Cambridge Journal of Mathematics

Volume 2 (2014)

Number 1

Integral Eisenstein cocycles on $\mathbf{GL}_n$, I: Sczech’s cocycle and $p$-adic $L$-functions of totally real fields

Pages: 49 – 90

DOI: http://dx.doi.org/10.4310/CJM.2014.v2.n1.a2

Authors

Pierre Charollois (Institut de Mathématiques de Jussieu, Université Paris, France)

Samit Dasgupta (Department of Mathematics, University of California, Santa Cruz, Calif., U.S.A.)

Abstract

We define an integral version of Sczech’s Eisenstein cocycle on $\mathbf{GL}_n$ by smoothing at a prime $\ell$. As a result we obtain a new proof of the integrality of the values at nonpositive integers of the smoothed partial zeta functions associated to ray class extensions of totally real fields. We also obtain a new construction of the $p$-adic $L$-functions associated to these extensions. Our cohomological construction allows for a study of the leading term of these $p$-adic $L$-functions at $s = 0$. We apply Spiess’s formalism to prove that the order of vanishing at $s = 0$ is at least equal to the expected one, as conjectured by Gross. This result was already known from Wiles’ proof of the Iwasawa Main Conjecture.

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