2006/01/30 by David Goss, Goss, David
Computer Science · Mathematics · #11M38 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.math/0601717
openalex publication_date 2006/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the classical theory of L-series, the exact order (of zero) at a trivial zero is easily computed via the functional equation. In the characteristic p theory, it has long been known that a functional equation of classical s↦ 1-s type could not exist. In fact, there exist trivial zeroes whose order of zero is ``too high;'' we call such trivial zeroes ``non-classical.'' This class of trivial zeroes was originally studied by Dinesh Thakur \citeth2 and quite recently, Javier Diaz-Vargas \citedv2. In the examples computed it was found that these non-classical trivial zeroes were correlated with integers having \it bounded sum of p-adic coefficients. In this paper we present a general conjecture along these lines and explain how this conjecture fits in with previous work on the zeroes of such characteristic p functions. In particular, a solution to this conjecture might entail finding the ``correct'' functional equations in finite characteristic.