p-Adic variation of motives

让我们固定一个数域F和一个素数p。数论中最神秘的猜想之一，被称为朗兰兹、克洛泽尔、方丹-马祖尔猜想，是关于L-函数的一个双射的预测，在F的绝对伽罗瓦群的不可约、连续、n维p-adic值、伽罗瓦表示之间，在p以上的有限个位置和de Rham之外是非分歧的，以及F的Adeles的GL_n的尖点代数自守表示.这个猜想似乎很难研究和证明，但已经观察到，出现在猜想中的对象可以被视为某些p-adic分析变种中的点，因此它们更容易理解。特别是这一点的观点已经作为后果建设一个p-adic L-功能重视要么伽罗瓦表示或自守形式和它们之间的比较。这种研究伽罗瓦表示和自守形式的方法，即通过使它们p进变形，被称为“动机的p进变化”，它已经在80年代由B开始。Mazur，H.希达河科尔曼。 本课题在研究了GL_2上模形式的p-dic变化、de Rham类和与自守形式相关的联系之后，提出了在新的情况下，将我们所获得的经验推广应用于GSp_4上的自守形式和Hilbert模形式。此外，回到GL_2，我们打算研究半整权模形式的p-adic变分。在这个项目中有趣的是，我们想研究的p-adic变分的模形式是非全纯的，因此还没有从算术的角度进行研究。

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.