We will speak about the Coleman-Oort conjecture on totally geodesic subvarieties of Ag, the moduli space of abelian varieties of dimension g. In order to understand the subject, we will summarize properties of Jacobians of curves, abelian varieties and the Torelli morphism.
The examples of totally geodesic subvarieties known so far are obtained as families of Jacobians of Galois coverings of curves f:C→ C'. All of them satisfy a sufficient condition, which we will denote by (∗). We will briefly explain why condition (∗) works and we will explicitly construct and study a particular example.
We will show that condition (∗) gives us a bound on the genus g' of C'. Computer calculations allow us then to say that, up to a certain genus bounded genus g of C, there are only 6 families: all of them describe Galois coverings of elliptic curves. We will quickly illustrate them.
Finally, we study the Prym maps of these families (which we will define accordingly): we will demonstrate that these families are fibered, via their Prym map, in totally geodesic curves.
A conjecture due to E. Artin stated that a system of $r$ homogeneous forms having degrees $d_1, d_2,\ldots, d_r$ in $n$ variables defined over a $p$-adic field $K$ should have a nontrivial common zero defined over $K$ as long as $n > d_1^2 + \ldots + d_r^2$.
Although this conjecture turned out to be false in general, the conjecture also turned out to be true in many cases. The determination of exactly when the conjecture is true and when it is false has become a fascinating and important question in number theory, as the question is directly related to questions of finding rational points on varieties.
This talk will focus on the motivation for Artin's conjecture, cases when the conjecture is true, some counterexamples, and some words on the techniques involved.