Formulas for roots of L-functions of the airy exponential sums

Abstract

We study the L-function of the Airy exponential sum associated to the polynomial f(X)=Xd+λX using Dwork's dual theory on cohomology level. We get explicit formulas for all the roots of the L-function in the ordinary case in terms of p-adic GKZ-hypergeometric functions. Our method can also be used to give explicit formulas for all the roots of the L-function of the exponential sum associated to f(X)=Xd+λd−1Xd−1+⋯+λ0.