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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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  • 1. G. V. Belyi, On extensions of the maximal cyclotomic field having a given classical Galois group. J. Reine Angew. Math. 341 (1983), 147-156. MR 84h:12010
  • 2. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups. Clarendon Press, Oxford, 1985. MR 88g:20025
  • 3. Ch. W. Curtis, The Steinberg character of a finite group with a $(B,\,N)$-pair. J. Algebra 4 (1966), 433-441. MR 34:1406
  • 4. D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simple groups, Number 3. Mathematical Surveys and Monographs 40.3, AMS, Providence, RI, 1998. MR 98j:20011
  • 5. R. W. Hartley, Determination of the ternary collineation groups whose coefficients lie in the $GF(2^n)$. Ann. of Math. 27 (1926), 140-158.
  • 6. A. R. Hoffer, On unitary collineation groups. J. Algebra 22 (1972), 211-218. MR 46:780
  • 7. J. E. Humphreys, The Steinberg representation. Bull. AMS (N.S.) 16 (1987), 247-263. MR 88c:20050
  • 8. Ch. Jansen, K. Lux, R. Parker, R. Wilson, An Atlas of Brauer characters. Clarendon Press, Oxford, 1995. MR 96k:20016
  • 9. N. Katz, Monodromy of families of curves: applications of some results of Davenport-Lewis. In: Séminaire de Théorie des Nombres, Paris 1979-80 (ed. M.-J. Bertin); Progress in Math. 12, pp. 171-195, Birkhäuser, Boston-Basel-Stuttgart, 1981. MR 83d:14012
  • 10. N. Katz, Affine cohomological transforms, perversity, and monodromy. J. Amer. Math. Soc. 6 (1993), 149-222. MR 94b:14013
  • 11. D. Masser, Specialization of some hyperelliptic jacobians. In: Number Theory in Progress (eds. K. Györy, H. Iwaniec, J.Urbanowicz), vol. I, pp. 293-307; de Gruyter, Berlin-New York, 1999. MR 2000j:11088
  • 12. Sh. Mori, The endomorphism rings of some abelian varieties. Japanese J. Math. 2 (1976), 109-130. MR 56:12013
  • 13. Sh. Mori, The endomorphism rings of some abelian varieties. II, Japanese J. Math. 3 (1977), 105-109. MR 80e:14009
  • 14. J.-P. Serre, Linear representations of finite groups, Springer-Verlag, 1977. MR 56:8675
  • 15. Yu. G. Zarhin, Hyperelliptic jacobians without complex multiplication. Math. Res. Letters 7 (2000), 123-132. MR 2001a:11097
  • 16. Yu. G. Zarhin, Hyperelliptic jacobians and modular representations. In: Moduli of abelian varieties (C. Faber, G. van der Geer, F. Oort, eds.), pp. 473-490, Progress in Math., Vol. 195, Birkhäuser, Basel-Boston-Berlin, 2001. MR 2002b:11082
  • 17. Yu. G. Zarhin, Hyperelliptic jacobians without complex multiplication in positive characteristic. Math. Res. Letters 8 (2001), 429-435.

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Additional Information

Yuri G. Zarhin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

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

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