Self-points on elliptic curves (Q2270636)

Let \(E/ \mathbb Q\) be an elliptic curve of conductor \(N\). There exists a surjective morphism \[ \phi_E : X_0(N) \rightarrow E \] defined over \(\mathbb Q\) called a modular parametrization. Let \(x_C\) be a point on the moduli space \(Y_0(N)\) representing \((E,C)\), where \(C\) is a cyclic subgroup of E of order \(N\). The point \(P_C=\phi_E(x_C)\) is called a \textit{self-point} of \(E\) and is defined over the field of definition \(\mathbb Q(C)\) of \(C\). There are \(N+1\) self-points corresponding to the \(N+1\) subgroups \(C\). The author conjectures that if \(E\) does not have complex multiplication, all self points are of infinite order, and the only relations between that are the ones induced by the degeneracy maps. In a previous paper, \textit{C. Delaunay} and the author [Int. J. Number Theory 5, No. 5, 911--932 (2009; Zbl 1238.11064)] completely prove this conjecture when \(N\) is prime, proving that every self-point is of infinite order and the self points generate a rank \(N\) group. In this paper the author considers the general case and proves the following two main results: if the \(j\)-invariant of \(E\) is not in \(\frac 1 2 \mathbb Z\), then at least one self-point is of infinite order and if \(E\) is a semi-stable elliptic curve of conductor \(N\neq 30\) or \(210\), then all the self points are of infinite order and they generate a group of rank \(N\). \textit{Higher self-points} are constructed on \(E\) from couples \((E',C')\), where \(E'\) is isogenous to \(E\). Using higher self points, under some assumptions on \(E\) and \(p\), the author manages to generate towers of points that generate a group of rank \(p^{n+1}+p^n\) over \(\mathbb Q(E[p^n])\). These are the only known towers of points of infinite order over \(\mathbb Q(E[p^\infty])\). Finally, lower bounds for Selmer groups of \(E\) in extensions are obtained by constructing cohomology classes from higher self points. This resembles Kolyvagin's construction of cohomology classes using Heegner points, but in that case upper bounds for the Selmer group were obtained. Examples of the obtained results are demonstrated on a curve of conductor 24 and on an isogeny class of curves with conductor 27.