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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Genus bounds for curves with fixed Frobenius eigenvalues
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by Noam D. Elkies, Everett W. Howe and Christophe Ritzenthaler PDF
Proc. Amer. Math. Soc. 142 (2014), 71-84

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 \[ r = \#(S\cap \{\pi \}) + 2 \! \sum _{\theta \in S\setminus \{0,\pi \}} \left \lceil \frac {\pi }{2\theta }\right \rceil . \] We show that if $C$ is a curve over $\mathbf {F}_q$ whose genus is greater than \[ \min \left ( 23 s^2 q^{2s} \log q, \ (\sqrt {q}+1)^{2r} \left (\frac { 1 + q^{-r}}{2} \right ) \right ), \] 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
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Additional Information
  • Noam D. Elkies
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138–2901
  • Email: elkies@math.harvard.edu
  • Everett W. Howe
  • Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121-1967
  • MR Author ID: 236352
  • ORCID: 0000-0003-4850-8391
  • Email: however@alumni.caltech.edu
  • Christophe Ritzenthaler
  • Affiliation: Institut de Mathématiques de Luminy, UMR 6206 du CNRS, Luminy, Case 907, 13288 Marseille, France
  • MR Author ID: 702917
  • Email: ritzenth@iml.univ-mrs.fr
  • 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
  • © Copyright 2013 Noam D. Elkies, Christophe Ritzenthaler, and the Institute for Defense Analyses
  • Journal: Proc. Amer. Math. Soc. 142 (2014), 71-84
  • MSC (2010): Primary 14G10; Secondary 11G20, 14G15, 14H25
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11839-3
  • MathSciNet review: 3119182