Research

During my PhD I studied modular forms mod p and their associated Galois representations, in particular I was interested into the question:

"How many mod p, two-dimensional, irreducible, odd Galois representations of Q of given conductor N are there?"

With the help of Panagiotis Tsaknias I created a database enconding quite a bit of information concerning congruences modulo p between caracteristic zero modular forms of level one, where p is a prime less than 2600. The database is basically a refined version of the table of this paper. If you happen to want to consult the database feel free to contact me.
During my Thesis I also got interested in supersingular elliptic curves in characteristic p (mainly thanks to a result of Serre connecting them to mod p modular forms). In particular, if one considers the supersingular isogeny class over the prime field attached to the Weil polynomial x^2+p then one notices that its rational endomorphism algebra is just an imaginary quadratic field. I then wondered whether these elliptic curves can be lifted to characteristic zero in a functorial way, just like ordinary once. Together with Filippo Nuccio we realized that indeed they can, and we plan to have a pre-print soon.

Next I looked at in some detail integral representations given by \ell-adic Tate modules arising from elliptic curves over finite fields of characteristic p\neq\ell. In this paper I suggest a way to read this integral information from the j-invariant of the curve. In the ordinary case this was well-known (but not-so-well advertized in the literature, I find). In the supersingular case it requires some observations, which might have some novelty. A global application is given: primes of a number field K which are completely split in the N-th torsion field of an ellitptic curve E over K can be characterized.

Recently with Jakob Stix we have been studying abelian varieties over finite fields. One result that we obtained is that there is a "very large" subcategory of all abelian varieties over the prime field F_p which can be described in terms of linear algebra data (with lattices and endomorphisms). This extends over F_p (with a completely different method) a description of the ordinary subcategory given by Deligne in the 60s.