"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."
One of the main goals of this project is to investigate the signs in the conjectural functional equations for L-functions attached to abelian varieties, the so-called root numbers. Root numbers are important fundamental invariants that arise in connection with several influential conjectures of number theory and representation theory, such as Birch--Swinnerton-Dyer and parity conjectures, conjectural functional equations for L-functions, and Langlands program. In studying root numbers attached to abelian varieties over extensions of the base field arise the so-called twists of the root numbers by Galois representations, or twisted root numbers for short. Twisted root numbers attached to elliptic curves have been studied extensively. The main objective of this proposal is to extend the results obtained in that direction to abelian varieties. The methods to be employed come from the combination of representation theory with the theory of Serre and Tate of abelian varieties with potential good reduction and the uniformization theory of abelian varieties. Another part of the project is devoted to studying questions arising naturally in the context of the stratification theory of reductive groups over the field of formal Laurent series with complex coefficients. This theory developed by Goresky, Kottwitz, and MacPherson is important in connection with the affine Springer fibers studied by Kazhdan, Luzstig, Bezrukavnikov, Spaltenstein and others.
The theory of L-functions is a generalization of the classical theory of the well-known Riemann and Dedekind zeta-functions that originated in works by Hecke and Hasse at the first half of the last century. As zeta-functions have been used to study the distribution of prime numbers, L-functions were introduced to study the distribution of prime ideals in number fields, i.e., finite extensions of the field of rational numbers. One of the main themes in arithmetic algebraic geometry is to study L-functions attached to algebraic varieties, in particular to abelian varieties, which are higher-dimensional analogues of elliptic curves. Inseparable from the theory of L-functions is the notion of a root number--the sign in the conjectural functional equations for L-functions. The importance of studying root numbers lies in the fact that according to several famous conjectures in number theory they are believed to have deep connections to the arithmetic of the subject in question; for example, they may predict the existence of rational solutions to systems of algebraic equations or, in other words, the existence of rational points. Studying rational points on algebraic varieties in turn has its applications to cryptography: some public key cryptosystems are based on rational points of elliptic curves. One of the goals of this project is to study root numbers attached to abelian varieties.
