\(K_ 2\) and L-functions of elliptic curves. Computer calculations. (English) Zbl 0629.14002

Bloch, S.; Grayson, D.

Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-ISM-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 79-88 (1986).

[For the entire collection see Zbl 0588.00014.]
The paper concerns some computer calculations done by Grayson in the fall of 1981 to compare the value of the regulator on \(K_ 2\) of an elliptic curve with the value of the L-function at \(s=2\). All curves on the Swinnerton-Dyer table [see “Modular functions of one Variable. IV” Proc. Internat. Summer School 1972, Univ. Antwerp, RUCA, Lect. Notes Math. 476 (1975; Zbl 0315.14014)] with Weil conductor \(\leq 180\), negative discriminant, and a rational torsion point of order \(\geq 5\) were considered. These computations are explained by a modified form of a conjecture advanced by Bloch and Beilinson. They provide evidence for a lot of exotic relations between special values of Eisenstein-Kronecker- Lerch series and values of Hasse-Weil L-functions for elliptic curves without complex multiplication.

Reviewer: Ş.A.Basarab

MSC:

14-04	Software, source code, etc. for problems pertaining to algebraic geometry
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H45	Special algebraic curves and curves of low genus
14H52	Elliptic curves

Keywords:

regulator; elliptic curve; L-function; Eisenstein-Kronecker-Lerch series