L-values, Galois Representations, and Modular Forms

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DMS-0245387Skinner, ChristopherAbstract:This proposal investigates connections between L-functions andGalois representations. More precisely, the investigator connectsspecial values of L-functions of varieties or motives to the orders ofassociated Selmer groups. In joint work with E. Urban the investigatorshowed that if the L-function of a modular form of weight 1 vanishes toodd order at its central critical point then the rank of the correspondingSelmer group is infinite. The investigator is exploring extensions torepresentations of higher dimension and to pursuing a ``mod p''version of this result relating the p-adic order of the L-value (iffinite) to that of its Selmer group. Additionally, the investigator iscompleting work in collaboration with M. Harris and J.-S. Li thatconstructs p-adic anti-cyclotomic L-function for automorphicrepresentations on unitary groups U(n) and which makes progress towardsanti-cyclotomic ``main conjectures'' for these groups. Doing so involvesrelating these L-functions to congruences between endoscopic and stableautomorphic forms on U(n). The investigator is also pursuing an approachto the ``main conjecture'' for modular forms on GL(2) by studying thearithmetic of certain Eisenstein series on U(2,2).Number theory is often divided into two branches: analytic and algebraic.The investigator is studying the connection between many seeminglyunrelated objects from these two branches. The primary focus of the firstbranch is the study of L-functions - complex functions built ofnumber-theoretically interesting data (such as the Riemann zeta functionwhich is built out of the prime numbers). The other branch focuses onalgebraic objects such as class groups. But it is now expected that theseare not unrelated: values of certain L-functions at special points(generally integers) should be the orders - the sizes - of certain of thealgebraic objects. Such connections are generally very signifigant.One instance was a crucial ingredient in Andrew Wiles' proof of Fermat'sLast Theorem, for example. Drawing from the theory of automorphic forms,the investigator's work establishes new instances of these conjectures.This work is motivated by the fact that special values of L-functionsappear in the fourier coefficients of modular forms.