Symmetric power 𝐿-functions for families of generalized Kloosterman sums

Source abstract

We construct relative p p -adic cohomology for a family of toric exponential sums fibered over the torus. The family under consideration here generalizes the classical Kloosterman sums. Under natural hypotheses such as quasi-homogeneity and nondegeneracy, this cohomology, just as in the absolute case, is acyclic except in the top dimension. Our construction gives us sufficiently sharp estimates for the action of Frobenius on relative cohomology so that we may obtain properties of L L -functions constructed by taking a suitable Euler product (over the family) of local factors using linear algebra operations (such as taking the k k -th symmetric power or other such operations) on the reciprocal zeros and poles of the L L -functions of each fiber.