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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Class groups of global function fields with certain splitting behaviors of the infinite prime


Author: Yoonjin Lee
Journal: Proc. Amer. Math. Soc. 137 (2009), 415-424
MSC (2000): Primary 11R29; Secondary 11R58
Published electronically: October 6, 2008
MathSciNet review: 2448559
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Abstract: For certain two cases of splitting behaviors of the prime at infinity with unit rank $ r$, given positive integers $ m, n$, we construct infinitely many global function fields $ K$ such that the ideal class group of $ K$ of degree $ m$ over $ \mathbb{F}(T)$ has $ n$-rank at least $ m-r-1$ and the prime at infinity splits in $ K$ as given, where $ \mathbb{F}$ denotes a finite field and $ T$ a transcendental element over $ \mathbb{F}$. In detail, for positive integers $ m$, $ n$ and $ r$ with $ 0 \le r \le m-1$ and a given signature $ (e_i, \mathfrak{f}_i)$, $ 1 \le i \le r+1$, such that $ \sum_{i=1}^{r+1}{e_i\mathfrak{f}_i} =m$, in the following two cases where $ e_i$ is arbitrary and $ \mathfrak{f}_i =1$ for each $ i$, or $ e_i =1$ and $ \mathfrak{f}_i$'s are the same for each $ i$, we construct infinitely many global function fields $ K$ of degree $ m$ over $ \mathbb{F}(T)$ such that the ideal class group of $ K$ contains a subgroup isomorphic to $ (\mathbb{Z}/n\mathbb{Z})^{m-r-1}$ and the prime at infinity $ {\wp_\infty}$ splits into $ r+1$ primes $ \mathfrak{P}_1, \mathfrak{P}_2, \cdots, \mathfrak{P}_{r+1}$ in $ K$ with $ e(\mathfrak{P}_i/{\wp_\infty}) = e_i$ and $ \mathfrak{f}(\mathfrak{P}_i/{\wp_\infty}) = \mathfrak{f}_i$ for $ 1 \le i \le r+1$ (so, $ K$ is of unit rank $ r$).


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Additional Information

Yoonjin Lee
Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 120-750, Republic of Korea
Email: yoonjinl@ewha.ac.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09581-6
PII: S 0002-9939(08)09581-6
Keywords: Class group, class number, rank of class group, imaginary function field
Received by editor(s): April 26, 2007
Published electronically: October 6, 2008
Additional Notes: This work was supported by the Ewha Womans University Research Grant of 2007
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.