We define an integral version of Sczech's Eisenstein cocycle on GL n 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. † See Remark 3.2 for a discussion of the condition that c has prime norm. Also, we note that Theorem 1 has the corollary that the "twice smoothed" zeta functionassumes integer values at nonpositive integers s when (Nb, Nc) = 1. Gross has recently provided an interpretation of these integers in terms of dimensions of certain spaces of automorphic forms [G2].
We define a cocycle on GL n (Q) using Shintani's method. This construction is closely related to earlier work of Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q. As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental domains, and give a cohomological reformulation of Shintani's proof of the Klingen-Siegel rationality theorem on partial zeta functions of totally real fields. Next we relate the Shintani cocycle to the Sczech cocycle by showing that the two differ by the sum of an explicit coboundary and a simple "polar" cocycle. This generalizes a result of Sczech and Solomon in the case n = 2. Finally, we introduce an integral version of our cocycle by smoothing at an auxiliary prime. Applying the formalism of the first paper in this series, we prove that certain specializations of the smoothed class yield the p-adic L-functions of totally real fields. Combining our cohomological construction with a theorem of Spiess, we show that the order of vanishing of these p-adic L-functions is at least as large as the expected one.
In this paper, we explicitly construct harmonic Maass forms that map to the holomorphic weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.
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