Mazur's conjecture on higher Heegner points.
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 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 \textit{V. Vatsal's} idea [Invent. Math. 148, 1--46 (2002; Zbl 1119.11035)] of using a theorem of \textit{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 \textit{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 \textit{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.
scientific article; zbMATH DE number 1777253