Uniform distribution of Heegner points.
type: Item
instance of: scholarly article
title: Uniform distribution of Heegner points.
zbMATH Open document ID: 1119.11035
author: Vinayak Vatsal
published in: Inventiones Mathematicae
publication date: 2002-08-08T00:00:00Z
review text: Let \(E\) be a modular elliptic curve over \(\mathbb Q\) of conductor \(N\), and let \(K\) be an imaginary quadratic field of discriminant \(D\) prime to \(N\). For any prime number \(p\) denote by \(K_\infty\) the unique \(\mathbb Z_p\)-extension of \(K\) such that \(\text{Gal}(K/\mathbb Q)\) acts non-trivially on \(\text{Gal}(K_\infty/K)\). As for the Mordell-Weil group \(E(K_\infty)\) of \(E\) over this so-called anticyclotomic \(\mathbb Z_p\)-extension \(K_\infty/K\) there is a fundamental conjecture by B. Mazur predicting that the size of \(E(K_\infty)\) is completely controlled by the prime factorization of \(N\) in \(K\). This famous conjecture, stated in 1984, also relates the size of \(E(K_\infty)\) to the sign in the functional equation of certain \(L\)-series. Mazur's conjecture was verified for elliptic curves with complex multiplication by \(K\) even so for certain more general Abelian varieties with complex multiplication, due to the results of \textit{D. E. Rohrlich} [Invent. Math. 75, 383--408 (1984; Zbl 0565.14008)], \textit{R. Greenberg} [Invent. Math. 79, 79--94 (1985; Zbl 0558.12005)], and \textit{K. Rubin} (1991) obtained in the sequel. The main goal of the paper under review is to investigate the so-called ``generic case'', that is, the situation when \(E\) has no complex multiplication at all, or when the field of complex multiplications differs from the ground field \(K\). In fact, under certain additional conditions on \(E\) and \(K\), Mazur's conjecture implies that the Mordell-Weil group \(E(K_\infty)\) is finitely generated, and the author's main result asserts that the latter statement is indeed true, at least when \(p\) is an ordinary prime for \(E\), or when the class number of \(K\) is prime to \(p\). This remarkable progress is achieved by a refined study of Heegner points on definite quaternion algebras. More precisely, the author describes the distribution of special points (Gross points) on special curves \(X\) which were original…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 1777237
zbMATH Keywords: arithmetic elliptic curves
zbMATH Keywords: modular curves
zbMATH Keywords: Heegner points
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Mazur's conjecture
zbMATH Keywords: \(p\)-adic Lie groups
MaRDI profile type: Publication
full work available at URL: s002220100183
OpenAlex ID: W2083946017
DOI: 10.1007/S002220100183
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description: scientific article; zbMATH DE number 1777237