Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Genus bounds for curves with fixed Frobenius eigenvalues

Authors: Noam D. Elkies, Everett W. Howe and Christophe Ritzenthaler
Journal: Proc. Amer. Math. Soc. 142 (2014), 71-84
MSC (2010): Primary 14G10; Secondary 11G20, 14G15, 14H25
Published electronically: September 18, 2013
MathSciNet review: 3119182
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For every finite collection $ \mathcal {C}$ of abelian varieties over $ \mathbf {F}_q$, we produce an explicit upper bound on the genus of curves over $ \mathbf {F}_q$ whose Jacobians are isogenous to a product of powers of elements of $ \mathcal {C}$.

Our explicit bound is expressed in terms of the Frobenius angles of the elements of $ \mathcal {C}$. In general, suppose that $ S$ is a finite collection of $ s$ real numbers in the interval $ [0,\pi ]$. If $ S = \{0\}$ set $ r = 1/2$; otherwise, let

$\displaystyle r = \char93 (S\cap \{\pi \}) + 2 \! \sum _{\theta \in S\setminus \{0,\pi \}} \left \lceil \frac {\pi }{2\theta }\right \rceil . $

We show that if $ C$ is a curve over $ \mathbf {F}_q$ whose genus is greater than

$\displaystyle \min \left ( 23\, s^2 q^{2s} \log q, \ (\sqrt {q}+1)^{2r} \left (\frac { 1 + q^{-r}}{2} \right ) \right ), $

then $ C$ has a Frobenius angle $ \theta $ such that neither $ \theta $ nor $ -\theta $ lies in $ S$.

We do not claim that this genus bound is best possible. For any particular set $ S$ we can usually obtain a better bound by solving a linear programming problem. For example, we use linear programming to give a new proof of a result of Duursma and Enjalbert: If the Jacobian of a curve $ C$ over $ \mathbf {F}_2$ is isogenous to a product of elliptic curves over $ \mathbf {F}_2$, then the genus of $ C$ is at most $ 26$. As Duursma and Enjalbert note, this bound is sharp, because there is an $ \mathbf {F}_2$-rational model of the genus-$ 26$ modular curve $ X(11)$ whose Jacobian splits completely into elliptic curves.

As an application of our results, we give a new proof of (and correct a small error in) a result of Yamauchi, which provides the complete list of positive integers $ N$ such that the modular Jacobian $ J_0(N)$ is isogenous over $ \mathbf {Q}$ to a product of elliptic curves.

References [Enhancements On Off] (What's this?)

