L-functions and Eisenstein series: p-adic aspects and applications

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This grant concerns several projects motivated by p-adic aspects of L-functions, a central object of study in number theory. The first part of the project involves the development of certain p-adic differential operators and p-adic measures (Eisensteinmeasures), technical tools that the PI will use to construct p-adic L-functions. These p-adic differential operators and p-adic measures build on the PI's prior results on these topics. These Eisenstein measures are also conjectured to give a homotopy-theoretic invariant of certain manifolds that generalizes the Witten genus. The second part of the project concerns applications of the tools developed in the first part of the proposal. The main application is the construction of p-adic L-functions that p-adically interpolate the special values of L-functions attached to families of automorphic forms. One portion of this application is joint with Michael Harris, Jian-Shu Li, and Christopher Skinner. This part of the project also has applications to Iwasawa theory, a p-adic theory for studying certain arithmetic data, which in turn is expected to relate to the Birch and Swinnerton-Dyer Conjecture. L-functions are certain complex-valued functions that play an important role in number theory. Their values at certain points satisfy striking congruence conditions and relate to many open problems. One approach to studying L-functions uses p-adic numbers (depending on a prime number p), an extension of the rational numbers analogous to but different from the real numbers. This proposal involves further developing techniques for studying p-adic aspects of L-functions. The proposed research has expected consequences in algebraic number theory, analytic number theory, and algebraic topology.