The p-adic Shintani cocycle and p-adic L-functions
OA Version
Citation
Abstract
The Shintani cocycle on GL_n(Q), as constructed by Hill, gives a cohomological interpretation of special values of Shintani zeta functions. We interpret this cocycle as taking values in a module, introduced by Stevens, of locally polynomial p-adic distributions localized with rational poles. Our first theorem shows that the Shintani cocycle takes values in the sub-module of locally analytic distributions with rational poles. Next, we give a simple criterion for the Shintani cocycle to specialize to true p-adic distributions. The theorem shows that such specializations are automatically p-adic measures.
We give two applications of these theorems: First, we recover the p-adic L-functions of totally real fields. Our construction greatly simplifies the original constructions of Deligne-Ribet, Cassou-Nogues, and Barsky, and we anticipate the methods will be useful for studying refinements of the Gross-Stark conjecture. Second, we give a simple construction of the p-adic L-function of critical slope Eisenstein series, proving a conjecture of Pasol and Stevens. The results of this construction were recently and independently proven by Bellaiche and Dasgupta, but our methods side-step many of the technical hurdles present in their construction.