Rohrlich, David E. 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 Cited in 4 Documents 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) Keywords:Eisenstein series; complex multiplication; L-function; Bloch-Beilinson regulator Citations:Zbl 0611.00005 × Cite Format Result Cite Review PDF