Langlands L-function arises naturally in the combinatorial zeta function for the 2-dimensional complex which is a finite quotient of either an apartment or the building attached to PGL(3) and PGSp(4).

November 5

Marie France Vigneras (Jussieu)
Representations modulo p of reductive p-adic groups, surprises, examples and questions

After the Abe-Henniart-Herzig-V. classification of admissible irreducible representations of modulo p of reductive p-adic groups in terms of supercuspidal representations and the Abe-V. classification of simple modules modulo p of pro-p Iwahori algebras, one wishes to understand the modulo p representations which are supercuspidal or not admissible irreducible, appearing in the p-adic local Langlands correspondence.

November 12


November 19

John Cremona (Warwick)
Black Box Galois Representations

The Serre-Faltings-Livne method concerns the comparison of two 2-dimensional 2-adic Galois representations of the absolute Galois group G_K of a number field K. Assuming that both are known to be unramified outside the same set S of primes of K and that they have the same determinant, the method provides a finite set T of primes of K, disjoint from S and depending only on K and S, such that if the two representations have the same trace at the Frobenius elements of all the primes in T then they are isomorphic. When we describe a Galois representation with data K, S as a a "Black Box", we mean that the only information we know about it is the trace and determinant of Frobenius at any given prime not in S; in these terms, the SFL method allows us to determine whether two black box representations are isomorphic. In this talk we consider black box representations from a slightly wider perspective, and consider how much information we can obtain about such a representation by asking only a finite number of questions, such as residual reducibility, the determinant character, and the size and structure of the isogeny class of the representation. This is joint work with Alejandro Argaez.

November 26


December 3