## STNB2017(31è any)

### The space of $\Lambda$-adic modular forms and the Eisenstein family.

Xavier Guitart

#### Resum

You have to cover [Laf, \S4.1] and [Laf, \S4.2]. More concretely, first determine the space of $\Lambda$-adic modular forms. Then, as an example of $\Lambda$-adic modular form, present the Eisenstein family. For this, you will require the existence of the $p$-adic $L$-function attached to a Dirichlet character. Show in detail its existence following the notes [Gui]. This is a basic construction. In the second course we will attach $p$-adic $L$-functions to elliptic curves.

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