Number Theory Days in Lille

Session

Arithmetic geometry and Galois theory

11 juil. 2019, 11:00

Salle de réunion (Université de Lille)

49. Density of specialization sets and Hasse principle in families of twisted Galois covers

Joachim König (KAIST)

11/07/2019 11:00

We discuss results on the structure of the set of all specializations of a Galois cover f:X->P^1 with group G over a number field k. Hilbert's irreducibility theorem yields that this set contains infinitely many G-extensions of k. A natural question is then how large this specialization set is compared to the set of all G-extensions. We present evidence for the following conjecture: If f is...

53. The Mumford--Tate conjecture implies the algebraic Sato--Tate conjecture

Victoria Cantoral-Farfan (ICTP)

11/07/2019 13:30

The famous Mumford—Tate conjecture asserts that, for every prime number $\ell$, Hodge cycles are ${\mathbb{Q}}_{\ell }$ linear combinations of Tate cycles, through Artin's caparisons theorems between Betti and étale cohomology. The algebraic Sato—Tate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized...

44. Counting points on algebraic varieties over the rational numbers

Philip Dittmann (KU Leuven)

11/07/2019 15:00

For both conceptual and practical reasons it is useful to have estimates on the number of points of algebraic varieties over Q, usually phrased in terms of asymptotics as the height of points increases. I will present a new such estimate, improving previous results by Bombieri, Pila, Heath-Brown, Browning, Salberger, Walsh and others. Time permitting, I will present an application to bounding...