143
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

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

&

References

  • [Adler and van Moerbeke 84] M. Adler and P. van Moerbeke. “Geodesic Flow on so(4) and the Intersections of Quadrics.” Proc.Natl. Acad. Sci. USA 81 (1984), 4613–4616.
  • [Adler and van Moerbeke 87] M. Adler and P. van Moerbeke. “The Intersection of Four Quadrics in P6. Abelian Surfaces and Their Moduli.” Math. Ann. 279: 1 (1987), 25–85.
  • [Adler and van Moerbeke 88] M. Adler and P. van Moerbeke. “The Kowalewski and Hénon-Heiles Motions as Manakov Geodesic Flows on SO(4) as Two-dimensional Family of Lax Pairs.” Commun. Math. Phys. 113:4 (1988), 659–700.
  • [Abarello et al. 84] E. Abarello, M. Cornalba, P. Griffiths, and J. Harris. Geometry of Algebraic Curves, Vol. I. New York: Springer-Verlag, 1985.
  • [Baker et al. 95] S. Baker, V. Z. Enolski, and A. P. Fordy. “Integrable Quartic Potentials and Coupled KdV Equations.” Phys. Lett. A 201:2–3 (1995), 167–174.
  • [Barth 85] W. Barth. Abelian Surfaces with (1,2)-Polarization. Algebraic Geometry, Sendai, 1985, 41–84, Adv. Stud. Pure Math. 10. Amsterdam: North-Holland, 1987.
  • [Beauville 77] A. Beauville. “Prym Varieties and the Schottky Problem.” Invent. Math. 41:2 (1977), 149–196.
  • [Belokolos and Enolski 01] E. D. Belokolos and V. Z. Enolski. “Reduction of Abelian Functions and Algebraically Integrable Systems, Part I.” J. Math. Sci. 106:6 (2001), 3395–3486. VINITI Publ. and Plenum Press.
  • [Belokolos et al. 94] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’sii, A. R. Its, and V. B. Matveev. Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer Series in Nonlinear Dynamics. Berlin: Springer–Verlag, 1994.
  • [Bobenko et al. 89] A. I. Bobenko, A. G. Reyman, and M. Semenov–Tian-Shansky. “The Kowalewski Top 99 Year Later: A Lax Pair, Generalizations and Explicit Solutions.” Commun. Math. Phys. 122 (1989), 321–354.
  • [Bolza 86] O. Bolza. “Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades.” Math. Ann. XXVIII (1886), 447–456.
  • [Bost and Mestre 88] J. B. Bost and J. F. Mestre. “Moyenne Arithmético-géometrique et Périodes des Courbes de genre 1 et 2.” Gaz.Math.S.M.F. 38 (1988), 36–64.
  • [Dalaljan 75] S. G. Dalaljan. “The Prym Variety of a Two-sheeted Covering of a Hyperelliptic Curve with Two Branch Points (Russian).” Mat. Sb. (N.S.) 98:2 (10) (140) (1975), 255–267, 334.
  • [Farkas and Kra 80] H. M. Farkas and I. Kra. Riemann Surfaces, Lectures Notes in Mathematics (Berlin), Vol. 71. Springer, 1980.
  • [Fay 73] J. D. Fay. Theta Functions on Riemann Surfaces, Lectures Notes in Mathematics (Berlin), Vol. 352. Springer, 1973.
  • [Fedorov 95] Yu. Fedorov. “Integrable Systems, Lax Representations, and Confocal Quadrics.” In Dynamical Systems in Classical Mechanics, 173–199, Amer. Math. Soc. Transl. Ser. 2, Vol. 168. Providence, RI: Am. Math. Soc., 1995.
  • [Fedorov 16] Yu. Fedorov, L. Garcá-Naranjo, and J.-C. Naranjo. “A Shortcut to the Kovalevskaya Curve.” arXiv:1606.08331v1. 2016.
  • [Frahm 74] F. Frahm. “Über gewisse Differentialgleichungen.” Math. Ann. 8 (1874), 35–44.
  • [Haine 83] L. Haine. “Geodesic Flow on so(4) and Abelian Surfaces.” Math. Ann. 263 (1983), 435–472.
  • [Horozov and van Moerbeke 89] E. Horozov and P. van Moerbeke. “The Full Geometry of Kowalewski’s Top and (1,2)-Abelian Surfaces.” Commun. Pure Appl. Math. 42:4 (1989), 357–407.
  • [Hurwitz and Courant 44] A. Hurwitz and R. Courant. Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. (German) New York: Interscience Publishers, Inc., 1944. pp. 534.
