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Counting cyclic extensions with local conditions and applications to L-values in the critical strip

Part of: Zeta and $L$-functions: analytic theory Algebraic number theory: global fields

Published online by Cambridge University Press:  08 August 2025

Peter Jaehyun Cho  and
Gyeongwon Oh
Peter Jaehyun Cho*
Affiliation:
Department of Mathematical Sciences, https://ror.org/017cjz748 Ulsan National Institute of Science and Technology, Ulsan 689-798, Republic of Korea
Gyeongwon Oh
Affiliation:
Department of Mathematics Education, https://ror.org/05kzjxq56 Chonnam National University, Gwangju 61469, Republic of Korea e-mail: wongyeong90@jnu.ac.kr
*
e-mail: petercho@unist.ac.kr
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Abstract

Let $\ell $ be an odd prime. We investigate the enumeration of cyclic extensions of degree $\ell $ over $\mathbb {Q}$ subject to specified local conditions. By ordering these extensions according to their conductors, we derive an asymptotic count with a power-saving error term. As a consequence of our results, we analyze the distribution of values of L-functions associated with these extensions in the critical strip.

Keywords

Cyclic extensionL-value

MSC classification

Primary: 11R42: Zeta functions and $L$-functions of number fields 11M06: $zeta (s)$ and $L(s, chi)$

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 39
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Peter J. Cho was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (Grant Nos. RS-2022-NR069491 and RS-2025-02262988) and Gyeongwon Oh was was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (Grant Nos. RS-2024-00415601 and RS-2024-00341372).

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