Alina Carmen Cojocaru
Title: Serre curves in one-parameter families
Abstract: For an elliptic curve E defined over the field of rational numbers, we consider its absolute Galois representation. By a celebrated theorem of Serre from 1972, if E is without complex multiplication, then its absolute Galois representation has open image. If the image has index 2 in the codomain, then E is called a Serre curve. I will discuss the occurrence of Serre curves as specializations of an elliptic surface. This is joint work with David Grant (University of Colorado) and Nathan Jones (Hausdorff Institute and Max Planck Institute).
Title: Euler characteristics of p-torsion Iwasawa modules
Abstract: It is possible to extract BSD-type information from a Selmer group of an elliptic curve over the rationals without complex multiplication: this is done by considering it as an Iwasawa module and by computing a K-theoretic invariant of this module called the Euler characteristic. I will discuss the general problem of computing Euler characteristics of Iwasawa modules, focusing in particular on the p-torsion ones.
Title: Two-dimensional Integration
Abstract: I will give a summary of the current state of integration over two-dimensional local fields, finite dimensional spaces and algebraic groups over two-dimensional local fields, including a discussion of the model-theoretic approach of Hrushovski and Kazhdan.
Title: Euler systems of rank r and Kolyvagin systems
Abstract: For a p-adic Galois representation T, I will devise an Euler system/Kolyvagin system machinery which as an input takes an Euler system of rank r (in the sense of Perrin-Riou), and gives a bound on the Bloch-Kato Selmer group in terms of an r×r determinant. I will give two fundamental applications of this refinement: The first with the (conjectural) Rubin-Stark elements; and the second with Perrin-Riou's (conjectural) p-adic L-functions.
Scattering theory for automorphic forms related to the Dirichlet series
Abstract: We construct a scattering system for automorphic forms that satisfies the axioms of Lax and Phillips. We obtain a spectral interpretation which relates the hyperbolic spinor Laplacian to the Dirichlet series associated to the real-analytic Eisenstein series.
Products of Hecke L-functions of holomorphic cusp forms
Explicit formulas in terms of the divisor function for the sum over the weights of products of Hecke L-functions of holomorphic cusp forms of the full modular group are given. The non-vanishing of Hecke L-functions at ½ and at any other given point for infinitely many holomorphic cusp forms is derived. Methods used to prove the above results are based on spectral theory of automorphic functions and Kuznietsov summation formulas.