School of Mathematics

Zagier's conjecture on zeta(F,4)

Alexander Goncharov
Yale University; Member, School of Mathematics
November 10, 2017

Abstract: We prove that the weight 4 Beilinson's regulator map can be expressed via the classical n-logarithms, $n \leq 4$.

This plus Borel's theorem implies Zagier's conjecture, relating the value of the Dedekind zeta functions at $s=4$ and the classical tetralogarithm.  Another application is to the values of L-functions of elliptic curves over $Q$ at $s=4$.

One of the new tools is a connection between cluster varieties and polylogarithms, generalising our work with V. Fock relating cluster varieties and the dilogarithm.

Pseudorepresentations and the Eisenstein ideal

Preston Wake
University of California, Los Angeles
November 9, 2017

Abstract:  In his ladmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences.  We'll also discuss some partial results in the composite-level case.  This is joint work with Carl Wang-Erickson.

Topological and arithmetic intersection numbers attached to real quadratic cycles

Henri Darmon
McGill University
November 8, 2017

Abstract:  I will discuss a recent conjecture formulated in an ongoing project with Jan Vonk relating the intersection numbers of one-dimensional topological cycles on certain Shimura curves to the arithmetic intersections of associated real multiplication points on the Drinfeld p-adic upper half-plane.  Numerical experiments carried out with Vonk and James Rickards supporting the conjecture will be described.

An Euler system for genus 2 Siegel modular forms

David Loeffler
University of Warwick
November 8, 2017

Abstract:  Euler systems are compatible families of cohomology classes for a global Galois represenation, which plan an important role in studying Selmer groups.  I will outline the construction of a new Euler system, for the Galois representation associated to a cohomological cuspidal automorphic representation on the symplectic group GSp(4).  This is joint work with Chris Skinner and Sarah Zerbes. 

A derived Hecke algebra in the context of the mod $p$ Langlands program

Rachel Ollivier
University of British Columbia
November 8, 2017

Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of smooth representations of G and the category of H-modules. When the field of coefficients has characteristic zero this link is well understood by work of Bernstein and Borel.

Time quasi-periodic gravity water waves in finite depth

Massimiliano Berti
International School for Advanced Studies
November 8, 2017
We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water waves solutions, namely periodic and even in the space variable $x$, of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a set of asymptotically full measure. This is a small divisor problem.