Local zeta factors and geometries under Spec Z

Abstract

The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez--Villegas transform ([Ro--V]) a polynomial satisfying the functional equation of zeta type and having nontrivial zeros only on the middle line of its critical strip. The second part discusses Chebyshev lambda--structure of the polynomial ring as Borger's descent data to ${\mathbf{F}}_1$ and suggests its role in possible relation of ${\Gamma }_{\mathbf{R}}$--factor to "real geometry over ${\mathbf{F}}_1$" (cf. also [CoCons2]).

arXiv comment9 pages

References in corpus (3)

• Lambda-rings and the field with one element
• Zeta functions, Grothendieck groups, and the Witt ring
• The canonical strip, I

Cited by in corpus (5)

• The Riemann Hypothesis For Period Polynomials Of Modular Forms
• Zeta-polynomials for modular form periods
• Special Values of Motivic $L$-Functions and Zeta-Polynomials for Symmetric Powers of Elliptic Curves
• Zeta-polynomials, Hilbert polynomials, and the Eichler-Shimura identities
• Modular forms of real weights and generalized Dedekind symbols