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Fiber products and class groups of hyperelliptic curves

Author: Jeffrey D. Achter
Journal: Proc. Amer. Math. Soc. 138 (2010), 3159-3161
MSC (2010): Primary 11R58; Secondary 11R29, 14H05
Published electronically: April 21, 2010
MathSciNet review: 2653940
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathbb{F}_q$ be a finite field of odd characteristic, and let $ N$ be an odd natural number. An explicit fiber product construction shows that if $ N$ divides the class number of some quadratic function field over $ \mathbb{F}_q$, then it does so for infinitely many such function fields.

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  • [Ach06] Jeffrey D. Achter, The distribution of class groups of function fields, J. Pure Appl. Algebra 204 (2006), no. 2, 316-333. MR 2184814 (2006h:11132)
  • [BJLS08] M. Bauer, M. J. Jacobson, Jr., Y. Lee, and R. Scheidler, Construction of hyperelliptic function fields of high three-rank, Math. Comp. 77 (2008), no. 261, 503-530 (electronic). MR 2353964 (2008i:11135)
  • [DH89] Bert Ditters and Simen J. Hoving, On the connected part of the covariant Tate $ p$-divisible group and the $ \zeta$-function of the family of hyperelliptic curves $ y^2=1+\mu x^N$ modulo various primes, Math. Z. 200 (1989), no. 2, 245-264. MR 978298 (90e:14047)
  • [EVW09] Jordan Ellenberg, Akshay Venkatesh, and Craig Westerberg, Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, December 2009, arXiv:0912.0325.
  • [FW89] Eduardo Friedman and Lawrence C. Washington, On the distribution of divisor class groups of curves over a finite field, Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 227-239. MR 91e:11138
  • [GP05] Darren Glass and Rachel Pries, Hyperelliptic curves with prescribed $ p$-torsion, Manuscripta Math. 117 (2005), no. 3, 299-317. MR 2154252 (2006e:14039)
  • [Hon66] Taira Honda, On the Jacobian variety of the algebraic curve $ y^{2}=1-x^{l}$ over a field of characteristic $ p>0$, Osaka J. Math. 3 (1966), 189-194. MR 0225777 (37:1370)
  • [Pac09] Allison M. Pacelli, Function fields with $ 3$-rank at least $ 2$, Acta Arith. 139 (2009), no. 2, 101-110. MR 2539539

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Additional Information

Jeffrey D. Achter
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523

Received by editor(s): December 18, 2009
Published electronically: April 21, 2010
Additional Notes: The author was partially supported by NSA grant H98230-08-1-0051.
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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