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Genus character L-functions of orders in quadratic function fields
Abstract
We define genus character L-functions of orders in quadratic function fields of odd characteristic. We express each as a product of two quadratic L-functions and explicit local factors at primes dividing the conductor, with a possible shift of the variable in one of the two quadratic L-functions. This product formula yields meromorphic continuation and functional equations. In the real case, we evaluate special values in terms of continued fractions and derive formulas for the product of class numbers of two imaginary quadratic function fields. These formulas are function field analogues of the Kaneko-Mizuno-Zagier formula.
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BibTeX
@misc{kim2026genus,
title = {{Genus character $L$-functions of orders in quadratic function fields}},
author = {Jigu Kim and Yoshinori Mizuno},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20214},
eprint = {2610.20214},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20214},
}
RIS
TY - UNPB
TI - Genus character L-functions of orders in quadratic function fields
AU - Jigu Kim
AU - Yoshinori Mizuno
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20214
UR - https://arxiv.org/abs/2610.20214
AB - We define genus character L-functions of orders in quadratic function fields of odd characteristic. We express each as a product of two quadratic L-functions and explicit local factors at primes dividing the conductor, with a possible shift of the variable in one of the two quadratic L-functions. This product formula yields meromorphic continuation and functional equations. In the real case, we evaluate special values in terms of continued fractions and derive formulas for the product of class numbers of two imaginary quadratic function fields. These formulas are function field analogues of the Kaneko-Mizuno-Zagier formula.
ER -