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Proceedings of the American Mathematical Society

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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
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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.

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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