The focus of this proposal is one of the central problems in number theory: relate special values of L-functions to the orders of associated algebraic quantities, especially Selmer groups. This is a problem whose origins lie in the celebrated class number formula from algebraic number theory and which includes the conjecture of Birch and Swinnerton-Dyer about the L-function of an elliptic curve. The research described in this proposal aims to prove such relations for various L-functions arising from automorphic forms. The theory of automorphic forms provides a rich source of L-functions (conjecturally all) and - through its connections with algebraic varieties attached to quotients of symmetric spaces - it is also closely connected to the objects of interest to algebraic number theorists (esp. Galois representations). The investigator aims to further develop and exploit the p-adic properties of these connections to relate L-values and Selmer groups.
The research described in this proposal aims to establish relations between certain analytic and algebraic objects. The analytic objects are L-functions- a special class of analytic functions built from number-theoretic data (this class includes the celebrated Riemann zeta function which is built from the prime numbers). For at least a century and a half L-functions have been central to efforts to tackle the most central problems in number theory (e.g., understanding the distribution of prime numbers). An important feature of L-functions is that their values at certain special points - these values are often called `special values' - are expected to be the orders of algebraic quantities associated to the data defining the L-function. The investigator aims to prove the existence of such relations for various classes of L-functions, drawing especially on the theory of automorphic forms. Automorphic forms are closely connected to both analysis (they are a rich source of L-functions, conjecturally all) and algebra. The investigator aims to prove such relations by systematically understanding the divisibility properties of the objects of interest (special values, automorphic forms, and Selmer groups) by powers of primes numbers.
