Number Theory: Benjamin Matschke (University of Bordeaux) "A database of rational elliptic curves with given bad reduction"

Thu, 21/04/201614:00-15:15
In this talk we present a database of rational elliptic curves with
good reduction outside certain finite sets of primes, including the
set {2, 3, 5, 7, 11}, and all sets whose product is at most 1000.
In fact this is a biproduct of a larger project, in which we construct
practical algorithms to solve S-unit, Mordell, cubic Thue, cubic
Thue--Mahler, as well as generalized Ramanujan--Nagell equations, and
to compute S-integral points on rational elliptic curves with given
Mordell--Weil basis.
Our algorithms rely on new height bounds, which we obtained using the
method of Faltings (Arakelov, Parshin, Szpiro) combined with the
Shimura--Taniyama conjecture (without relying on linear forms in
logarithms), as well as several improved and new sieves.
In addition we used the resulting data to motivate several conjectures
and questions, such as Baker's explicit abc-conjecture, and a new
conjecture on the number of S-integral points of rational elliptic
This is joint work with Rafael von Känel.