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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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References

Aisteleitner, C., Mahatab, K., Munsch, M. and Peyrot, A., On the proportion of characters with large values of $L(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D712})$ . Q. J. Math. (to appear). Preprint, 2018, arXiv:1803.00760.Google Scholar
Andrade, J., A note on the mean value of L-functions over function fields. Int. J. Number Theory 8(7) 2012, 1725–1740.Google Scholar
Andrade, J. and Keating, J., The mean value of L ([[()[]mml:mfrac[]()]][[()[]mml:mrow []()]]1[[()[]/mml:mrow[]()]] [[()[]mml:mrow []()]]2[[()[]/mml:mrow[]()]][[()[]/mml:mfrac[]()]], 𝜒) in the hyperelliptic ensemble. J. Number Theory 132(12) 2012, 2793–2816.Google Scholar
Artin, E., Quadratische Körper in Geibiet der Höheren Kongruzzen I and II. Math. Z. 19 1924, 153–296.Google Scholar
Bui, H. and Florea, A., Hybrid Euler–Hadamard product for quadratic Dirichlet L-functions in function fields. Proc. Lond. Math. Soc. 117 2018, 65–99.Google Scholar
Dahl, A. and Lamzouri, Y., The distribution of class numbers in a special family of real quadratic fields. Trans. Amer. Math. Soc. 370(9) 2018, 6331–6356.Google Scholar
Granville, A. and Soundararajan, K., The distribution of values of L (1, 𝜒 d ). Geom. Funct. Anal. 13(5) 2003, 992–1028.Google Scholar
Granville, A. and Soundararajan, K., Extreme values of |𝜁(1 + it)|. In The Riemann Zeta Function and Related Themes: Papers in Honour of Professor K. Ramachandra (Lecture Notes Series 2 ), Ramanujan Mathematical Society (2006), 65–80.Google Scholar
Hoffstein, J. and Rosen, M., Average values of L-series in function fields. J. Reine Angew. Math. 426 1992, 117–150.Google Scholar
Jung, H., A remark on average of class numbers of function fields. Korean J. Math. 21(4) 2013, 365–374.Google Scholar
Jung, H., A note on the mean value of L (1, 𝜒) in the hyperelliptic ensemble. Int. J. Number Theory 10(4) 2014, 859–874.Google Scholar
Lamzouri, Y., Extreme values of class numbers or real quadratic fields. Int. Math. Res. Not. IMRN 2015(22) 2015, 11847–11860.Google Scholar
Lamzouri, Y., The distribution of Euler-Kronecker constants of quadratic fields. J. Math. Anal. Appl. 432(2) 2015, 632–653.Google Scholar
Rosen, M., A generalization of Mertens’ theorem. J. Ramanujan Math. Soc. 14(1) 1999, 1–19.Google Scholar
Rosen, M., Number Theory in Function Fields (Graduate Texts in Mathematics 210 ), Springer (New York, 2002).Google Scholar
Siegel, C. L., The average measure of quadratic forms with a given determinant and signature. Ann. Math. 45 1949, 667–685.Google Scholar
Weil, A., Sur les courbes Algébriques et les Variétés qui s’en Déduisent, Hermann (Paris, 1984).Google Scholar