Rational Points on Rational Curves

2019/11/28 by Beers, Brecken, Sung, Yih
#11F67 #11M41 #14G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.12551

Abstract

For a given elliptic curve, its associated L-function evaluated at 1 is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use all the counting information to define two L-type series. Then we consider special values of these series at 1. One of the L-type series matches the Dirichlet L-series of modulo 4, so the evaluation at 1 is π/4; the special evaluation at 1 of the other L-type series is equal to a real period associated to the rational curve. This identity confirms the general principle that an L-type series associated to a variety can reflect its geometry.

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