Abstracts for Number theory lunch seminar

I defined the Stickelberger splitting map in the case of abelian extensions $F/\mathbb{Q}$ in my Ph.D thesis in 1990. The construction used the classical Stickelberger's theorem. For abelian extensions $F/K,$ with an arbitrary totally real base field $K,$ the construction cannot be generalized since Brumer's conjecture (the analogue of  Stickelberger's theorem) is not proved yet at that level of generality.

In 1980, Pomerance conjectured that if $k$ is a positive integer such that the first $\phi(k)$ primes co-prime to $k$ form a reduced residue system mod$k$, then $k \leq 30.$ This conjecture is still open. We shall discuss some old and new results towards this conjecture and how the Jacobsthal function plays an important role in this problem.

I will report on my recent joint work with S. Arias-de-Reyna and S. Petersen. We have proven conjecture of Geyer and Jarden on torsion part of the Mordell-Weil group for abelian varieties with big monodromy. By definition, an abelian variety over a fin. gen. field has big monodromy, if the image of Galois representation attached to its l-torsion points contains the full symplectic group, for almost all primes l. In addition, we have also found a new, large family of abelian varieties with big monodromy defined over fin. gen. fields of arbitrary characteristic.

Let $F$ be a binary form over the integers and consider the exponential Diophantine equation $F(x,y)=z^n$ with $x$ and $y$ coprime. In general it seems very difficult to study this equation, but as we will explain in this talk, for so-called Klein forms $F$, the modular method can provide a good starting point. By combining this with a new method for solving infinite families of Thue equations, we can show in particular that there exist infinitely many (essentially different) cubic (Klein) forms $F$ for which the equation above has no solutions for large enough exponent $n$.