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 Zeroes of $L$-series in characteristic $p$

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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 $s\mapsto 1-s$ 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 \cite{th2} and quite recently, Javier Diaz-Vargas \cite{dv2}. 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.

Autor: David Goss


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