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]).

9 pages

References in corpus (2)

• Lambda-rings and the field with one element
• The canonical strip, I