### 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

Date | Speaker | Title | Abstract |
---|---|---|---|

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 | Counting twin prime polynomials | Twin prime polynomials are the function field analog of twin prime numbers. The talk presents joint work with Prof. Bary-Soroker about asymptotics and explicit formulas for the number of twin prime polynomials in light of a function field version of the Hardy-Littlewood prime tuple conjecture. | |

14.12.2017 | Non-archimedean hyperbolicity | I will explain a non-archimedean analogue of Brody hyperbolicity introduced by Cherry in the 90's, and explain a strategy for proving the non-archimedean Brody hyperbolicity of the moduli space of abelian varieties. This is ongoing work with Alberto Vezzani. | |

21.12.2017 | Singular units | A singular moduli is the j-invariant of an elliptic curve with complex multiplication. Since the 19th century, it is well-known that singular moduli are algebraic integers. Bilu, Masser, and Zannier asked whether there exist singular moduli that are even algebraic units. In earlier work, Habegger was able to show non-effectively that there are at most finitely many such singular moduli. The non-effectivity in his proof is due to a contingent Siegel zero within the proof of Duke's equidistribution theorem. In joint work of Bilu, Habegger, and myself, we were able to give an effective proof, with an explicit bound on the discriminant of possible singular units. |