On isogenous principally polarized abelian surfaces

We study a relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of p-torsion points with respect to the Weil pairing. For p = 3 we find an explicit relationship between the set of Weierstrass points of the two curves extending the classical results of F. Richelot (1837) and G. Humbert (1901) in the case p = 2.

[1]  G. C. Shephard,et al.  Finite Unitary Reflection Groups , 1954, Canadian Journal of Mathematics.

[2]  Serge Lang,et al.  Abelian varieties , 1983 .

[3]  A. Ortega,et al.  On the moduli space of rank 3 vector bundles on a genus 2 curve and the Coble cubic , 2005 .

[4]  Christophe Ritzenthaler,et al.  An Explicit Formula for the Arithmetic–Geometric Mean in Genus 3 , 2007, Exp. Math..

[5]  F. Richelot,et al.  De transformatione integralium Abelianorum primi ordinis commentatio. , 1837 .

[6]  Herbert Lange,et al.  Complex Abelian Varieties , 1992 .

[7]  N. Yui.  On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2 , 1978 .

[8]  E. V. Flynn,et al.  Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Index rerum et personarum , 1996 .

[9]  Aufstellung des vollen Formensystems einer quaternären Gruppe von 51840 linearen Substitutionen , 1889 .

[10]  George Salmon.  Lessons introductory to the modern higher algebra , 1885 .

[11]  David A. Cox.  The Arithmetic-Geometric Mean of Gauss , 2004 .

[12]  J. Borwein,et al.  Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity , 1998 .

[13]  Point sets and allied Cremona groups. III , 1915 .

[14]  R. Manni.  On the differential of applications defined on moduli spaces of p.p.a.v. with level theta structure , 1996 .

[15]  Point Sets and Allied Cremona Groups. , 1915, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Formulas for the arithmetic geometric mean of curves of genus 3 , 2004, math/0403182.

[17]  Michele Bolognesi,et al.  On Weddle surfaces and their moduli , 2006, math/0601251.

[18]  VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO , 2007, math/0702724.

[19]  Ron Livne,et al.  The arithmetic-geometric mean and isogenies for curves of higher genus , 1997 .

[20]  Gerard van der Geer.  Note on abelian schemes of level three , 1987 .

[21]  Eberhard Freitag,et al.  The Burkhardt Group and Modular Forms , 2004 .

[22]  G. Geer,et al.  On the geometry of a Siegel modular threefold , 1982 .

[23]  Jun-ichi Igusa,et al.  Equations Defining Abelian Varieties , 1972 .

[24]  An application of Moore’s cross-ratio group to the solution of the sextic equation , 1911 .

[25]  G. Kempf,et al.  LINEAR SYSTEMS ON ABELIAN VARIETIES , 1989 .

[26]  Eberhard Freitag,et al.  The Burkhardt Group And Modular Forms 2 , 2004 .

[27]  J. Igusa.  ON THE GRADED RING OF THETA-CONSTANTS.* , 1964 .

[28]  Bruce Hunt,et al.  The Geometry of some special Arithmetic Quotients , 1996 .