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.
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.
In the third talk we will continue the discussion of arithmetic Newton-Okounkov bodies and more.
Perfectoid spaces are specially interesting instances of adic spaces.
The p-stabilization of certain Eisenstein weight one forms are cuspidal-overconvergent forms, and they belong to the cuspidal locus of the p-adic eigencurve.
Estudiarem com definir un candidat a feix estructural per a un espai afinoide (de la forma Spa(A,A^+)) i en quins casos surt un feix.
The theory of Heegner points, which form a supply of algebraic points on a given elliptic curve, are one of the main tools used in proving the known cases of the Birch and Swinnerton-Dyer conjecture.