Elliptic subcovers of a curve of genus 2. I. The isogeny defect
- PDF / 660,211 Bytes
- 23 Pages / 439.37 x 666.142 pts Page_size
- 102 Downloads / 213 Views
Elliptic subcovers of a curve of genus 2. I. The isogeny defect Ernst Kani1 Received: 25 October 2017 / Accepted: 27 August 2018 © Fondation Carl-Herz and Springer International Publishing AG, part of Springer Nature 2018
Abstract Let C/K be a curve of genus 2 over an arbitrary field K. The purpose of this paper is to relate the set of equivalence classes of elliptic subcovers of C/K to the set of primitive representations of an intrinsic quadratic form qC called the refined Humbert invariant. Furthermore, the discriminant of this quadratic form is determined here by relating it to a quantity called the isogeny defect. Résumé Soit C/K une courbe de genre 2 sur un corps K quelconque. Le but de cet article est d’établir un rapport entre l’ensemble des classes d’equivalence des sous-revêtements de C/K et l’ensemble des representations primitives d’une forme quadratique intrinséque qC quelle s’appelle l’invariant raffiné de Humbert. De plus, le discriminant de cette forme quadratique est determiné ici. Pour cela, on le relie avec une quantité quelle s’appelle le défaut d’isogénie. Mathematics Subject Classification 14H30 · 14H40 · 14H05 · 14H25
1 Introduction Let C be a curve of genus 2 over an arbitrary field K . An elliptic subcover is a finite morphism f : C → E to an elliptic curve E/K which does not factor over a non-trivial isogeny of E. If f : C → E is another elliptic subcover, then f is said to be equivalent to f if there is ∼ an isomorphism ϕ : E → E such that f = ϕ ◦ f . The purpose of this paper is to study the set E (C) of all equivalence classes of elliptic subcovers of a given genus 2 curve C. This is equivalent to studying the set of maximal elliptic subfields of the function field F = κ(C) of C. It is well-known that E (C) has either 0 or 2 or infinitely many elements; this fact (for K = C) was first noticed by Picard and Bolza; cf. Krazer [12], pp. 487–489. Here we want to study the case that C has infinitely many elliptic subcovers. This happens precisely when the Jacobian JC of C is K -isogenous to E × E, for some elliptic curve E/K . A first step towards the problem of understanding E (C) was taken in [4] in the case that K = K is algebraically closed. In that paper it was explained that a certain positive quadratic
B 1
Ernst Kani [email protected] Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada
123
E. Kani
form qC , called the refined Humbert invariant in [8], gives a (theoretical) description of E (C), for it establishes a bijection between the set En (C) of elliptic subcovers of degree n and the set of primitive solutions x of qC (x) = n 2 , for n ∈ N; cf. [4], Theorem 4.5. Here we extend this result to an arbitrary base field; cf. Theorem 20. In particular, we have Theorem 1 The curve C/K has an elliptic subcover of degree n if and only if the refined Humbert invariant qC primitively represents n 2 . However, in order to be able to apply this theorem to concrete situations, it is necessary to be able to compute the refined Humbert in
Data Loading...
