On the structure of Artin L-functions

摘要

We consider the generating Artin L-function L(z) = L(z, f) = exp(∞∑v=1 (Tv)/v z~v) for the character sums T_v=∑(X_1,...,x_n∈F_qv) ψv(f(x1_, . . ., x_n)), where F_q is a finite field, F_(q~v) is a finite extension of F_q, ψv(α) is a non-trivial additive character of F_(q~v), and ∈ 2 F_q[x_1, . . ., x_n] is a polynomial of degree d ≥ 2, and give an elementary proof of Bombieri’s conjecture on the algebraic structure of L(z) in the case n = 2.