Computation of p-Adic Heights and Log Convergence

This paper is about computational and theoretical ques- tions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algo- rithm for computing these quantities, i.e., for computing the value of E2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E2 is log convergent.

[1]  K. Rubin,et al.  Organizing the arithmetic of elliptic curves , 2005 .

[2]  Bernadette Perrin-Riou.  Arithmétique des courbes elliptiques à réduction supersingulière en p , 2003, Exp. Math..

[3]  F. Gouvêa,et al.  Arithmetic of p-adic Modular Forms , 1988 .

[4]  K. Rubin,et al.  Pairings in the Arithmetic of Elliptic Curves , 2004 .

[5]  Barry Mazur,et al.  Refined conjectures of the “Birch and Swinnerton-Dyer type” , 1987 .

[6]  Verification of the Birch and Swinnerton - Dyer Conjecture for Specific Elliptic Curves , 2005 .

[7]  R. Greenberg.  Galois Theory for the Selmer Group of an Abelian Variety , 2003, Compositio Mathematica.

[8]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[9]  Alan G. B. Lauder.  Rigid cohomology and $p$-adic point counting , 2005 .

[10]  J. Nekovář.  On p-adic height pairings , 2002 .

[11]  P. Schneider.  p-adic height pairings. II , 1982 .

[12]  Sinnou David,et al.  Séminaire de Théorie des Nombres, Paris, 1990–91 , 1993 .

[13]  David Joyner,et al.  SAGE: system for algebra and geometry experimentation , 2005, SIGS.

[14]  Alain Robert,et al.  Introduction to modular forms , 1976 .

[15]  B. Mazur,et al.  The $p$-adic sigma function , 1991 .

[16]  p-adic height pairings on abelian varieties with semistable ordinary reduction , 2002, math/0209362.

[17]  Daqing Wan,et al.  Dwork’s conjecture on unit root zeta functions , 1999, math/9911270.

[18]  Barry Mazur,et al.  Onp-adic analogues of the conjectures of Birch and Swinnerton-Dyer , 1986 .

[19]  The p-adic height pairings of Coleman-Gross and of Nekovar , 2002, math/0209006.

[20]  N. M. Katz.  p-adic Interpolation of Real Analytic Eisenstein Series , 1976 .

[21]  K. Kedlaya.  Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology , 2001, math/0105031.

[22]  B. Mazur,et al.  Canonical Height Pairings via Biextensions , 1983 .

[23]  R. Coleman.  The universal vectorial Bi-extension andp-adic heights , 1991 .

[24]  C. Wuthrich.  On p‐Adic Heights in Families of Elliptic Curves , 2004 .

[25]  Séminaire de Théorie des Nombres, Paris 1987–88 , 1982 .