p-Adic variation of motives

设F是一个数域F和一个素数p.数论中最神秘的猜想之一，称为朗兰兹猜想，Clozel猜想，Fontaine-Mazur猜想，是关于不可约的，连续的，n维p-ad值的L函数，F的绝对伽罗华群的伽罗华群的伽罗华群表示之间的一个双射的预测，它在有限个位置外未分支，在p以上的步长de Rham，和F的Adeles的尖面代数自同构表示之间.这个猜想似乎很难研究和证明.但已经观察到，猜想中出现的对象可以被视为某些p-进分析变种中的点，因此它们更容易理解。特别地，作为推论，这一观点构造了附加于伽罗瓦表示或自同构型的p元L函数，并对它们进行了比较。这种研究伽罗瓦表示和自同构形式的方法，即通过对它们进行p-adgate变形，被称为“动机的p-进变分”，它早在80年代由B.Mazur，H.Hida，R.Coleman开始。在本项目中，在研究了与GL_2上的自同构形式有关的模形式、de Rham类和联系的p-ic变式之后，建议在新的情况下将我们所获得的经验应用于GSP_4和Hilbert模形式上的自同构形式。此外，回到GL_2，我们打算研究半整权模形式的p-进变分。这个项目的有趣之处在于，我们要研究的模形式是非全纯的，因此还没有从算术的角度来研究它。

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.