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; Journal]

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.

The aim of this thesis is to study the Diophantine equations \[\frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} = N \quad\text{and}\quad \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = N,\qquad N\in\mathbb{Z}.\] These equations are investigated by relating them to elliptic curves defined over \(\mathbb{Q}\) and studied via their local properties over \(\mathbb{Q}_p\). For each equation, an explicit correspondence is established between the set of integer solutions and the set of rational points on an elliptic curve whose coefficients depend on \(N\). This reduction allows the application of the classical theory of elliptic curves to study the rational points and, consequently, the integer solutions of the original equations. The same method is also employed to investigate special families of solutions, such as those with \( x, y, z > 0 \).