** Lectures will take place in room 1AD/100, Torre Archimede, Via Trieste 63, Padova. **

**Aims of the Summer School**

The aim of the Summer School is to introduce master students and Ph.D. students to the arithmetic theory of modular forms.

We split the summer school in two parts.

The first part (from August 27 to September 2) is meant to be an introduction to the main topics in the subject,

especially modular curves, classical holomorphic modular forms and p-adic modular forms, a' la Serre and a' la Katz.

The second part (September 5 to September 6) is a miniworkshop on more advanced topics.

More precisely, the two parts will be as follows:

a) Summer school from 28th august to 2nd september (around 30 hours divided in 6 classes ). An

intensive one week summer school and a more advanced 3/4 days advanced one in such a topic. The

first week is aimed to master students in algebra, number theory and geometry. The second even for PhD

students, but we believe that after the first week the master students will be able to attend the second

part. We consider the first week as an introduction to the different ways in which modular forms can be

understood. At the end of that week we will introduce the definition given by Katz and later by Coleman in

the overconvergent case. We hope to address also the definition of the eigencurve.

b) Advanced school/workshop from 2nd september to 6th september (more 15/16 hours more advanced

in a seminar style). In the second part of the summer school (the more advanced one) we will use all the

techniques which have been introduced to understand recent results of Iovita, Andreatta and Pilloni by

which overconvergent modular forms can be understood in terms of Katz’s sheaves methods (this at least

in the elliptic case). We will also try to give the perfectoid point of view (after the recent works of Chojecki,

Hansen,Johansson and Birkbeck)