Horizontal Factorizations of Certain Hasse-Weil Zeta Functions - a Remark on a Paper by Taniyama

Abstract

The paper [T] by Taniyama contains several results in the arithmetic theory of CM abelian varieties which have become well known. Moreover the notion of a compatible system ρ = (ρl) of l-adic representations is introduced there for the first time. Under suitable conditions on ρ, Taniyama proves an interesting formula for the alternating product of the L-functions of the exterior powers of ρ: it is given by an infinite product of Artin L-functions each of which modified by a change of finitely many Euler factors. As an application, Taniyama obtains such a formula for the Hasse–Weil zeta function of an abelian scheme over a localized number ring. These infinite product formulas of Taniyama seem to be little known but they motivated the works [JY] and [JR]. In [JY] Joshi and Yogananda show that the Hasse–Weil zeta function ζGm(s) = ζ(s−1)/ζ(s) of Gm = specZ[T, T −1] is an infinite product of Dirichlet L-series. This is a special case of Taniyama’s formulas but they give an elementary direct argument. In [JR] Joshi and Raghunathan express quotients of very general Dirichlet series with Euler products as infinite products of “twisted Dirichlet series”. The method is purely local. It applies in particular to the Hasse–Weil zeta function of X×Gm where X is a scheme of finite type over Z. In this case the formula has a geometric proof which was suggested by Serre, c.f. [JR] § 3. Joshi and Raghunathan also obtain certain infinite product formulas for quotients of automorphic L-functions. Recently these have been related to canonical bases by Kim and Lee, [KL] Remark 2.21.