p-adic L-functions, geometry of eigenvarieties, Selmer groups

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This project is about p-adic L-functions, eigenvarieties, and Selmer groups. Eigenvarieties are the universal p-adic families of automorphic forms for a given reductive group. Individual automorphic forms are conjectured to have a p-adic L-function, a p-adic counterpart of their usual complex L-function, and it is natural to expect that the p-adic L-functions of individual automorphic forms for a given reductive group will fit in an analytic family carried by the eigenvariety. However, besides the case of modular forms, very little is known on the existence of p-adic L-functions, and virtually nothing on their families. Even for modular forms, many important questions remain, such as the de finition and computation of many critical p-adic L-functions. This project proposes a strategy to address some of those questions. It focuses on the most arithmetically significant situation: the case of "critical" automorphic forms. The strategy is to consider automorphic forms in families, in which a critical automorphic forms may have milder siblings. The ultimate aim of the project is to relate the geometry of the Eigenvariety at some point to the order of vanishing of the p-adic L-function of the corresponding automorphic form. This should be done in such a way that, combined with earlier work of the PI and Chenevier, could lead to a proof of an inequality in the equality conjectured by Bloch and Kato between rank of Selmer groups, and order of vanishing of L-functions.The discovery, by the pioneers of mathematics of modern times, of some very remarkable equalities, like that the sum of the reciprocals of the square of all positive integers is equal to one sixth of the square of the area of a unit disc (Euler) have opened a trend of mathematical research which is still very active today. Those equalities relate an analytic side (the sum of an in finite series , an object of calculus) to a side which is a product of a number of geometric nature times a rational number (hence an object of study for number theorists). Those equalties, and a very great number of famous results obtained since then, as well as many more still to be proved, are all contained in a vast framework of conjectures built by Deligne, Beilinson, Bloch, Kato and Perrin-Riou. In their modern and general forms, those conjectures still relate an analytic object, called a L-function, and a number- theoretical one, called a Selmer group. The project of the PI intends to shed some light of one important aspect of those conjectures, the one concerning the order of the zeros of the L-functions. The PI proposes to do so by relating the two sides to a third object, whose appearance is much more recent, the Eigenvarieties, which are the universal families of automorphic forms.