Linear relations in families of powers of elliptic curves (Q5963092)

In this article, the authors prove the following result on the Legendre elliptic curve \(E_{\lambda}\) of equation \(Y^2 = X(X-1)(X-\lambda)\) ``we prove that given \(n\) linearly independent points \(P_1(\lambda)\), \(P_2(\lambda)\), \(\dots\) , \(P_n(\lambda)\) on \(E_\lambda\) with coordinates in \(\overline{\mathbb{Q}(\lambda)}\), there are at most finitely many complex numbers \(\lambda_0\) such that the points \(P_1(\lambda_0), \dots, P_n(\lambda_0)\) satisfy two independent relations on \(E_{\lambda_0}\).'' The authors used mainly results of algebraic geometry and complex analysis.NEWLINENEWLINEThe authors first make a recap of the state of the art and motivate their research. They first establish a consequence of their result on the Zilber-Pink conjecture. Then they prove in 6 steps their main theorem stated in the first part. The authors recall some results on o-minimal structures and then they state a result on the number of connected components. Next, using holomorphic functions, they formulate a result on the bound of the solutions of their theorem. Then they explicit the holomorphic functions linked to the elliptic curve to which they apply the results in the previous section. Next, they state boundary results to finally conclude in the last part.NEWLINENEWLINERemark: there is a typo in the paper, in the 15 page over 24, there is written: ``In fact in the first integral on the right'' which should be replaced by ``In fact in the first integral on the left''.