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Integral Eisenstein cocycles on \(\mathrm{GL}_n\). I: Sczech’s cocycle and \(p\)-adic \(L\)-functions of totally real fields. (English) Zbl 1353.11074
The goal of the present paper is to provide an integral refinement of Sczech’s Eisenstein cocycle on \(\mathrm{GL}_n\) by smoothing at a prime \(l\). As a result, the authors 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. They also obtain a new construction of the \(p\)-adic \(L\)-functions associated to these extensions. Their cohomological construction allows for a study of the leading term of these \(p\)-adic \(L\)-functions at \(s=0\). The authors 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. The method presented in the current paper contrasts with that of Wiles in that it is purely analytic.

MSC:
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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