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${K}_{2}$ and L-functions of elliptic curves. Computer calculations. (English) Zbl 0629.14002
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 $\le 180$, negative discriminant, and a rational torsion point of order $\ge 5$ 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 Machine computation, programs (algebraic geometry) 14C35 Applications of methods of algebraic $K$-theory 14G10 Zeta-functions and related questions 14H45 Special curves and curves of low genus 14H52 Elliptic curves