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

 

Hyperelliptic curves over $\mathbb{F} _2$ of every $2$-rank without extra automorphisms


Author: Hui June Zhu
Journal: Proc. Amer. Math. Soc. 134 (2006), 323-331
MSC (2000): Primary 11G10, 14G15
Published electronically: August 25, 2005
MathSciNet review: 2175998
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Abstract: We prove that for any pair of integers $0\leq r\leq g$ such that $g\geq 3$ or $r>0$, there exists a (hyper)elliptic curve $C$ over $\mathbb{F} _2$ of genus $g$ and $2$-rank $r$ whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties $(A,\lambda)$ over $\mathbb{F} _2$ of dimension $g$ and $2$-rank $r$ such that $\operatorname{Aut}(A,\lambda)=\{\pm 1\}$.


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Additional Information

Hui June Zhu
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email: zhu@cal.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08294-8
PII: S 0002-9939(05)08294-8
Keywords: Automorphism group, hyperelliptic curve
Received by editor(s): July 20, 2004
Published electronically: August 25, 2005
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.