On unit root formulas for toric exponential sums
Alan Adolphson, Steven Sperber

TL;DR
This paper generalizes the computation of p-adic unit roots for exponential sums on tori using Dwork's dual theory and hypergeometric functions, extending previous results to more general, nondegenerate sums without cohomological methods.
Contribution
It introduces a noncohomological approach to express p-adic unit roots of exponential sums on tori via hypergeometric functions, broadening applicability beyond prior nondegeneracy assumptions.
Findings
Expressed p-adic unit roots in terms of hypergeometric functions
Extended results to arbitrary exponential sums without nondegeneracy conditions
Used Dwork's relative dual theory and classical Bessel function series
Abstract
Starting from a classical generating series for Bessel functions due to Schlomilch, 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 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
