Hyperelliptic jacobians and simple groups $\mathbf {U}_3(2^m)$
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- by Yuri G. Zarhin PDF
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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.References
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Additional Information
- Yuri G. Zarhin
- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 200326
- Email: zarhin@math.psu.edu
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 95-102
- MSC (2000): Primary 14H40; Secondary 14K05
- DOI: https://doi.org/10.1090/S0002-9939-02-06689-3
- MathSciNet review: 1929028