Algebraic description of Jacobians isogenous to certain Prym varieties with polarization \((1,2)\). (Algebraic description of Jacobians isogeneous to certain Prym varieties with polarization \((1,2)\).) (English) Zbl 1412.14022
The motivation of this study comes from an algebraic integrable system known as the Clebsch integrable case of the Krichhoff equations, which was first integrated in terms of theta-functions of a genus 2 curve \(\Gamma\) by F. Kötter [J. Reine Angew. Math. 109, 51–81, 89–111 (1892; JFM 23.0977.01)]. The authors consider a class of generally non-hyperelliptic genus 3 curves which are twofold coverings of elliptic curves \(E\); for the affine standard coordinate \((x, y)\in E\), they study the covering \(\pi : C \to E\) such that \(\pi(x, y,w) = (x, y) \in E\), then \(C\) has the natural involution \(\sigma : (x,w) \to (x,-w)\). The involution implies the existence of 2-dimensional Prym variety \(\mathrm{Prym}(C,\sigma)\) in the Jacobian of \(C\), \(\mathrm{Jac}(C)\). Using the previous results of Kötter and E. Horozov and P. Van Moerbeke [Commun. Pure Appl. Math. 42, No. 4, 357–407 (1989; Zbl 0689.58020)], the authors provide explicit algebraic descriptions of all birationally nonequivalent genus 2 curves whose Jacobians are degree 2 isogenous to the Prym variety and its dual \(\mathrm{Prym}^*(C,\sigma)\); in total there appear six such genus 2 curves. It should be noticed that an alternative but equivalent description of the genus 2 curves have been obtained in [C. Ritzenthaler and M. Romagny, Épijournal de Géom. Algébr., EPIGA 2, Article No. 2, 8 p. (2018; Zbl 1471.14068)].
In application to the Clebsch integrable system, the authors show that when \(C\) is the spectral curve of the corresponding Lax pair, then one of these six genus 2 curves coincides with the curve \(\Gamma\) of Kötter. They also consider some special cases of the covering \(\pi \colon C \to E\) when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian \(\mathrm{Jac}(C)\) is isogenous (but not isomorphic) to the product of three different elliptic curves. The description is accompanied with explicit algorithms of calculation of periods of the Prym varieties and of absolute invariants of genus 2 curves. By computing these results numerically, they show that their main results are experimentally confirmed.
In application to the Clebsch integrable system, the authors show that when \(C\) is the spectral curve of the corresponding Lax pair, then one of these six genus 2 curves coincides with the curve \(\Gamma\) of Kötter. They also consider some special cases of the covering \(\pi \colon C \to E\) when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian \(\mathrm{Jac}(C)\) is isogenous (but not isomorphic) to the product of three different elliptic curves. The description is accompanied with explicit algorithms of calculation of periods of the Prym varieties and of absolute invariants of genus 2 curves. By computing these results numerically, they show that their main results are experimentally confirmed.
Reviewer: Shigeki Matsutani (Kanazawa)
MSC:
| 14H40 | Jacobians, Prym varieties |
| 14H45 | Special algebraic curves and curves of low genus |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 14K02 | Isogeny |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 32G20 | Period matrices, variation of Hodge structure; degenerations |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
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