Intercity Number Theory SeminarDecember 18, Leiden. In Snellius 412; PhD defense Erik Visse in the Academiegebouw in the center of Leiden.
|10:00-11:00||Erik Visse (Leiden), PhD defense|
|13:45-14:45||Rachel Newton (Reading), Number fields with prescribed norms|
Abstract. Let G be a finite abelian group, let k be a number field, and let x be an element of k. We count Galois extensions K/k with Galois group G such that x is a norm from K/k. In particular, we show that such extensions always exist. This is joint work with Christopher Frei and Daniel Loughran.
|15:00-16:00||Daniel Loughran (Manchester), Integral points on Markoff surfaces|
Abstract. Integral solutions to Markoff-type equations of the form x2 + y2 + z2 - xyz = m were studied by Ghosh and Sarnak. In this talk we explain how to re-interpret their work using the Brauer-Manin obstruction, and quantify the number of such surfaces which fail the integral Hasse principle. This is joint work with Vlad Mitankin.
|16:15-17:15||Efthymios Sofos (Bonn), The size of the primes p for which a Diophantine equation is not soluble modulo p|
Abstract. The set of the primes p for which a variety over the rational numbers has no p-adic point plays a fundamental role in arithmetic geometry. This set is deterministic, however, we prove that if we choose a typical variety from a family then the set has random behaviour. We furthermore prove that this behaviour is modelled by a random walk in Brownian motion. This has several consequences, the main one being the description of the finer properties of the distribution of the primes in this set via the Feynman-Kac formula.
Belgian-Dutch Algebraic Geometry seminarFebruary 1, Utrecht.
Intercity Number Theory SeminarMarch 15, Utrecht.