On unit root formulas for toric exponential sums

Abstract

Starting from a classical generating series for Bessel functions due to Schlömilch, we use Dwork’s relative dual theory to broadly generalize unit-root results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In particular, we express the (unique) $p$-adic unit root of an arbitrary exponential sum on the torus ${\mathbb{T}}^n$ in terms of special values of the $p$-adic analytic continuation of a ratio of $A$-hypergeometric functions. In contrast with the earlier works, we use noncohomological methods and obtain results that are valid for arbitrary exponential sums without any hypothesis of nondegeneracy.