Local zeta factors and geometries under Spec Z

2014/07/18 by Yuri I. Manin, Manin, Yuri I.
Mathematics · #14H10 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H10

paper · pdf · doi:10.48550/arxiv.1407.4969

9 pages

arxiv created 2014/07/18 · arxiv updated 2014/07/21

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 \boldF₁ and suggests its role in possible relation of Γ_\boldR--factor to "real geometry over \boldF₁" (cf. also [CoCons2]).

Related

• Zeta-polynomials for modular form periods
• Local factors and Cuntz-Pimsner algebras
• The Riemann Hypothesis for period polynomials of cusp forms
• Local Zeta Functions From Calabi-Yau Differential Equations
• Zeta function factorisation, Dwork hypersurfaces, hypergeometric hypersurfaces