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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Zeros of quadratic Dirichlet $ L$-functions in the hyperelliptic ensemble


Authors: H. M. Bui and Alexandra Florea
Journal: Trans. Amer. Math. Soc. 370 (2018), 8013-8045
MSC (2010): Primary 11M06, 11M38; Secondary 11M50
DOI: https://doi.org/10.1090/tran/7317
Published electronically: June 7, 2018
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Abstract: We study the $ 1$-level density and the pair correlation of zeros of quadratic Dirichlet $ L$-functions in function fields, as we average over the ensemble $ \mathcal {H}_{2g+1}$ of monic, square-free polynomials with coefficients in $ \mathbb{F}_q[x]$. In the case of the $ 1$-level density, when the Fourier transform of the test function is supported in the restricted interval $ (\frac {1}{3},1)$, we compute a secondary term of size $ q^{-\frac {4g}{3}}/g$, which is not predicted by the Ratios Conjecture. Moreover, when the support is even more restricted, we obtain several lower order terms. For example, if the Fourier transform is supported in $ (\frac {1}{3}, \frac {1}{2})$, we identify another lower order term of size $ q^{-\frac {8g}{5}}/g$. We also compute the pair correlation, and as for the $ 1$-level density, we detect lower order terms under certain restrictions; for example, we see a term of size $ q^{-g}/g^2$ when the Fourier transform is supported in $ (\frac {1}{4},\frac {1}{2})$. The $ 1$-level density and the pair correlation allow us to obtain non-vanishing results for $ L(\frac 12,\chi _D)$, as well as lower bounds for the proportion of simple zeros of this family of $ L$-functions.


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Additional Information

H. M. Bui
Affiliation: School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom
Email: hung.bui@manchester.ac.uk

Alexandra Florea
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: alexandra.m.florea@gmail.com

DOI: https://doi.org/10.1090/tran/7317
Received by editor(s): February 28, 2017
Published electronically: June 7, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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