tbranzov

A seminar talk by Boaz Moerman, Utrecht University

Abstract: The Chinese remainder theorem states that given coprime integers p_1, ..., p_n and integers a_1, ..., a_n, we can always find an integer m such that m ~ a_i mod p_i for all i. Similarly given distinct numbers x_1,..., x_n and y_1, ..., y_n we can find a polynomial f such that f(x_i)=y_i. These statements are two instances of strong approximation for the affine line (over the integers Z and the polynomials k[x] over a field k). In this talk we will consider when an analogue of this holds for special subsets of Z and k[x], such as squarefree integers or polynomials without simple roots, and different varieties. We give a precise description for which subsets this holds on a toric variety.

An ICMS seminar talk by Mladen Dimitrov, University of Lille

Abstract: An amazing feature of the p-adic L-functions is their ability to live in families, thus their laws are governed by the geometry of p-adic eigenvarieties. In this lecture we will illustrate this philosophy through examples coming from classical modular forms and the Coleman-Mazur eigencurve.

A seminar talk by Marta Pieropan, Utrecht University

Abstract: In joint work with Damaris Schindler we develop a new version of the hyperbola method for counting rational points of bounded height that generalizes the work of Blomer and Brüdern for products of projective spaces. The hyperbola method transforms a counting problem into an optimization problem on certain polytopes. For rational points on subvarieties of toric varieties, the polytopes have a geometric meaning that reflects Manin's conjecture, and the same holds for counts of Campana points of bounded height. I will present our results as well as some general heuristics.