We will recall the definition and some properties of Huber rings and we will define and study spaces of continuous valuations.
We will explain in detail some examples of affinoid adic spaces including the unit ball (for rank 1 valuations). If time permits we will talk on completion and rational subsets.
We will review the construction of Newton-Okounkov bodies by Kaveh-Khovanskii and Lazarsfeld-Mustata, within a general framework of semigroups and cones associated to filtrations.
The use of modular symbols to attach p-adic Lfunctions to Hecke eigenforms goes back to the work of Manin et al in the 70s.
We will address the construction of arithmetic Newton-Okounkov bodies, and more generally functions on Newton-Okounkov bodies arising from filtrations, following ideas of Boucksom and collaborators.
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.
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.
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).
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.
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.