Galois representations and torsion classes
I will describe joint work in progress with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne on potential modularity for elliptic curves over imaginary quadratic fields.
Fermat’s Last Theorem over some totally real number fields
In the past few years, numerous progress have been achieved about the Fermat equation over number fields. We have in particular a criterion, due to Freitas and Siksek, allowing to establish the asymptotic Fermat’s Last Theorem over certain totally real number fields.
Surjectivity of Galois representations of quadratic Q-curves
One initial tool in Wiles’ proof of Fermat’s last theorem is using the Galois representation on the p-torsion of an elliptic curve and proving that this representation is irreducible except for p very small (with an absolute bound).
Which integers are sums of seven cubes?
In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454.
On the exceptional zeros of p-adic L-functions of Hilbert modular forms
The use of modular symbols to attach p-adic L-functions to Hecke eigenforms goes back to the work of Manin et al in the 70s.
Bounds for Mordell-Weil ranks via Fourier coefficients of automorphic forms on Mp(2)
Let E be an elliptic curve of conductor N defined over the rationals, K a quadratic extension of absolute discriminant D, and e the sign of the Hasse-Weil L-function of E over K.
Lifts of Hilbert modular forms and application to a conjecture of Gross
In this talk, we prove the existence of certain lifts of Hilbert modular forms to general odd spin groups.
Ramification of the Eigencurve at classical RM points
J.Bellaïche and M.Dimitrov have shown that the p-adic eigencurve is smooth but not étale over the weight space at p-regular theta series attached to a character of a real quadratic field F in which p splits.
p-adic modular forms over unitary Shimura curves
We study some p-adic properties of modular forms over unitary Shimura curves, including a description of their relation with automorphic representations via p-adic Hodge theory, overconvergence and classicality, and p-adic families.
Triple product p-adic L-functions
We consider three Coleman families of modular forms and we would like to construct a p-adic L-function interpolating the algebraic part of central values of the triple product L-function attached to these families and classical weights.