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arXiv · 2604.07330

$p$-adic Theory for Partial Toric Exponential Sums

C. Douglas Haessig
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Abstract

Wan proved the rationality of partial toric $L$-functions using $\ell$-adic techniques. In this paper, we present a $p$-adic proof in the spirit of Dwork. We demonstrate that partial $L$-functions can be expressed as an alternating product of twisted Fredholm determinants. These twisted determinants appear to be intrinsic to the analytic structure of partial $L$-functions, and unlike their classical counterparts, twisted Fredholm determinants of completely continuous operators are not automatically $p$-adic entire functions. However, for partial $L$-functions they will be $p$-adic meromorphic. After proving rationality, we construct a $p$-adic cohomology theory and give a $p$-adic cohomological formula for partial toric $L$-functions. Last, we show they have a unique $p$-adic unit root which may be explicitly written in terms of $A$-hypergeometric series.

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C. Douglas Haessig. 2026-04-08. $p$-adic Theory for Partial Toric Exponential Sums. https://arxiv.org/abs/2604.07330

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