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

We define an integral version of Sczech’s Eisenstein cocycle on GLn by smoothing at a prime l. 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 Lfunctions 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.