Genus bounds for curves with fixed Frobenius eigenvalues

Christophe Ritzenthaler (1) , Noam Elkies , Everett Howe

1 IML - Institut de mathématiques de Luminy

For every finite collection C of abelian varieties over F-q, we produce an explicit upper bound on the genus of curves over F-q whose Jacobians are isogenous to a product of powers of elements of C. Our explicit bound is expressed in terms of the Frobenius angles of the elements of 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 boolean AND {pi}) + 2 Sigma(theta is an element of S\{0,pi}) inverted right perpendicular pi/2 theta inverted left perpendicular. We show that if C is a curve over F-q whose genus is greater than min(23 s(2)q(2s) log q, (root q + 1)(2r) (1 + q(-r)/2)), 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 F-2 is isogenous to a product of elliptic curves over F-2 then the genus of C is at most 26. As Duursma and Enjalbert note, this bound is sharp, because there is an 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 reprove (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 Q to a product of elliptic curves.

Domaines: Mathématiques [math]

Christophe Ritzenthaler, Noam Elkies, Everett Howe. Genus bounds for curves with fixed Frobenius eigenvalues. Proceedings of the American Mathematical Society, 2014, 142 (1), pp.71-84. ⟨10.1090/S0002-9939-2013-11839-3⟩. ⟨hal-01099960⟩