Local zeta factors and geometries under

Abstract

The first part of this note shows that the odd-period polynomial of each Hecke cusp eigenform for the full modular group produces via the Rodriguez-Villegas transform ([1]) a polynomial satisfying the functional equation of zeta type and having non- trivial zeros only in the middle line of its critical strip. The second part discusses the Chebyshev lambda- structure of the polynomial ring as Borger's descent data to and suggests its role in a possible relation of the -factor to `real geometry over ' (cf. [2]).