Uniform distribution of Heegner points.

*(English)*Zbl 1119.11035Let \(E\) 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 D. E. Rohrlich [Invent. Math. 75, 383–408 (1984; Zbl 0565.14008)], R. Greenberg [Invent. Math. 79, 79–94 (1985; Zbl 0558.12005)], and 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 originally introduced and studied by B. H. Gross [Heights and the special values of \(L\)-series, Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 115–187 (1987; Zbl 0623.10019)], thereby showing that those are “uniformly distributed” on the various components of a Gross curve \(X\) in a well-defined sense. This, combined with a special value formula for certain \(L\)-series due to B. H. Gross, allows then to conclude that the special values of anticyclotomic \(L\)-functions are almost always different from zero. From this very fact, the stated conjecture on the finite generatedness of the Mordell-Weil groups \(E(K_\infty)\) is verified by applying the powerful machinery of Euler systems as developed by M. Bertolini and H. Darmon [Ann. Math. (2) 146, No. 1, 111–147 (1997; Zbl 1029.11027)]. Another basic ingredient of the author’s proof is a deep theorem by M. Ratner on unipotent flows of Lie groups [Duke Math. J. 77, No. 2, 275–382 (1995; Zbl 0914.22016)].

All together, the author’s method of proof is highly ingenious, versatile, subtle, involved, and pioneering in a like manner.

It should be mentioned that an independent proof of Mazur’s conjecture was almost simultaneously given by C. Cornut [Invent. Math. 148, No. 3, 495–523 (2002; Zbl 1111.11029)] from a slightly different viewpoint, but by using a quite similar analysis.

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 D. E. Rohrlich [Invent. Math. 75, 383–408 (1984; Zbl 0565.14008)], R. Greenberg [Invent. Math. 79, 79–94 (1985; Zbl 0558.12005)], and 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 originally introduced and studied by B. H. Gross [Heights and the special values of \(L\)-series, Number theory, Proc. Conf., Montreal/Can. 1985, CMS Conf. Proc. 7, 115–187 (1987; Zbl 0623.10019)], thereby showing that those are “uniformly distributed” on the various components of a Gross curve \(X\) in a well-defined sense. This, combined with a special value formula for certain \(L\)-series due to B. H. Gross, allows then to conclude that the special values of anticyclotomic \(L\)-functions are almost always different from zero. From this very fact, the stated conjecture on the finite generatedness of the Mordell-Weil groups \(E(K_\infty)\) is verified by applying the powerful machinery of Euler systems as developed by M. Bertolini and H. Darmon [Ann. Math. (2) 146, No. 1, 111–147 (1997; Zbl 1029.11027)]. Another basic ingredient of the author’s proof is a deep theorem by M. Ratner on unipotent flows of Lie groups [Duke Math. J. 77, No. 2, 275–382 (1995; Zbl 0914.22016)].

All together, the author’s method of proof is highly ingenious, versatile, subtle, involved, and pioneering in a like manner.

It should be mentioned that an independent proof of Mazur’s conjecture was almost simultaneously given by C. Cornut [Invent. Math. 148, No. 3, 495–523 (2002; Zbl 1111.11029)] from a slightly different viewpoint, but by using a quite similar analysis.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11R23 | Iwasawa theory |

11G05 | Elliptic curves over global fields |

11S40 | Zeta functions and \(L\)-functions |

11R52 | Quaternion and other division algebras: arithmetic, zeta functions |

22E35 | Analysis on \(p\)-adic Lie groups |