Local zeta factors and geometries under Spec Z

摘要

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 F-1 and suggests its role in a possible relation of the R-factor to 'real geometry over F-1' (cf. [2]).