Algebraic Description of Jacobians Isogeneous to Certain Prym Varieties with Polarization (1,2)

ABSTRACT

For a class of non-hyperelliptic genus 3 curves C which are twofold coverings of elliptic curves E, we give an explicit algebraic description of all birationally nonequivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated with such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties. We also consider some special cases of the covering C → E, in particular, when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian of C is isogeneous (but not isomorphic) to the product of three different elliptic curves. Our description is accompanied with explicit algorithms of calculation of periods of the Prym varieties and of absolute invariants of genus 2 curves. They are followed by numerical examples, which experimentally confirm main results of the article.

KEYWORDS:

• algebraic curves
• Prym varieties
• algebraic integrable systems

2000 AMS SUBJECT CLASSIFICATION:

• 70H06
• 14H40
• 14H45
• 14H70
• 14K02
• 30E20
• 32G20

Acknowledgments

The authors are grateful to H. Braden for making independently a series of hard calculations which confirmed the results of their numerical examples, as well as for continuous stimulating discussions. They are also grateful to A. Levin, J. C. Naranjo, and T. Shaska for valuable remarks, as well as to the anonymous referee whose useful comments and suggestions helped in improving the manuscript. The figures of the paper have been generated with a compact, but powerful and easy-to-use IPE extensible drawing editor.

Funding

The authors thank the Department of Physics of the Oldenburg University and the School of Mathematics of the University of Edinburgh for funding their research visits to these institutions, which allowed the completion of the present article.

Y.F acknowledges support of the Spanish MINECO-FEDER Grants MTM2015-65715-P, MTM2012-37070, MTM2016-80276-P, and the Catalan Grant 2014SGR504. The work of V.E was partially supported by the School of Mathematics, University of Edinburgh, under the certificate of sponsorship C5E7V94128U.

Notes

1 More precisely, according to [Bobenko et al. 89], complexified invariant tori of the Kovalevskaya top are given by two copies of such open subsets.

2 Although the curve is also rather special: two of the corresponding branch points Q₁, …, Q₄ are in the hyperelliptic involution on the elliptic curve E.

3 A coordinate-free definition of the dual Prym variety can be found, e.g., in [Haine 83] p. 466; [Milne 08].

4 One should stress that C and K are not birationally equivalent.

5 Here we identify conformally equivalent two-dimensional Abelian varieties obtained one from another by duplication of all of the four periods.

6 This term is due to the fact that S^(α)_(ij)(kl) are algebraic functions of the resolvents of the quartic equation ψ(x) = 0.

7 Note the difference in the last terms in (4–45(4—45) ) and (4–48(4—48) ).

8 That is, the corresponding Jacobian is isogeneous (but not isomorphic) to the direct product of the elliptic curves.

9 Note that the images of the cycles on E in Σ are just the horizontal an vertical straight line segments passing through the half-period . Then the cuts between q₁, q₂ and q₃, q₄ made within the parallelogram Σ cross the above segments, and, therefore, the corresponding images of would cross , which is not allowed because of canonicity of the cycles. In order to avoid this, we replaced q₁, q₂, q₄ inside Σ by their translations by full periods of E.