# Arithmetic, Modular Forms And p-Adic Variation

1. Plan

**Lecture 1: Arithmetic and Modular forms.** General introduction and motivation to the mini course. We introduce Modular forms: definitions, examples, *L*-functions. We explain interesting arithmetic problems related with modular forms.

**Lecture 2: Modular forms and algebraic geometry.** Modular curves, modular sheaves, Modular forms as section of sheaves.

**Lecture 3: ***p***-adic study of the Riemann zeta function.** We consider our first example of a *p*-adic family and the connection with the construction of the *p*-adic analogue of the Riemann zeta function. The goal of this lecture is to give a motivation to the introduction of *p*-adic families of modular forms.

**Alternative lecture 3: Elements of rigid geometry.** The idea is to introduce some basic definitions and facts about rigid geometry. This language will be strongly used in the rest of these lectures.

**Lecture 4: Families of Banach spaces.** We explain the spectral theory for certain operators on Banach A-modules. For us the basic example of the algebra A will be the rigid functions on the natural rigid variety of “p-adic weights of modular forms”.

**Lecture 5: ***p***-adic families of modular forms.** Following [2] we construct *p*-adic families of modular forms.

2. Original goal and technical explanations

I would like to explain the goals and the possibilities of the plan above. The original motivation of this mini course was to explain the new ideas of the preprint [1]. In that work the authors consider three *p*-adic families of modular forms and construct certains *p*-adic analytic functions which are related to certain values of some L-series attached to triples of modular forms. The main task of Andreatta-Iovita’s approach is to construct *p*-adic families of modular and de Rham sheaves. Moreover, we need to *p*-adically interpolate certain connections on these de Rham sheaves. These geometric constructions were the original motivation to this serie of lectures.

These constructions are technicals and require a considered background regarding the duration of the mini course and the audience expected. For that reason I decided to cover some more classical background which is crucial in the direction of the original goal. Moreover, in private discutions I will explain the Andreatta- Iovita’s new ideas to the interested people. More precisely, the new goal of this mini-course is to explain the arithmetic importance and construction of p-adic families of finite slope modular forms.

I consider the new goal could make the course not that technical. The goal of lectures 1 and 2 is to introduce modular forms and describe them in geometric terms. A good reference is [3]. Lecture 3 can be considered as a motivation to the *p*-adic variation of arithmetic objects. I also propose “alternative lecture 3” instead of “lecture 3”, more technical but useful for the rest. In lectures 4 and 5 I will explain the construction of *p*-adic families of modular forms carried out by R. Colemam. In fact, the main ideas of R. Coleman are crucial in [1].

**References:**

[1] F. Andreatta, A. Iovita, – Triple product p-adic L-functions associated to finite slope p-adic families of modular forms, preprint 2017.

[2] Coleman, R. Coleman, – P-adic Banach spaces and families of modular forms, Invent. Math. 127, 417-479 (1997).

[3] F. Diamond, J. Shurman, – A first course in Modular forms, Grad. Texts Math. 228, (2005) Springer-Verlag.