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COMPLEX MOMENTS AND THE DISTRIBUTION OF VALUES OF $L(1,\unicode[STIX]{x1D712}_{D})$ OVER FUNCTION FIELDS WITH APPLICATIONS TO CLASS NUMBERS

Part of: Arithmetic problems. Diophantine geometry Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  05 December 2018

Allysa Lumley
Allysa Lumley*
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada email alumley@yorku.ca
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Abstract

In this paper we investigate the moments and the distribution of $L(1,\unicode[STIX]{x1D712}_{D})$, where $\unicode[STIX]{x1D712}_{D}$ varies over quadratic characters associated to square-free polynomials $D$ of degree $n$ over $\mathbb{F}_{q}$, as $n\rightarrow \infty$. Our first result gives asymptotic formulas for the complex moments of $L(1,\unicode[STIX]{x1D712}_{D})$ in a large uniform range. Previously, only the first moment has been computed due to the work of Andrade and Jung. Using our asymptotic formulas together with the saddle-point method, we show that the distribution function of $L(1,\unicode[STIX]{x1D712}_{D})$ is very close to that of a corresponding probabilistic model. In particular, we uncover an interesting feature in the distribution of large (and small) values of $L(1,\unicode[STIX]{x1D712}_{D})$, which is not present in the number field setting. We also obtain $\unicode[STIX]{x1D6FA}$-results for the extreme values of $L(1,\unicode[STIX]{x1D712}_{D})$, which we conjecture to be the best possible. Specializing $n=2g+1$ and making use of one case of Artin’s class number formula, we obtain similar results for the class number $h_{D}$ associated to $\mathbb{F}_{q}(T)[\sqrt{D}]$. Similarly, specializing to $n=2g+2$ we can appeal to the second case of Artin’s class number formula and deduce analogous results for $h_{D}R_{D}$, where $R_{D}$ is the regulator of $\mathbb{F}_{q}(T)[\sqrt{D}]$.

MSC classification

Primary: 11G20: Curves over finite and local fields 11R29: Class numbers, class groups, discriminants 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)
Type
Research Article
Information
Mathematika , Volume 65 , Issue 2 , 2019 , pp. 236 - 271
Copyright
Copyright © University College London 2018 

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