Let F be a number fields
and K be a commutative algebra over F of degree n. A basic question in number
theory is whether the ratio ζK(s)ζF(s) of the two Dedekind zeta functions is an entire
function in the complex variable s. From the point of view of the trace formula, the
above basic question is expected to be equivalent to a basic question in automorphic
L-functions, which asks whether or not the ratio LS(Π⊗Π∨,s)ζFS(s) is entire for all
irreducible cuspidal automorphic representation of GL(n, 𝔸F) with trivial central
character, where LS(Π ⊗ Π∨,s) is the standard tensor product L-function of Π with
its contragredient Π∨, see for example the work of Jacquet and Zagier [JaZa]. The
main idea in this paper is to develop two intrinsically related methods to
attack the above two questions. The work of Siegel [Sie], and of Shimura
[Shi] (and of Gelbart and Jacquet [GeJa]) provided an evidence for this
approach for the case of n = 2. Combined with the work of Ginzburg [Gin], the
main result of this paper shows that our approach works for the case of
n = 3. It is hoped that such an approach extends to at least the case of
n = 5.
