Eisenstein cocycles over imaginary quadratic fields and special values of L-functions

Flórez, Jorge; Karabulut, Cihan; Wong, Tian An

doi:10.1016/j.jnt.2019.04.016

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…Our viewpoint is well suited to study integrality features of the cohomology classes associated to the above complex cocycles. This leads to a new proof of the general conjecture 6 A recent work by Florez, Karabulut and Wong [24], generalising Sczech's Eisenstein cocycle to extensions of an imaginary quadratic fields, provides a different construction of a similar degree N − 1 cocycle. 7 Compare to (7.8) there is a missing factor −(4π) 2 = (2πi) 2 ; this normalisation is made to have integral periods and is reminiscent of the factor 2πi factor in equation (11.6).…”

Section: Eisenstein Theta Correspondence For the Dual Pair (Gl A Glmentioning

confidence: 93%

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

2020

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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 GL N (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 SL N (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 (GL N , GL 1). This suggests looking to reductive dual pairs (GL N , GL k) 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 don't deal with the most general cases and put a lot of emphasis on various examples that are often classical.

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Section: Eisenstein Theta Correspondence For the Dual Pair (Gl A Glmentioning

confidence: 93%

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

2020

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“…In the case N = 2 Theorem 1.1 is proved by Sczech [31] and Ito [23], in case (p, q) = (0, 0), and Obaisi [28] proved the corresponding Theorem 1.2. In general, partial results towards both Theorem 1.2 and Corollary 1.3 are obtained by Colmez in [11]; see also [16] for related works.…”

Section: Introductionmentioning

confidence: 97%

“…Let k ≥ 0, x ∈ X and γ ∈ Γ 0 (p, Λ(I)) k+1 with k < N . In this paragraph we define a homotopy between the simplices ∆(γ, x) and ∆(γ); that is, a mapH(γ, x) : |∆ k | × [0, 1] → Q X T such that H(γ, x)| |∆ k |×{0} = ∆(γ, x) H(γ, x)| |∆ k |×{1} = ∆(γ) (4 16).…”

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confidence: 99%

Eisenstein cohomology classes for GL_N over imaginary quadratic fields

Bergeron

Charollois

Garcia

2023

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We study the arithmetic of degree N - 1 {N-1} Eisenstein cohomology classes for the locally symmetric spaces attached to GL N {\mathrm{GL}_{N}} over an imaginary quadratic field k. Under natural conditions we evaluate these classes on ( N - 1 ) {(N-1)} -cycles associated to degree N extensions L / k {L/k} as linear combinations of generalized Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of L-functions attached to Hecke characters of L as polynomials in Kronecker–Eisenstein series evaluated at torsion points on elliptic curves with complex multiplication by k. We recover in particular the algebraicity of these critical values.

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“…Such a cocycle is often called the Eisenstein cocycle. Actually many different kinds of Eisenstein cocycles have been constructed and studied by Harder [11], Sczech [23], Nori [21], Solomon [26], Hill [15], Vlasenko-Zagier [28], Charollois-Dasgupta-Greenberg [6], Beilinson-Kings-Levin [3], Bergeron-Charollois-Garcia [4], Flórez-Karabulut-Wong [9], Lim-Park [20], Bannai-Hagihara-Yamada-Yamamoto [1], Sharifi-Venkatesh [24], and so on, and various applications have been obtained.…”

Section: Introductionmentioning

confidence: 99%

Shintani-Barnes cocycles and values of the zeta functions of algebraic number fields

Bekki¹

2021

Preprint

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In this paper, we construct a new Eisenstein cocycle called the Shintani-Barnes cocycle which specializes in a uniform way to the values of the zeta functions of general number fields at positive integers. Our basic strategy is to generalize the construction of the Eisenstein cocycle presented in the work of Vlasenko and Zagier by using some recent techniques developed by Bannai, Hagihara, Yamada, and Yamamoto in their study of the polylogarithm for totally real fields. We also closely follow the work of Charollois, Dasgupta, and Greenberg. In fact, one of the key ingredients in this paper which enables us to deal with general number fields is the introduction of a new technique called the "exponential perturbation" which is a slight modification of the Q-perturbation studied in their work. Contents 1. Introduction 1 2. Preliminaries 3 3. The space Y • and the sheaves F d , F Ξ d 7 4. Equivariant cohomology 12 5. Cones and the exponential perturbation 20 6. Construction of the Shintani-Barnes cocycle 26 7. Integration 30 8. Specialization to the zeta values 36 References 48

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Eisenstein cocycles over imaginary quadratic fields and special values of L-functions

Cited by 5 publications

References 11 publications

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

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

Eisenstein cohomology classes for GL_N over imaginary quadratic fields

Shintani-Barnes cocycles and values of the zeta functions of algebraic number fields

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Eisenstein cocycles over imaginary quadratic fields and special values of L-functions

Cited by 5 publications

References 11 publications

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

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

Eisenstein cohomology classes for GL N over imaginary quadratic fields

Shintani-Barnes cocycles and values of the zeta functions of algebraic number fields

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Product

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Eisenstein cohomology classes for GL_N over imaginary quadratic fields