L-functions associated with families of toric exponential sums

Details L-functions associated with families of toric exponential sums L-functions associated with families of toric exponential sums

2011-09-16

arxiv.org/abs/1109.3682v1

Mathematics

Number Theory

43 pages

Scientific paper

We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponential sums over a finite field. We construct a relative polytope, the aid of which we define a ring of coefficients consisting of p-adic analytic functions with polyhedral growth prescribed by the relative polytope. Using this we compute relative cohomology for such families and calculate sharp estimates for the relative Frobenius map. In applications one is interested in L-functions associated with linear algebraic operations (symmetric powers, tensor powers, exterior powers and combinations thereof) applied to relative Frobenius. Using methods pioneered by Ax, Katz and Bombieri we prove estimates for the degree and total degree of the associated L-function and p-divisibility of the reciprocal zeros and poles. Similar estimates are then established for affine families and pure archimedean weight families (in the simplicial case).

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