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Elliptic curves and values of L-functions. (English) Zbl 0632.14020

Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 371-387 (1987).
[For the entire collection see Zbl 0611.00005.]
Let E be an elliptic curve defined over \({\mathbb{Q}}\) with complex multiplication by the ring of integers of a field with class number 1, and let L(s,E) denote its L-function over \({\mathbb{Q}}\). The author gives an essentially explicit formula for the derivative of L(s,E) at 0 in terms of the value of the Bloch-Beilinson regulator map at \(f\otimes g\), where f and g are \({\mathbb{Q}}\)-rational functions on E whose divisors are supported by torsion points on E. The proof relies on the theory of complex multiplication and on Kronecker’s second limit formula.
Reviewer: D.Bertrand

MSC:

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14K22 Complex multiplication and abelian varieties
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Citations:

Zbl 0611.00005