Elliptic curves and special values of \(L\)-series
Let \(E/ \mathbb{Q}\) be a modular elliptic curve, and let \(L(E/ \mathbb{Q},s)\) be its Hasse-Weil \(L\)-series. Following the methods of joint work of the author with Henri Darmon, and working under certain simplifying assumptions, it is proved that \(L(E/ \mathbb{Q}, 1)\neq 0\) implies that \(E(\mathbb{Q})\) is finite. The method of proof uses a result of B. Gross on the special value of the \(L\)-series of \(E\) over an imaginary quadratic field, and a result of B. Edixhoven on the image of Heegner points in the groups of connected components of modular Jacobians.
scientific article; zbMATH DE number 937448