  • [1] Peter Bending, Alan Camina, and Robert Guralnick, Automorphisms of the modular curve, Progress in Galois theory, Dev. Math., vol. 12, Springer, New York, 2005, pp. 25-37. MR 2148458 (2006c:14050),
  • [2] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, Computational algebra and number theory (London, 1993), J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. MR 1484478,
  • [3] Henri Cohen, Sur les $ N$ tels que $ J_0(N)$ soit $ \mathbf {Q}$-isogène à un produit de courbes elliptiques. Undated preprint, available at cohen/serretrace.dvi.
  • [4] János A. Csirik, Joseph L. Wetherell, and Michael E. Zieve, On the genera of $ X_0(N)$ (2000)., arXiv:math/0006096v2 [math.NT].
  • [5] Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272 (German). DOI 10.1007/BF02940746. MR 0005125 (3,104f)
  • [6] Stephen A. DiPippo and Everett W. Howe, Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory 73 (1998), no. 2, 426-450. MR 1657992 (2000c:11101),
  • [7] Stephen A. DiPippo and Everett W. Howe, Corrigendum: ``Real polynomials with all roots on the unit circle and abelian varieties over finite fields'' [J. Number Theory 73 (1998), no. 2, 426-450; MR1657992 (2000c:11101)], J. Number Theory 83 (2000), no. 1, 182. MR 1767658,
  • [8] Iwan Duursma and Jean-Yves Enjalbert, Bounds for completely decomposable Jacobians, Finite fields with applications to coding theory, cryptography and related areas (Oaxaca, 2001) Springer, Berlin, 2002, pp. 86-93. Electronic version, with an addendum, available at arXiv:1007.3344v1 [math.NT]. MR 1995329 (2005d:14032)
  • [9] Torsten Ekedahl and Jean-Pierre Serre, Exemples de courbes algébriques à jacobienne complètement décomposable, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 509-513 (French, with English and French summaries). MR 1239039 (94j:14029)
  • [10] Daniel Goldstein, Robert M. Guralnick, Everett W. Howe, and Michael E. Zieve, Nonisomorphic curves that become isomorphic over extensions of coprime degrees, J. Algebra 320 (2008), no. 6, 2526-2558. MR 2437513 (2009h:14051),
  • [11] Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721-724 (1982). MR 656048 (84c:14016)
  • [12] Gérard Ligozat, Courbes modulaires de niveau $ 11$, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., vol. 601, Springer, Berlin, 1977, pp. 149-237 (French). DOI 10.1007/BFb0063948. MR 0463118 (57 #3079)
  • [13] Christophe Ritzenthaler, Automorphismes des courbes modulaires $ X(n)$ en caractéristique $ p$, Manuscripta Math. 109 (2002), no. 1, 49-62 (French, with English summary). MR 1931207 (2003g:11067),
  • [14] Hans-Georg Rück and Henning Stichtenoth, A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math. 457 (1994), 185-188. MR 1305281 (95h:11059),
  • [15] Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l'opérateur de Hecke $ T_p$, J. Amer. Math. Soc. 10 (1997), no. 1, 75-102 (French). MR 1396897 (97h:11048),
  • [16] C. J. Smyth, Some inequalities for certain power sums, Acta Math. Acad. Sci. Hungar. 28 (1976), no. 3-4, 271-273. DOI 10.1007/BF01896789. MR 0424710 (54 #12669)
  • [17] William Stein and others, Sage Mathematics Software (Version 4.0.1), the Sage Development Team, 2009.
  • [18] Michael A. Tsfasman, Some remarks on the asymptotic number of points, Coding theory and algebraic geometry (Luminy, 1991) Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992, pp. 178-192. MR 1186424 (93h:11064),
  • [19] M. A. Tsfasman and S. G. Vlăduţ, Asymptotic properties of zeta-functions, Algebraic geometry, 7, J. Math. Sci. (New York) 84 (1997), no. 5, 1445-1467. MR 1465522 (98h:11079),
  • [20] William C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4) 2 (1969), 521-560. MR 0265369 (42 #279)
  • [21] Takuya Yamauchi, On $ \mathbb{Q}$-simple factors of Jacobian varieties of modular curves, Yokohama Math. J. 53 (2007), no. 2, 149-160. MR 2302608 (2008k:11062)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14G10, 11G20, 14G15, 14H25

Retrieve articles in all journals with MSC (2010): 14G10, 11G20, 14G15, 14H25

Additional Information

Noam D. Elkies
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138–2901

Everett W. Howe
Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121-1967

Christophe Ritzenthaler
Affiliation: Institut de Mathématiques de Luminy, UMR 6206 du CNRS, Luminy, Case 907, 13288 Marseille, France

Keywords: Curve, Jacobian, Weil polynomial, Frobenius eigenvalue, genus, linear programming
Received by editor(s): February 28, 2012
Published electronically: September 18, 2013
Additional Notes: The third author was partially supported by grant MTM2006-11391 from the Spanish MEC and by grant ANR-09-BLAN-0020-01 from the French ANR
Communicated by: Ken Ono
Article copyright: © Copyright 2013 Noam D. Elkies, Christophe Ritzenthaler, and the Institute for Defense Analyses

American Mathematical Society