FOURIER COEFFICIENTS OF EISENSTEIN SERIES OF THE EXCEPTIONAL GROUP OFTYPE G(2)

Let F be a number fields and K be a commutative algebra over F of degr ee n. A basic question in number theory is whether the ratio zeta K(s) /zeta F(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 quest ion in automorphic L-functions, which asks whether or not the ratio L- S(Pi X Pi(V),s)/zeta(F)(S)(s) is entire for all irreducible cuspidal a utomorphic representation of GL(n,A(F)) with trivial central character , where L-S(Pi X Pi(V),s) is the standard tensor product L-function of Pi with its contragredient Pi(V), see for example the work of Jacquet and Zagier [JaZa]. The main idea in this paper is to develop two intr insically related methods to attack the above two questions. The work of Siegel [Sie], and of Shimura [Shi] (and of Gelbart and Jacquet [GeJ a]) provided an evidence for this approach for the case of n = 2. Comb ined with the work of Ginzburg [Gin], the main result of this paper sh ows that our approach works for the case of n = 3. It is hoped that su ch an approach extends to at least the case of n = 5.