L-functions of Exponential Sums over Finite Fields

Let F q be the finite field of q elements with characteristic p and F q m its extension of degree m. Fix a nontrivial additive character ψ of F p. For any Laurent polynomial −1 n ], we form the exponential sum S * m (f) := The corresponding L-function L * (f, t) is defined by L * (f, t) := exp (∞ ∑ m=0 S * m (f) t m m). The corresponding L-function L(f, t) is defined as follows L(f, t) := exp (∞ ∑ m=0 S m (f) t m m). A fundamental and important problem in number theory is to understand the sequence S * m (f) and S m (f) (1 ≤ m ≤ ∞) of algebraic integers, each of them lying in the p-th cyclotomic field Q(ζ p), where ζ p means a fixed primitive p-th root of unity in the complex numbers. The well-known Dwork-Grothendieck theorem tells us that the L-functions L * (f, t) and L(f, t) are rational functions. Write L * (f, t) = ∏ d1 i=1 (1 − α i t) ∏ d2 j=1 (1 − β j t) and L(f, t) = ∏ d3 i=1 (1 − γ i t) ∏ d4 j=1 (1 − δ j t) , 1