  • [Igusa 62] J. Igusa. “On Siegel Modular Forms of Genus Two.” Am. J. Math. 84 (1962), 175–200. ibid: II 88 (1966) 221–236.
  • [Knörrer 80] H. Knörrer. “Geodesics on the Ellipsoid.” Invent. Math. 59 (1980), 119–143.
  • [Kötter 92] F. Kötter. “Uber die Bewegung eines festen Körpers in einer Flüssigkeit. I, II.” J. Reine Angew. Math. 109 (1892), 51–81, 89–111.
  • [Kowalewski 89] S. Kowalewski. “Sur le probleme de la rotation d’un corps solide autour d’un point fixe.” Acta Math. 12:1 (1889), 177–232.
  • [Krazer 03] A. Krazer. Lehrbuch der Thetafunktionen. Lepzig: Teubner, 1903.
  • [Krishnamoorthy et al. 05] V. Krishnamoorthy, T. Shaska, and H. Völklein. “Invariants of Binary Forms.” In Progress in Galois Theory, pp. 101–122, Dev. Math., 12, New York: Springer, 2005.
  • [Leprövost and Markushevich 99] F. Leprövost and D. Markushevich. “A Tower of Genus Two Curves Related to the Kowalewski Top.” J. Reine Angew. Math. 514 (1999), 103–111.
  • [Levin 12] A. Levin. “Siegel’s Theorem and the Shafarevich Conjecture.” J. Théor. Nombres Bordeaux 24:3 (2012), 705–727.
  • [Levin] A. Levin. A private communication.
  • [Magri and Skrypnyk 16] F. Magri and T. Skrypnyk. “The Clebsch System.” arXiv:1512.04872v1, 2016.
  • [Milne 08] J. S. Milne. Abelian Varieties (v 2.00). Available online (www.jmilne.org/math/CourseNotes/AV.pdf), 2008.
  • [Moser 80] J. Moser. “Various Aspects of Integrable Hamiltonian Systems.” In “Proc. CIME Conference. Bressanone, Italy, 1978.” Prog. Math. 8 (1980), 233–290.
  • [Mumford 74] D. Mumford. “Prym Varieties I.” In Contributions to Analysis, edited by L. V. Ahlfors, I. Kra, B. Maskit, L. Nirenberg, pp. 325–350. New York: Academic Press, 1974.
  • [Pantazis 86] S. Pantazis. “Prym Varieties and the Geodesic Flow on SO(n).” Math. Ann. 273:2 (1986), 297–315.
  • [Marcucci and Pirola 12] V. Marcucci and G. P. Pirola. “Generic Torelli Theorem for Prym Varieties of Ramified Coverings.” Compos. Math. 148:4 (2012), 1147–1170.
  • [Rosenhain 51] G. Rosenhain. “Abhandlung über die Funktionen zweier Variabler mit vier Perioden.” Mém. prés. l’Acad. de Sci. de France des savants XI (1851), 361–455. The paper is dated 1846. German Translation: H. Weber (Ed.), Engelmann-Verlag, Leipzig 1895.
  • [Shaska and Völklein 04] T. Shaska and H. Völklein. “Elliptic Subfields and Automorphisms of Genus 2 Function Fields.” In Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), edited by C. Christensen, G. Sundaram, A. Sathaye and C. Bajaj, pp. 703–723. Berlin: Springer, 2004.
  • [Schottky 91] F. Schottky. “Über das analytische Problem der Rotation eines starren Körpers in Raume von vier Dimensionen.” Sitzungsber., König. Preuss. Akad. Wiss., Berlin 12 (1891), 227–232.
  • [Schottky 26] F. Schottky. “Über die analytische Aufgabe der Bewegung eines starren Körpers im vierdimensionalen Raume.” Sitzungsber., König. Preuss. Akad. Wiss., Berlin 19 (1926), 215–241.
  • [Tretkoff and Tretkoff 84] C. L. Tretkoff and M. D. Tretkoff. “Combinatorial Group Theory, Riemann Surfaces and Differential Equations. Contributions to Group Theory, 467–519, Contemp. Math., 33.” Providence, RI: Amer. Math. Soc., 1984.
  • [Weber 78] H. Weber. “Anwendung der Thetafunctionen zweir Veranderlicher auf die Theorie der Bewegung eines festen Körpers in einer Flüssigkeit.” Math. Ann. 14 (1878), 173–206.
  • [Weng 02] A. Weng. “Constructing Hyperelliptic Curves of Genus 2 Suitable for Cryptography.” Math. Comput. 72:241 (2002), 435–458.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.