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Mazur’s conjecture on higher Heegner points. (English) Zbl 1111.11029
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. 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 the finite extension $$K[p^\infty]:= \bigcup_{n\geq 0} K[p^n]$$ of the anticyclotomic $$\mathbb{Z}_p$$-extension $$H_\infty$$ of $$K$$, there should exist a number $$n\geq 0$$ such that $\text{Tr}_{K[p^\infty]/H_\infty} (y_{p^n})\not\in E(H_\infty)_{\text{tors}}$ for the Heegner point $$y_{p^n}\in E(K[p^n])$$.
This non-triviality statement for Heegner points emphasizes the importance of Heegner points with respect to the arithmetic of the elliptic curve $$E(K)$$, and that’s why it has often been used as a working hypothesis in the past.
The paper under review is devoted to a complete proof of this particular conjecture of B. Mazur’s. The author’s subtle and intricate proof is significantly based on V. Vatsal’s idea [Invent. Math. 148, 1–46 (2002; Zbl 1119.11035)] of using a theorem of M. Ratner [Duke Math. J. 77, No. 2, 275–382 (1995; Zbl 0914.22016)] to deduce an equidistribution statement for Gross points on the connected components of a definite Shimura curve. The author’s analogue of Vatsal’s result is a very powerful variant, which is also of independent interest. Furthermore, combining these methods with an earlier theorem of Y. Ihara [cf. On modular curves over finite fields. Discrete Subgroups of Lie Groups Appl. Moduli, Pap. Bombay Colloq. 1973, 161–202 (1975; Zbl 0434.14007)], two refined and generalized versions of Mazur’s conjecture are derived.
An independent proof of Mazur’s conjecture was almost simultaneously given by V. Vatsal [Duke Math. J. 116, No. 2, 219–261 (2003; Zbl 1065.11048)] from a slightly different viewpoint, but by using a quite similar analysis.

##### MSC:
 11G18 Arithmetic aspects of modular and Shimura varieties 11G50 Heights 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23 Iwasawa theory 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols
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