Non-supersingular hyperelliptic jacobians

Yuri G. Zarhin

Bulletin de la Société Mathématique de France (2004)

  • Volume: 132, Issue: 4, page 617-634
  • ISSN: 0037-9484

Abstract

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Let K be a field of odd characteristic p , let f ( x ) be an irreducible separable polynomial of degree n ≥ 5 with big Galois group (the symmetric group or the alternating group). Let C be the hyperelliptic curve y 2 = f ( x ) and J ( C ) its jacobian. We prove that J ( C ) does not have nontrivial endomorphisms over an algebraic closure of K if either n ≥ 7 or p ≠ 3 .

How to cite

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Zarhin, Yuri G.. "Non-supersingular hyperelliptic jacobians." Bulletin de la Société Mathématique de France 132.4 (2004): 617-634. <http://eudml.org/doc/272307>.

@article{Zarhin2004,
abstract = {Let $K$ be a field of odd characteristic $p$, let $f(x)$ be an irreducible separable polynomial of degree $n \ge 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^2=f(x)$ and $J(C)$ its jacobian. We prove that $J(C)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n \ge 7$ or $p \ne 3$.},
author = {Zarhin, Yuri G.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {hyperelliptic jacobians; endomorphisms of abelian varieties; supersingular abelian varieties},
language = {eng},
number = {4},
pages = {617-634},
publisher = {Société mathématique de France},
title = {Non-supersingular hyperelliptic jacobians},
url = {http://eudml.org/doc/272307},
volume = {132},
year = {2004},
}

TY - JOUR
AU - Zarhin, Yuri G.
TI - Non-supersingular hyperelliptic jacobians
JO - Bulletin de la Société Mathématique de France
PY - 2004
PB - Société mathématique de France
VL - 132
IS - 4
SP - 617
EP - 634
AB - Let $K$ be a field of odd characteristic $p$, let $f(x)$ be an irreducible separable polynomial of degree $n \ge 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^2=f(x)$ and $J(C)$ its jacobian. We prove that $J(C)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n \ge 7$ or $p \ne 3$.
LA - eng
KW - hyperelliptic jacobians; endomorphisms of abelian varieties; supersingular abelian varieties
UR - http://eudml.org/doc/272307
ER -

