Luca Ferrigno

Let \(E_{\lambda}\) be the Legendre family of elliptic curves with equation \(Y^2=X(X-1)(X-\lambda)\). Given a curve \(\mathcal{C}\), satisfying a condition on the degrees of some of its coordinates and parametrizing \(m\) points \(P_1, \ldots, P_m \in E_{\lambda}\) and \(n\) points \(Q_1, \ldots, Q_n \in E_{\mu}\) and assuming that those points are generically linearly independent over the generic endomorphism ring, we prove that there are at most finitely many points \(\mathbf{c}_0\) on \(\mathcal{C}\), such that there exists an isogeny \(\phi: E_{\mu(\mathbf{c}_0)} \rightarrow E_{\lambda(\mathbf{c}_0)}\) and the \(m+n\) points \(P_1(\mathbf{c}_0), \ldots, P_m(\mathbf{c}_0), \phi(Q_1(\mathbf{c}_0)), \ldots, \phi(Q_n(\mathbf{c}_0)) \in E_{\lambda(\mathbf{c}_0)}\) are linearly dependent over \(\mathrm{End}(E_{\lambda(\mathbf{c}_0)})\).

[Arxiv]

In preparation

In preparation

This thesis investigates several instances of the Zilber–Pink conjecture, originally formulated by Zilber and independently by Bombieri, Masser and Zannier for tori, and later generalized by Pink in the framework of mixed Shimura varieties. The conjecture, which unifies and extends fundamental results in arithmetic geometry, has been intensively studied over the last two decades.
Our focus lies on the case of curves in families of abelian varieties. First, for curves in products of fibered powers of elliptic schemes, we prove that such a curve contains only finitely many points lying in an algebraic subgroup of a fiber for which there exist non-trivial homomorphisms between the two powers, extending previous results of Masser-Zannier and Barroero-Capuano.
Second, for curves in non-isotrivial abelian schemes \(A \to S\), we prove that if \(C \subset A\) is not contained in a fiber nor in a translate of a flat subgroup scheme, then \(C\) meets the union of all proper algebraic subgroups of the CM fibers of \(A\) in at most finitely many points. This generalizes a previous result by Barroero for fibered powers of elliptic schemes to higher-dimensional abelian varieties.
Both results are obtained using the Pila–Zannier strategy, combining functional transcendence and o-minimality with new arithmetic estimates for canonical heights of images of points under endomorphisms in abelian varieties.

In 1922, Mordell conjectured that every curve of genus at least 2 has only finitely many rational points and this was later proved by Faltings in 1983. However, in 1941, Chabauty proved that Mordell's conjecture is true with the additional hypothesis that the Mordell-Weil rank of the Jacobian variety over \(\mathbb{Q}\) is strictly less that the genus of the curve. Although this doesn't prove Mordell's conjecture in full generality, Chabauty's ideas lead to the creation of new methods for finding rational points on curves. After proving Chabauty's result and its improvement by Coleman (1985), we will explicitely solve some diophantine equations.

After introducing the general theory of elliptic curves over \( \mathbb{Q}\) and \(\mathbb{Q}_p\), I investigated the existence and the number of integer solutions of the equations \(\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=N\) and \(\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = N\) with \(N \in \mathbb{Z}\). This was done by exhibiting explicit bijections between the integer solutions of the equations and the set of rational points of two elliptic curves (whose coefficients depend on \(N\)), and then using the well-known theory to find the rational points of those curves. Furthermore, the existence of special solutions (e.g. solutions with \(x, y, z > 0\)) is studied with the same method.