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Uniform distribution of Heegner points. (English) Zbl 1119.11035
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 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.

##### 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
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