p-Adic variation of motives

点击翻译按钮获取中文摘要

Let us fix a number field F and a prime integer p. One of the most mysterious conjectures in Number Theory, known as the Langlands, Clozel, Fontaine-Mazur conjecture is the prediction of a bijection, respecting L-functions, between irreducible, continuous, n-dimensional p-adic valued, Galois representations of the absolute Galois group of F, which are unramified outside a finite number of places and de Rham at the paces above p, and cuspidal algebraic automorphic representations of GL_n of the adeles of F. This conjecture seems very difficult to study and prove, but it has been observed that the objects appearing in the conjecture can be seen as points in certain p-adic analytic varieties, and as such they are easier to understand. In particular this point of view has had as consequence constructions a p-adic L-functions attached to either Galois representations or automorphic forms and comparisons between them. This method of study Galois representations and automorphic forms, i.e. by deforming them p-adically, is called "p-adic variation of motives" and it has been started back in the 80's by B. Mazur, H. Hida, R. Coleman.    In the present project, after having studied p-dic variations of modular forms, de Rham classes and connections related to automorphic forms on GL_2, proposes to move forward and apply the experience we gained to automorphic forms on GSp_4 and Hilbert modular forms, in new situations. Moreover, back to GL_2, we intend to study p-adic variation of half integral weight modular forms. What is interesting in this project is that the modular forms whose p-adic variation we would like to study are non-holomorphic and therefore have not yet been studied from an arithmetic point of view.