The Hodge Realization of the Polylogarithm and the Shintani Generating Class for Totally Real Fields

Bannai, Kenichi; Bekki, Hohto; Hagihara, Kei; Ohshita, Tatsuya; Yamada, Kazuki; Yamamoto, S.

doi:10.48550/arxiv.2207.03285

Cited by 1 publication

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , α g (recall that Pα is defined by (3), depending on a fixed τ 0 ∈ I). Then G σ (t) corresponds to the function G σ (z) of Definition 2.7 through the uniformization (13).…”

Section: Shintani Generating Classmentioning

confidence: 99%

See 1 more Smart Citation

Canonical equivariant cohomology classes generating zeta values of totally real fields

Bannai

Hagihara

Yamada

et al. 2023

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

It is known that the special values at nonpositive integers of a Dirichlet L L -function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L L -functions of totally real fields. Hecke L L -functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393–417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shintani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.

show abstract

“…. , α g (recall that Pα is defined by (3), depending on a fixed τ 0 ∈ I). Then G σ (t) corresponds to the function G σ (z) of Definition 2.7 through the uniformization (13).…”

Section: Shintani Generating Classmentioning

confidence: 99%

“…In the forthcoming article [3], we will consider a de Rham cohomology class with coefficients in the logarithm sheaf, which is similar to this G.…”

Section: Considering the Relation Among The Characteristic Functions ...mentioning

confidence: 99%

Canonical equivariant cohomology classes generating zeta values of totally real fields

Bannai

Hagihara

Yamada

et al. 2023

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

show abstract

The Hodge Realization of the Polylogarithm and the Shintani Generating Class for Totally Real Fields

Cited by 1 publication

References 0 publications

Canonical equivariant cohomology classes generating zeta values of totally real fields

Canonical equivariant cohomology classes generating zeta values of totally real fields

Contact Info

Product

Resources

About