NTD 2022 Titles and Abstracts

Torsion points in families of abelian varieties

  Abstract: Given an abelian scheme defined over \overline Q and an irreducible subvariety X which dominates the base, the Relative Manin-Mumford Conjecture (inspired by S. Zhang and proposed by Zannier) predicts how torsion points in closed fibers lie on X. The conjecture says that if such torsion points are Zariski dense in X, then the dimension of X is at least the relative dimension of the abelian scheme, unless X is contained in a proper subgroup scheme. In this talk, I will present a proof of this conjecture. As a consequence this gives a new proof of the Uniform Manin-Mumford Conjecture for curves (recently proved by Kühne) without using equidistribution. This is joint work with Philipp Habegger.

Distributions of Manin’s iterated integrals.

  Abstract: We recall the definition of Manin’s iterated integrals of a given length. We then explain how these generalise modular symbols and certain aspects of the theory of multiple zeta-values. In length one and two we determine the limiting distribution of these iterated integrals. Maybe surprisingly, even if we can compute all moments also in higher length we cannot determine a distribution for length three or higher. This is joint work with Y. Petridis and with N. Matthes. 

Congruences between modular forms and applications

Moments of Dirichlet L-functions 

Abstract: In recent decades there has been much interest and measured progress in the study of moments of L-functions. Despite a great deal of effort spanning over a century, asymptotic formulas for moments of L-functions remain stubbornly out of reach in all but a few cases. I will begin this talk by reviewing what is known for moments of the Riemann zeta-function on the critical line, and we will then consider the problem for the family of all Dirichlet L-functions of even primitive characters of bounded conductor. A heuristic of Conrey, Farmer, Keating, Rubenstein, and Snaith gives a precise prediction for the asymptotic formula for the general 2kth moment of this family. I will outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of approximations of this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees with the prediction of CFKRS. Moreover, it provides the first rigorous evidence beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions. This is joint work with Siegfred Baluyot and a product of the NSF Focused Research Group “Averages of L-functions and Arithmetic Stratification.”

Euler systems and the Birch–Swinnerton-Dyer conjecture for abelian surfaces

Abstract: Euler systems are one of the most powerful tools for proving cases of the Bloch–Kato conjecture, and other related problems such as the Birch and Swinnerton-Dyer conjecture. I will recall a series of recent works (variously joint with Loeffler, Pilloni, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4), and an explicit reciprocity law relating this to values of L-functions. I will then explain work in progress with Loeffler, in which we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces over Q, and modular elliptic curves over imaginary quadratic fields.