Mazur's conjecture on higher Heegner points.
type: Item
instance of: scholarly article
title: Mazur's conjecture on higher Heegner points.
zbMATH Open document ID: 1111.11029
author: Christophe Cornut
published in: Inventiones Mathematicae
publication date: 2002-08-08T00:00:00Z
review text: Let \(E\) be an elliptic curve over \(\mathbb{Q}\) of conductor \(N\) equipped with a modular parametrization \(\pi: X_0(N)\to E\), and let \(K=\mathbb{Q}(\sqrt{d})\) be an imaginary quadratic field with discriminant \(d-0\). If all prime factors of \(N\) split in \(K\) then it is said that the so-called ``Heegner hypothesis'' is satisfied for the whole setup. For any ideal \({\mathcal N}\) in the ring \(O_K\) of integers of \(K\) such that \(O_K/{\mathcal N}\) is isomorphic to \(\mathbb{Z}/N\mathbb{Z}\), the two complex tori \(\mathbb{C}/O_K\) and \(\mathbb{C}/{\mathcal N}^{-1}\) define elliptic curves, which are related by a cyclic \(N\)-isogeny corresponding to a complex point \(x_1\in X_0(N)\). More generally,if \(c\) is an arbitrary integer prime to \(N\), then the ring \(O_c:= \mathbb{Z}+ cO_K\) and the ideal \({\mathcal N}_0:= O_c\cap{\mathcal N}\) analogously define a complex point \(x_c\in X_0(N)\) corresponding to the isogeny \(\mathbb{C}/O_c\to \mathbb{C}/{\mathcal N}^{-1}\). Under the validity of the Heegner hypothesis, the theory of complex multiplication yields that such a so-called ``Heegner point'' \(x_c\) is rational over \(K[c]\) the rung class field of conductor \(c\) of the groundfield \(K\). Heegner points and their images \(y_c:=\pi(x_c)\in E(K[c])\) play a significant role in the study of elliptic curves with fixed modular or Shimura curve parametrization, which are defined over ring class fields of imaginary qudratic fields, in particular with a view to their \(L\)-functions and the related famous conjecture of Birch and Swinnerton-Dyer. In his programmatic paper ``Modular Curves and Arithmetic'', Barry Mazur formulated several conjectures on the variation of Mordell-Weil groups in towers of ring class fields with restricted variation [cf. \textit{B. Mazur}, Proc. Int. Congr. Math., Warsaw 1983, Vol. 1, 185--211 (1984; Zbl 0597.14023)]. One of these conjectures stated the following: Let \(p\) be a prime number not dividing \(N\). Then, for th…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 1777253
zbMATH Keywords: arithmetic elliptic curves
zbMATH Keywords: Heegner points
zbMATH Keywords: modular curves
zbMATH Keywords: Shimura curves
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Mazur's conjecture
MaRDI profile type: Publication
full work available at URL: s002220100199
OpenAlex ID: W2024965120
Wikidata QID: Q122980753
DOI: 10.1007/S002220100199
description: scientific article; zbMATH DE number 1777253