Hyperelliptic curves over $\mathbb {F}_2$ of every $2$-rank without extra automorphisms
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- by Hui June Zhu PDF
- Proc. Amer. Math. Soc. 134 (2006), 323-331 Request permission
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 $\operatorname {Aut}(A,\lambda )=\{\pm 1\}$.References
- Gerard van der Geer and Marcel van der Vlugt, Reed-Muller codes and supersingular curves. I, Compositio Math. 84 (1992), no. 3, 333–367. MR 1189892
- Nicholas M. Katz and Peter Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999. MR 1659828, DOI 10.1090/coll/045
- Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
- Manohar L. Madan, On a theorem of M. Deuring and I. R. Šafarevič, Manuscripta Math. 23 (1977/78), no. 1, 91–102. MR 460335, DOI 10.1007/BF01168587
- Sh\B{o}ichi Nakajima, Equivariant form of the Deuring-Šafarevič formula for Hasse-Witt invariants, Math. Z. 190 (1985), no. 4, 559–566. MR 808922, DOI 10.1007/BF01214754
- J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167–212. MR 861976
- Shigefumi Mori, The endomorphism rings of some Abelian varieties, Japan. J. Math. (N.S.) 2 (1976), no. 1, 109–130. MR 453754, DOI 10.4099/math1924.2.109
- Frans Oort, Endomorphism algebras of abelian varieties, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 469–502. MR 977774
- Bjorn Poonen, Varieties without extra automorphisms. I. Curves, Math. Res. Lett. 7 (2000), no. 1, 67–76. MR 1748288, DOI 10.4310/MRL.2000.v7.n1.a6
- Bjorn Poonen, Varieties without extra automorphisms. I. Curves, Math. Res. Lett. 7 (2000), no. 1, 67–76. MR 1748288, DOI 10.4310/MRL.2000.v7.n1.a6
- Bjorn Poonen, Varieties without extra automorphisms. III. Hypersurfaces, Finite Fields Appl. 11 (2005), no. 2, 230–268. MR 2129679, DOI 10.1016/j.ffa.2004.12.001
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- Henning Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe, Arch. Math. (Basel) 24 (1973), 527–544 (German). MR 337980, DOI 10.1007/BF01228251
- Yuri G. Zarhin, Hyperelliptic Jacobians without complex multiplication in positive characteristic, Math. Res. Lett. 8 (2001), no. 4, 429–435. MR 1849259, DOI 10.4310/MRL.2001.v8.n4.a3
Additional Information
- Hui June Zhu
- Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
- Email: zhu@cal.berkeley.edu
- Received by editor(s): July 20, 2004
- Published electronically: August 25, 2005
- Communicated by: David E. Rohrlich
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 323-331
- MSC (2000): Primary 11G10, 14G15
- DOI: https://doi.org/10.1090/S0002-9939-05-08294-8
- MathSciNet review: 2175998