Gluing curves of genus 1 and 2 along their 2-torsion. (English) Zbl 1472.14036
Let \(A\) be an abelian variety over a field \(k\). It is well-known that there is an isogeny decomposition (Poincaré’s Complete Reducibility Theorem) in terms of simple abelian subvarieties \[A \sim B_1^{n_1} \times \cdots \times B_r^{n_r}\] that are pairwise non-isogenous over \(k\). This decomposition is unique up to reordering the factors.
In the case that \(A\) equals the Jacobian variety of a curve \(Z\), there exist algorithms to calculate the aforementioned decomposition in terms of the Jacobians of curves over small extensions of \(k\) whenever possible. The decomposition of the Jacobian of curves has been extensively studied.
The article under review consider a different approach to this problem. The authors aim to develop algorithms to construct an abelian variety \(A\) given factors \(B_i\) as before, in some special cases.
Let X be a curve of genus 1 and let \(Y\) be a curve of genus 2, defined over a base field \(k\) whose characteristic is different from 2. The main result of the paper provides criteria for the existence of a curve \(Z\) over \(k\) whose Jacobian is, up to a special isogeny, isogenous to the products of the Jacobians of \(X\) and \(Y\). The authors also developed algorithms to construct the curve \(Z\) once equations for \(X\) and \(Y\) are given.
In the case that \(A\) equals the Jacobian variety of a curve \(Z\), there exist algorithms to calculate the aforementioned decomposition in terms of the Jacobians of curves over small extensions of \(k\) whenever possible. The decomposition of the Jacobian of curves has been extensively studied.
The article under review consider a different approach to this problem. The authors aim to develop algorithms to construct an abelian variety \(A\) given factors \(B_i\) as before, in some special cases.
Let X be a curve of genus 1 and let \(Y\) be a curve of genus 2, defined over a base field \(k\) whose characteristic is different from 2. The main result of the paper provides criteria for the existence of a curve \(Z\) over \(k\) whose Jacobian is, up to a special isogeny, isogenous to the products of the Jacobians of \(X\) and \(Y\). The authors also developed algorithms to construct the curve \(Z\) once equations for \(X\) and \(Y\) are given.
Reviewer: Sebastián Reyes-Carocca (Temuco)
MSC:
| 14H40 | Jacobians, Prym varieties |
| 14H25 | Arithmetic ground fields for curves |
| 14H30 | Coverings of curves, fundamental group |
| 14H45 | Special algebraic curves and curves of low genus |
| 14H50 | Plane and space curves |
| 14K20 | Analytic theory of abelian varieties; abelian integrals and differentials |
| 14K30 | Picard schemes, higher Jacobians |
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