Transgressions of the Euler class and Eisenstein cohomology of GLN(Z)

These notes were written to be distributed to the audience of the first author’s Takagi Lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh. In this work-in-progress we give a new construction of some Eisenstein classes for GLN (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SLN (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GLN, GL1). This suggests looking to reductive dual pairs (GLN, GLk) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms. In these notes we do not deal with the most general cases and put a lot of emphasis on various examples that are often classical

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

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

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

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

Stark units and special Gamma values

Versión postprintIn this paper we develop an effective procedure for expressing Stark units in real quadratic extensions of totally real fields as values of the Barnes multiple Gamma function at algebraic points. This procedure is used to explicitly generate non-abelian extensions of ℚ by special Gamma values. As a main component of our work, we develop an algorithm to compute Shintani sets in all dimensions.UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de MatemáticaUCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigaciones en Matemáticas Puras y Aplicadas (CIMPA