A refined conjecture of Mazur-Tate type for Heegner points (Q1209128)

This paper deals with the problem of relating the rank of a modular elliptic curve to data involving Heegner points. Such points have been used very successfully by \textit{B. H. Gross} and \textit{D. B. Zagier} [Invent. Math. 84, 225-320 (1986; Zbl 0608.14019)] who used them to prove parts of the Birch and Swinnerton-Dyer conjectures for modular elliptic curves of rank at most 1, and also by \textit{V. A. Kolyvagin} [Math. USSR, Izv. 32, 523-541 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, 522-540 (1988; Zbl 0662.14017)] who was able to prove finiteness of the Tate-Shafarevich group in a similar setting. One feature of the present paper is that contrary to the quoted literature, here certain sums of Heegner points are considered which allow the author to formulate conjectures and prove some evidence for them also for curves of higher rank. Although this leads to rather technical statements, we will try to give an impression. Let \(E\) be a modular elliptic curve over \(\mathbb{Q}\) of conductor \(N\), with a modular parametrization \(\varphi\) defined over \(\mathbb{Q}\). Fix a quadratic imaginary field \(K\) of discriminant \(D\) in which all the primes dividing \(N\) split and choose an ideal of norm \(N\) in \(K\) involving for each prime in \(N\) only one prime of \(K\) above it. If \({\mathcal O}_ T\) denotes the order of \(K\) of conductor \(T\), then for \(T\) and \(ND\) relatively prime, \[ \mathbb{C}/{\mathcal O}_ T \to \mathbb{C}/({\mathcal O}_ T\cap {\mathcal N})^{-1} \] defines a cyclic \(N\)-isogeny between elliptic curves, hence a point on the modular curve \(X_ 0(N)\), hence via \(\varphi\) a point \(\alpha(T)\) on \(E\). This is by definition a Heegner point. Given a square free integer \(S\) prime to \(ND\), the author defines certain weighted sums over divisors \(T\) of \(S\) of the points \(\alpha(T)\). These sums define points on \(E\) over the ring class field \(K_ S\) of conductor \(S\) of \(K\). They are used to define an element \(\theta'(E,S)\) in \(E(K_ S)\otimes E(K_ S) \otimes\mathbb{Z}[\Gamma_ S]\), where \(\Gamma_ S\) is the Galois group of \(K_ S\) over \(K\). Denote by \(I\) the augmentation ideal of \(\mathbb{Z}[\Gamma_ S]\), which is the ideal generated by all \(\sigma-1\) for \(\sigma\in\Gamma_ S\). Then the basic conjecture in this paper is the statement that the element \(\theta'(E,S)\) belongs to \(E(K_ S)\otimes E(K_ S)\otimes I^{r-1}\), where \(r\) is the rank of \(E(K)\). Under the condition that all primes in \(S\) are inert in \(K\), after inverting a finite number of primes the author is able to prove this statement. In fact, he even shows that in such a case \(\theta'\) is in a submodule involving an even higher power of \(I\). This power is easily described in terms of the ranks of \(E\) over \(\mathbb{Q}\) and over \(K\).