### Number Theory Seminar FS 2015

Unless otherwise stated, all talks start at 14.15 and take place in the Alte Universität

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

21.04.15 | Abelian varieties and maximal orders | I study abelian varieties whose endomorphism ring is a maximal order. The algebraic properties of these orders, which can be seen as non-commutative analogs of Dedekind domains, allow to prove structure theorems for the aforesaid abelian varieties. For example, they are always products of simple varieties. Furthermore, this kind of results also give information for an arbitrary abelian variety, since it is isogenous to another one as above. I give also an application to the size of the torsion part of the group of rational points over a number field. | |

28.04.15 | Singular modular that are S-units / Generalized Jacobians and additive extensions of elliptic curves | ||

07.05.15 | Unlikely Intersections in certain families of abelian varieties and the polynomial Pell equation. | Let E_t be the Legendre elliptic curve of equation Y^2=X(X-1)(X-t). In 2010 Masser and Zannier proved that, given two points on E_t with coordinates algebraic over Q(t), there are at most finitely many specializations of t such that the two points become simultaneously torsion on the specialized elliptic curve, unless they were already generically linearly dependent. This fits inside the framework of the so-called Unlikely Intersections. As a natural higher-dimensional analogue, we considered the case of n generically independent points on E_t with coordinates algebraic over Q(t). Then there are at most finitely many specializations of t such that two independent relations hold between the specialized points. We recently also dealt with the case of points on Jacobians of genus two curves. This has applications in the study of solvability of the polynomial (almost) Pell equation. This is joint work with Laura Capuano. | |

21.05.15 | Frobenius distribution for pairs of elliptic curves and exceptional
isogenies | We will discuss a proof of the following result: if E and E' are two elliptic curves over a number field K, then there exist infinitely many primes p of K such that the reductions of E and E' modulo p are geometrically isogenous. The proof relies on the arithmetic dynamics of Hecke correspondences. | |

28.05.15 | Bad reduction of curves with CM jacobians | An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves. We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over the algebraic closure of the rationals with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence, except for finitely many examples, such a curve will always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit infinite families of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of the rationals of degree 4 that contains F for some specific F | |

19.06.15 | Solving S-unit and Mordell equations via Shimura-Taniyama conjecture | Joint work with Benjamin Matschke. In the first part of this talk, we shall present new practical algorithms which solve S-unit and Mordell equations by combining the method of Faltings (Arakelov, Parshin, Szpiro) with the Shimura-Taniyama conjecture. Our algorithms do not use lower bounds for linear forms in logarithms and they considerably improve the actual best algorithms. In the second part we plan to discuss in detail the construction of a sieve used in the algorithm for Mordell equations. Our sieve settles in particular the open problem of efficiently enumerating integral points of bounded height on elliptic curves. |