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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 jacobians and simple groups $\mathbf{U}_3(2^m)$


Author: Yuri G. Zarhin
Journal: Proc. Amer. Math. Soc. 131 (2003), 95-102
MSC (2000): Primary 14H40; Secondary 14K05
Published electronically: May 22, 2002
MathSciNet review: 1929028
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Abstract: In a previous paper, the author proved that in characteristic zero the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure $K_a$ of the ground field $K$ if the Galois group $\operatorname{Gal}(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $\mathbf{S}_n$ or the alternating group $\mathbf{A}_n$. Here $n>4$ is the degree of $f$. In another paper by the author this result was extended to the case of certain ``smaller'' Galois groups. In particular, the infinite series $n=2^r+1, \operatorname{Gal}(f)=\mathbf{L}_2(2^r):=\operatorname{PSL}_2 (\mathbf{F}_{2^r})$ and $n=2^{4r+2}+1, \operatorname{Gal}(f)=\mathbf{Sz}(2^{2r+1})$were treated. In this paper the case of $\operatorname{Gal}(f)=\mathbf{U}_3(2^m):=\operatorname{PSU}_3 (\mathbf{F}_{2^m})$ and $n=2^{3m}+1$ is treated.


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Additional Information

Yuri G. Zarhin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: zarhin@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06689-3
PII: S 0002-9939(02)06689-3
Keywords: Hyperelliptic jacobians, endomorphisms of abelian varieties, Steinberg representations, unitary groups, Hermitian curves
Received by editor(s): August 30, 2001
Published electronically: May 22, 2002
Additional Notes: This work was partially supported by NSF grant DMS-0070664
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2002 American Mathematical Society