L-values, Galois Representations, and Modular Forms

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

DMS-0245387 Skinner, Christopher Abstract: This proposal investigates connections between L-functions and Galois representations. More precisely, the investigator connects special values of L-functions of varieties or motives to the orders of associated Selmer groups. In joint work with E. Urban the investigator showed that if the L-function of a modular form of weight 1 vanishes to odd order at its central critical point then the rank of the corresponding Selmer group is infinite. The investigator is exploring extensions to representations of higher dimension and to pursuing a ``mod p'' version of this result relating the p-adic order of the L-value (if finite) to that of its Selmer group. Additionally, the investigator is completing work in collaboration with M. Harris and J.-S. Li that constructs p-adic anti-cyclotomic L-function for automorphic representations on unitary groups U(n) and which makes progress towards anti-cyclotomic ``main conjectures'' for these groups. Doing so involves relating these L-functions to congruences between endoscopic and stable automorphic forms on U(n). The investigator is also pursuing an approach to the ``main conjecture'' for modular forms on GL(2) by studying the arithmetic 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 seemingly unrelated objects from these two branches. The primary focus of the first branch is the study of L-functions - complex functions built of number-theoretically interesting data (such as the Riemann zeta function which is built out of the prime numbers). The other branch focuses on algebraic objects such as class groups. But it is now expected that these are not unrelated: values of certain L-functions at special points (generally integers) should be the orders - the sizes - of certain of the algebraic objects. Such connections are generally very signifigant. One instance was a crucial ingredient in Andrew Wiles' proof of Fermat's Last 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-functions appear in the fourier coefficients of modular forms.