Summer School:
p-adic Cohomology and Explicit Arithmetic

Mini-courses

• Dragoș Frățilă (Strasbourg)
  Comparison theorems, periods and applications Abstract

  In algebraic geometry there are several possible cohomology theories, each one with its own features. Comparison theorems relate them, this enables one to pass informations from one to the other. The most classical comparison theorem is the one relating de Rham cohomology with singular cohomology. The coefficients appearing in this this comparison are called periods. In this course we will recall the classical complex story and then focus on several p-adic variants. We will focus especially on applications of those, namely on transcendence, field of definition and algebraic cycles.

• Davide Lombardo (Pisa)
  Explicit p-adic Hodge theory for elliptic curves over Qp Abstract
  One of the main aims of p-adic Hodge theory is the classification of p-adic Galois representations in terms of semilinear algebraic data. For elliptic curves over Q_p, this goal has essentially been achieved: the Galois action on their rational p-adic Tate modules admits a precise description in p-adic Hodge theoretic terms. However, this is not the end of the story, since extracting concrete arithmetic information from this classification is itself a nontrivial task. In this mini-course, I will first introduce the relevant linear-algebraic structures and explain how they encode the Galois representations in question. I will then focus on the delicate case of potentially supersingular elliptic curves. In this setting, I will show how to write down simple, explicit polynomials realising the torsion representations of such curves and how to make the abstract correspondence between elliptic curves and filtered (\varphi, G_{K/Q_p})-modules computable. In particular, this yields a simple recipe to determine the images of these Galois representations.

• Margherita Pagano (Imperial College, London)
  Brauer-Manin obstruction via refined Swan conductors Abstract
  The Brauer--Manin obstruction is one of the most powerful tools available to study the image of rational points inside the set of p-adic points for varieties over number fields. In this course, I will explain how the interplay between elements of the Brauer group in characteristic zero and differential forms in positive characteristic can be used to study the Brauer--Manin obstruction, and hence rational points on algebraic varieties. To this end, we will introduce the notion of Swan conductor. This will allow us to define a natural filtration on the Brauer group whose graded pieces are, on the one hand, closely related to de Rham cohomology in positive characteristic, and on the other hand tell us which Brauer classes can contribute to the Brauer--Manin obstruction.