Mazur’s conjecture on higher Heegner points.

*(English)*Zbl 1111.11029Let \(E\) 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.

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.

Reviewer: Werner Kleinert (Berlin)

##### 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 |