Number Theory Seminar HS 2017
Unless otherwise stated, all talks start at 14.15 and take place in Seminarraum 05.001, Spiegelgasse 1
Information for speakers
|05.10.2017||A statistical version of a conjecture of Lang||Adopting a statistical point of view on diophantine problems often allows to gain insight into the questions at hand. This idea was for instance illustrated over the last years in the context of the arithmetic of elliptic curves by the astonishing achievements of Bhargava and his collaborators. The goal of this talk will be to provide another illustration of the above principle by investigating the conjecture of Lang predicting a lower bound for the canonical height of non-torsion points on elliptic curves defined over number fields.|
|26.10.2017||Volumes of quasi-arithmetic hyperbolic lattices||I will recall some connections between hyperbolic geometry and number theory, in particular concerning the volumes of arithmetic manifolds. Then I will present some of my work concerning quasi-arithmetic lattices.|
|02.11.2017||Expansions of quadratic numbers in a p-adic continued fraction.||It goes back to Lagrange that a real quadratic irrational always has a periodic continued fraction. Starting from decades ago, several authors generalised proposed different definitions of a p-adic continued fraction, and the definition depends on the chosen system of residues mod p. It turns out that the theory of p-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagrange's theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known. In recent work wth F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the p-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion relies on deep theorems in transcendence and diophantine analysis, and, somewhat surprisingly, depends on the “real” value of the p-adic continued fraction.|
|09.11.2017||Torsion in subvarieties of abelian varieties||A theorem of Raynaud shows that a given subvariety of an abelian variety only contains a finite number of (maximal) torsion cosets; that is, translates of abelian subvarieties by a point of finite order. This result is also known as the Manin-Mumford conjecture for abelian varieties.
In this talk, we focus on bounding the number of maximal torsion cosets. We present an interpolation method via Galois representations which allow us to give an explicit bound with a "good" dependence on the degree of the subvariety. This is joint work with Aurélien Galateau.
|23.11.2017||On Faltings' delta-invariant||Faltings' delta-invariant of compact Riemann surfaces plays a crucial role in Arakelov theory of arithmetic surfaces. It is the archimedean contribution of the arithmetic Noether formula. We will give a new formula for this invariant in terms of integrals of theta functions. As applications, we obtain a lower bound for delta only in terms of the genus, a canonical extension of delta to abelian varieties and an upper bound for the Arakelov-Green function in terms of delta.|
|30.11.2017||Some recent results on the arithmetic of elliptic curves||This talk will describe some recent progress on aspects of the Birch--Swinnerton-Dyer conjecture for elliptic curves over Q.|
|07.12.2017||Anabelian Geometry with étale homotopy types||Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. I will report on progress obtained in this direction in joint work with Alexander Schmidt.|