Contents Online

# 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

#### 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.