On Zagier’s conjecture for L(E, 2): A number field example

Jeffrey Stopple
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 4, Pages 47–53
Full paper (PDF, 164 Kb)

Details

Authors and affiliations

Jeffrey Stopple
Mathematics Department, UC Santa Barbara
Santa Barbara CA 93106, United States

Abstract

We compute, for a CM elliptic curve E defined over a real quadratic field F, the details of an example of Zagier’s conjecture. This relates L(E, 2) to values of the elliptic dilogarithm function at a divisor in the Jacobian of E which arises from K-theory.

Keywords

• Diophantine equation
• Factorial
• Fibonacci
• Brocard-Ramanujan

AMS Classification

• Primary 11G40
• Secondary 11G05 11G55 19F27

References

1. Cremona, J., E. Whitley, Periods of cusp forms and elliptic curves over imaginary quadratic fields, Math. Comp., Vol. 62, 1994, 407–427.
2. Dokchitser, T., R. de Jeu, D. Zagier, Numerical verification of Beilinson’s conjecture for K₂ of hyperelliptic curves, Compos. Math., Vol. 142, 2006, 339–373.
3. Goncharov, A., A. Levin, Zagier’s conjecture on L(E, 2), Invent. Math., Vol. 132, 1998, 393–432.
4. Gross, B., Arithmetic on Elliptic Curves with Complex Multiplication, Springer Lecture Notes in Mathematics, Springer, Berlin, Vol. 776, 1980.
5. Ramakrishnan, D., Regulators, algebraic cycles, and values of L-functions, in Algebraic K theory and algebraic number theory, Contemporary Mathematics, Vol. 83, 1987.
6. Silverman, J., Computing heights on elliptic curves, Math. Comp., Vol. 51, 1988, 339–358.
7. Silverman, J., Advanced Topics in the Arithmetic of Elliptic Curves, Springer Graduate Texts in Mathematics, Vol. 151, 1994.
8. Stopple, J. Stark conjectures for CM elliptic curves over number fields, J. Number Theory, Vol. 103, 2003, 163–196.
9. Zagier, D., Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields, in Arithmetic Algebraic Geometry (Texel, 1989) Progr. Math., Birkhäuser Boston, Vol. 89,
   1991, 391–430.