Large values of quadratic Dirichlet $L$-functions over monic irreducible polynomial in $\mathbb {F}_q[t]$
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- by Pranendu Darbar and Gopal Maiti;
- Proc. Amer. Math. Soc. 152 (2024), 3243-3254
- DOI: https://doi.org/10.1090/proc/16828
- Published electronically: June 14, 2024
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Abstract:
We prove an $\Omega$-result for the quadratic Dirichlet $L$-function $|L(1/2, \chi _P)|$ over irreducible polynomials $P$ associated with the hyperelliptic curve of genus $g$ over a fixed finite field $\mathbb {F}_q$ in the large genus limit. In particular, we showed that for any $\epsilon \in (0, 1/2)$, \[ \max _{\substack {P\in \mathcal {P}_{2g+1}}}|L(1/2, \chi _P)|\gg \exp \left (\left (\sqrt {\left (1/2-\epsilon \right )\ln q}+o(1)\right )\sqrt {\frac {g \ln _2 g}{\ln g}}\right ), \] where $\mathcal {P}_{2g+1}$ is the set of all monic irreducible polynomials of degree $2g+1$. This matches with the order of magnitude of the Bondarenko–Seip bound.References
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Bibliographic Information
- Pranendu Darbar
- Affiliation: Max-Planck Institute for Mathematics, Vivatsgasse 7, Bonn 53111, Germany
- MR Author ID: 1230063
- Email: darbar@mpim-bonn.mpg.de
- Gopal Maiti
- Affiliation: Max-Planck Institute for Mathematics, Vivatsgasse 7, Bonn 53111, Germany
- MR Author ID: 1376895
- Email: maiti@mpim-bonn.mpg.de
- Received by editor(s): April 27, 2023
- Received by editor(s) in revised form: November 15, 2023, and February 5, 2024
- Published electronically: June 14, 2024
- Additional Notes: The first author was funded by Grant 275113 of the Research Council of Norway through the Alain Bensoussan Fellowship Programme of the European Research Consortium for Informatics and Mathematics. The second author was funded by the joint FWF-ANR project Arithrand: FWF: I 4945-N and ANR-20-CE91-0006.
- Communicated by: Amanda Folsom
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3243-3254
- MSC (2020): Primary 11R59, 11G20, 11T06
- DOI: https://doi.org/10.1090/proc/16828
- MathSciNet review: 4767259