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Integral Eisenstein cocycles on GLn, II: Shintani's method

Publication ,  Journal Article
Charollois, P; Dasgupta, S; Greenberg, M
Published in: Commentarii Mathematici Helvetici
January 1, 2015

We define a cocycle on GLn(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 l. This integral refinement has strong arithmetic consequences. We showed in previous work that certain specializations of the smoothed class yield the p-adic L-functions of totally real fields. Furthermore, combining our cohomological construction with a theorem of Spiess, one deduces that that the order of vanishing of these p-adic L-functions is at least as large as the expected one.

Duke Scholars

Published In

Commentarii Mathematici Helvetici

DOI

EISSN

1420-8946

ISSN

0010-2571

Publication Date

January 1, 2015

Volume

90

Issue

2

Start / End Page

435 / 477

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics
 

Citation

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Charollois, P., Dasgupta, S., & Greenberg, M. (2015). Integral Eisenstein cocycles on GLn, II: Shintani's method. Commentarii Mathematici Helvetici, 90(2), 435–477. https://doi.org/10.4171/CMH/360
Charollois, P., S. Dasgupta, and M. Greenberg. “Integral Eisenstein cocycles on GLn, II: Shintani's method.” Commentarii Mathematici Helvetici 90, no. 2 (January 1, 2015): 435–77. https://doi.org/10.4171/CMH/360.
Charollois P, Dasgupta S, Greenberg M. Integral Eisenstein cocycles on GLn, II: Shintani's method. Commentarii Mathematici Helvetici. 2015 Jan 1;90(2):435–77.
Charollois, P., et al. “Integral Eisenstein cocycles on GLn, II: Shintani's method.” Commentarii Mathematici Helvetici, vol. 90, no. 2, Jan. 2015, pp. 435–77. Scopus, doi:10.4171/CMH/360.
Charollois P, Dasgupta S, Greenberg M. Integral Eisenstein cocycles on GLn, II: Shintani's method. Commentarii Mathematici Helvetici. 2015 Jan 1;90(2):435–477.

Published In

Commentarii Mathematici Helvetici

DOI

EISSN

1420-8946

ISSN

0010-2571

Publication Date

January 1, 2015

Volume

90

Issue

2

Start / End Page

435 / 477

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 0101 Pure Mathematics