1-Motives, Equivariant Iwasawa Theory and Special Values of L-functions

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The theory of special values of global and p-adic L-functions is of central importance in number theory. It establishes subtle links between the analytic aspects of the theory and the arithmetic-algebraic-geometric aspects, shedding new light on both and leading to striking solutions to outstanding open problems in number theory. The PI has developed a conjectural program which generalizes and refines the Gross-Rubin-Stark Conjectures and the Coates-Sinnott Conjectures on special values of global L-functions at non-positive integers and generalizes and refines the Main Conjecture in Iwasawa Theory and Gross's Conjectures on special values of p-adic L-functions at non-positive integers. The PI will employ techniques of equivariant Iwasawa Theory of 1-motives and the Iwasawa theoretic analogues of their l-adic realizations, Weil-etale cohomology, crystalline cohomology and the theory of t-motives to provide new evidence for his conjectures. He will also attempt to establish links between his conjectural program and those of Burns-Flach (on the Equivariant Tamagawa Number Conjecture for Artin Motives) and Ritter-Weiss (on non-abelian equivariant versions of the Main Conjecture in Iwasawa Theory). The PI will also study applications of his conjectural program to the construction of explicit Euler Systems, explicit generation of class-fields over an arbitrary global field (Hilbert's 10th problem) and refined class-number formulas. The PI will offer graduate courses and organize seminars and conferences bringing together students and experts in the field of special values of L-functions.An L-function is a gadget of analytic (continuous) nature which encodes a tremendous amount of arithmetic-algebraic-geometric (discrete) data of interest to experts working in the fields of number theory and arithmetic-algebraic geometry, as well as cryptographers, coding theorists, telecommunications engineers etc. The PIs conjectural program builds upon classical conjectures due to Gross, Rubin, Stark and Iwasawa among others and aims for determining (decoding) the discrete data out of special values of L-functions at the integral points on the real axis. The PI will employ techniques coming from various areas of mathematics, especially number theory and arithmetic geometry to prove his conjectures in several significant cases and to establish links between his conjectural program and those of Burns-Flach and Ritter-Weiss. Aside from its many far reaching applications to number theory and algebraic geometry (which will be explored by the PI), this research project has significant potential impact upon practical fields such as cryptography and coding theory.