References

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  1. [1] S. Abhyankar – « Galois theory on the line in nonzero characteristic », Bull. Amer. Math. Soc.27 (1992), p. 68–133. Zbl0760.12002MR1118002
  2. [2] L. Dornhoff – Group Representation Theory, Part A, Marcel Dekker, Inc., New York, 1972. Zbl0236.20004MR347960
  3. [3] W. Feit – « The computations of some Schur indices », Israel J. Math.46 (1983), p. 274–300. Zbl0528.20009MR730344
  4. [4] D. Gorenstein – Finite Simple Groups, An Introduction to their Classification, Plenum Press, New York and London, 1982. Zbl0483.20008MR698782
  5. [5] T. Ibukiyama, T. Katsura & F. Oort – « Supersingular curves of genus two and class numbers », Compositio Math.57 (1986), p. 127–152. Zbl0589.14028MR827350
  6. [6] I. Isaacs – Character theory of finite groups, Pure and Applied Mathematics, vol. 69, Academic Press, New York-San Francisco-London, 1976. Zbl0337.20005MR460423
  7. [7] A. Ivanov & C. Praeger – « On finite affine 2-arc transitive graphs », Europ. J. Combinatorics14 (1993), p. 421–444. Zbl0794.05045MR1241910
  8. [8] G. Janusz – « Simple components of ℚ [ SL ( 2 , q ) ] », Commun. Algebra1 (1974), p. 1–22. Zbl0281.20003MR344323
  9. [9] N. Katz – « Monodromy of families of curves: applications of some results of Davenport-Lewis », Séminaire de Théorie des Nombres (Paris 1979–1980) (M.-J. Bertin, éd.), Progress in Math., vol. 12, Birkhäuser, Boston-Basel-Stuttgart, 1981, p. 171–195. Zbl0475.14025MR633896
  10. [10] —, « Affine cohomological transforms, perversity, and monodromy », J. Amer. Math. Soc.6 (1993), p. 149–222. Zbl0815.14011MR1161307
  11. [11] N. Katz & P. Sarnak – Random matrices, Frobenius eigenvalues and Monodromy, Amer. Math. Soc., Providence, RI, 1999. Zbl0958.11004MR1659828
  12. [12] M. Klemm – « Über die Reduktion von Permutationsmoduln », Math. Z.143 (1975), p. 113–117. Zbl0291.20007MR379639
  13. [13] Y. Manin – « The theory of commutative formal groups over fields of finite characteristic », Russian Math. Surveys18 (1963), p. 1–83. Zbl0128.15603MR157972
  14. [14] D. Masser – « Specialization of some hyperelliptic jacobians », Number Theory in Progress, volI (K. Györy, H. Iwaniec & J. Urbanowicz, éds.), de Gruyter, Berlin-New York, 1999, p. 293–307. Zbl0942.14015MR1689511
  15. [15] S. Mori – « The endomorphism rings of some abelian varieties », Japanese J. Math.2 (1976), p. 109–130. Zbl0339.14016MR453754
  16. [16] —, « The endomorphism rings of some abelian varieties, II », Japanese J. Math.3 (1977), p. 105–109. Zbl0379.14011MR529440
  17. [17] B. Mortimer – « The modular permutation representations of the known doubly transitive groups », Proc. London Math. Soc. 41 (1980), no. 3, p. 1–20. Zbl0393.20002MR579714
  18. [18] D. Mumford – Abelian varieties, 2nd éd., Oxford University Press, London, 1974. Zbl0326.14012MR282985
  19. [19] N. Nygaard – « Slopes of powers of Frobenius on crystalline cohomology », Ann. Sci. École Norm. Sup. 14 (1981), no. 4, p. 369–401. Zbl0519.14012MR654203
  20. [20] J.-P. Serre – « Revêtements des courbes algébriques », Séminaire Bourbaki 1991–92, Astérisque, vol. 206, Société Mathématique de France, Paris, 1992, Exposé no 749, p. 177–182; Œuvres, vol.IV, 157, pp.252–264. Zbl0798.14014
  21. [21] —, Topics in Galois Theory, Jones and Bartlett Publishers, Boston-London, 1992. MR1162313
  22. [22] A. Silverberg – « Fields of definition for homomorphisms of abelian varieties », J. Pure Applied Algebra77 (1992), p. 253–262. Zbl0808.14037MR1154704
  23. [23] A. Silverberg & Y. Zarhin – « Variations on a theme of Minkowski and Serre », J. Pure Applied Algebra111 (1996), p. 285–302. Zbl0885.14006MR1394358
  24. [24] N. Yui – « On the jacobian varieties of hyperelliptic curves over fields of characteristic p g t ; 2 », J. Algebra52 (1978), p. 378–410. Zbl0404.14008MR491717
  25. [25] Y. Zarhin – « Hyperelliptic jacobians without complex multiplication », Math. Res. Letters7 (2000), p. 123–132. Zbl0959.14013MR1748293
  26. [26] —, « Hyperelliptic jacobians and modular representations », Moduli of abelian varieties (G. van der Geer, C. Faber & F. Oort, éds.), Progress in Math., vol. 195, Birkhäuser, Basel-Boston-Berlin, 2001, p. 473–490. MR1827030
  27. [27] —, « Hyperelliptic jacobians without complex multiplication in positive characteristic », Math. Research Letters8 (2001), p. 429–435. Zbl1079.14512MR1849259
  28. [28] —, « Hyperelliptic Jacobians without Complex Multiplication, Doubly Transitive Permutation Groups and Projective Representations », Algebraic Number Theory and Algebraic Geometry (Parshin Festschrift), Contemporary Math., vol. 300, American Mathematical Society, Providence, RI, 2002, p. 195–210. Zbl0886.14001MR1936373
  29. [29] —, « Very simple 2 -adic representations and hyperelliptic jacobians », Moscow Math. J. 2 (2002), no. 2, p. 403–431. Zbl1082.11039MR1944511
  30. [30] —, « Hyperelliptic jacobians and simple groups U 3 ( 2 m ) », Proc. Amer. Math. Soc. 131 (2003), no. 1, p. 95–102. Zbl1001.14012MR1929028
  31. [31] —, « Very simple representations: variations on a theme of Clifford », Progress in Galois Theory (H. Völklein & T. Shaska, éds.), Developments in Math., Kluwer, 2004, pp.151–168, to appear. Zbl1129.20006MR2148465

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