### Number Theory Seminar HS 2016

Unless otherwise stated, all talks start at 14.15 and take place in lecture hall 05.001, Spiegelgasse 5

Information for speakers

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

06.10.2016 | Weakly admissible lattices, o-minimality, and Diophantine approximation | We present new estimates for the number of lattice points in sets such as aligned boxes which, in certain cases, improve on Skriganov's celebrated counting results. We establish a criterion under which our error term is sharp, and we provide examples in dimensions $2$ and $3$ using continued fractions. Moreover, we use o-minimality to describe large classes of sets to which our counting results apply. If time permits we also present a similar counting result for primitive lattice points, and apply the latter to the classical problem of Diophantine approximation with primitive points as studied by Chalk, Erd\H{o}s and others. | |

27.10.2016 | A New Northcott Property for Faltings' Heights | We develop explicit techniques using tools from integral $p$-adic Hodge theory to study the change in Faltings' height within an isogeny class of CM abelian varieties. Assuming the Colmez conjecture, this results in a new Northcott property for Faltings' heights for CM points. On the Hilbert modular variety we are moreover able to develop a Colmez-type formula for the Faltings' height of all CM points. | |

10.11.2016 | The height of subvarieties | Heights are a fundamental tool in diophantine geometry. I will show how to extend the definition of the Weil height for points in the projective space to subvarieties of (multi)projective spaces and what properties can be proved about it. | |

17.11.2016 | Bounding the height of certain algebraic numbers | Especially since a paper of Bombieri-Masser-Zannier in 1999, bounding the height became the standard first step in getting finiteness for certain problems. Therefore, bounding the height of certain families of algebraic numbers or points from above is an active area of current research. In my master thesis I showed that the height of an algebraic number $\alpha$ satisfying $\alpha^n + (1-\alpha)^n + (1+\alpha) =1$ for some integer $n\ge2$ or satisfying $\alpha^r + (1-\alpha)^s =1$ for some integers $r\ge1$ and $s\ge1$, not both $1$, is bounded. The proofs are fully effective and I gave concrete values for bounds in these two situations. Both cases are special ones of a much more general result by Amoroso-Masser-Zannier (to appear). In this talk I will present my approach to prove these two results using a version of Siegel's Lemma and I will focus on how I dealt with the ``non-vanishing`` problem usually arising in this context. The latter is done differently and much simpler than by Amoroso-Masser-Zannier due to the special cases here, while the general method is the same. | |

24.11.2016 | On division fields of CM abelian varieties | Let K be a number field and A/K be an abelian variety with complex multiplication. We consider the extensions of K generated by torsion points of A and give uniform bounds for their degrees in terms of K and of the dimension of A. This refines a result of Ribet and has applications to some cases of the (uniform) Rasmussen-Tamagawa conjecture. | |

08.12.2016 | Simultaneous torsion in the Legendre family of elliptic curves | ||

22.12.2016 | The Bounded Height Conjecture for Semiabelian Varieties | The Bounded Height Conjecture of Bombieri, Masser and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian variety G there is an upper bound on the Weil height of the points contained in its intersection with the union of all algebraic subgroups having (at most) complementary dimension in G. After partial work of many authors, Habegger proved the conjecture completely for both tori and abelian varieties in 2009. In my talk, I will discuss how to prove the conjecture for general semiabelian varieties. |