p-adic modular forms

The aim of this thesis will be introducing an analogue of the classical modular forms that can work in the p-adic environment. To do so, we will first try to make sense of a modulo-p concept of modular forms. As the classical object is defined over the complex number there is not an immediate way to make this reduction. In order to do so, we have to utilize the q-expansion principle to obtain an “integral" object (we use the quote-on-quote to remind that the q -expansion of a modular from lives in the localization of ℤ at a prime). So the first idea will be to work with those object, to do so we will follow [8]. Once speaking of modular forms modulo p , and modulo pn makes sense, we will start talking about the p-adic theory as described in [6]. This first construction will be quite easy, but it will have important consequences on the notion of weight of a modular form. While the approach described above is quite natural and efficient in order to have something to work with (we will end up with q -expansion of modular forms automatically), to retrive the geometrical nature of those object will be much harder if we proceed on this path. Therefore we look at the theory of modular forms as section of the sheaf of invariant differentials on the modular curve, following [3]. In the end we will end up with two different definitions, one which gives us objects that are easier to grasp (and to compute), the other which has a more clear geometric nature (which is the reason why we study modular forms in first place). The last section of this thesis show the relation between those two, proving that we can recover one object in the first form by object defined in the second way and vice-versa