Singular torsion points on elliptic curves (Q1429388)

Let \(K\) be a number field and \(\overline{K}\) an algebraic closure of \(K\). Given an elliptic curve \(E\) defined over \(K\) and an integer \(n\geq1\), let us denote by \(E[n]\) the subgroup of points of \(E(\overline{K})\) of order dividing \(n\) and by \(E_{\text{tors}}\) the subgroup of all torsion points of \(E(\overline{K})\). By definition, a point \(P\in E[n]\) is a singular \(n\)-torsion point if the following condition is satisfied: for any local parameter \(t\) at the origin 0 of \(E\) satisfying \([-1]^*t=-t\), any function \(f\in K(E)\) with divisor \(n(P_0)\) has a Laurent expansion at 0 of the form \[ f= \frac{a}{t^n}+ O\biggl(\frac{1}{t^{n-2}}\biggr), \quad a\in\overline{K},\;a\neq 0. \] A point \(P\in E_{\text{tors}}\) is said to be a singular torsion point if \(P\) is a singular \(n\)-torsion point when \(n\) is the order of \(P\). Let \(E_{\text{sing}}\) be the set of all singular torsion points of \(E\). It is a finite set. In this paper, the authors show that \(E_{\text{sing}}\) can be effectively determined. They prove that the orders of the singular torsion points of \(E\) can be bounded in an effective way. After a local study, they deduce that the orders of points in \(E_{\text{sing}}\) are bounded by a constant which only depends on the degree of the field \(K\). Let us mention the two following results proved by the authors: 1) Let \(d\geq 1\) be an integer. There exists an explicit integer \(N_d\), such that for any elliptic curve \(E\) defined over a number field of degree at most \(d\), the set \(E_{\text{sing}}\) is contained in \(E[N_d]\). 2) If \(E\) is a semi-stable elliptic curve defined over \(\mathbb Q\), the set \(E_{\text{sing}}\) is contained in \(E[24]\).