On a refinement of Mordell's conjecture for curves

Question

Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \emptyset$ and thus we may embed $C$ into its Jacobian variety $J$, also definable over $K$. We abuse notation and continue to call this embedded image $C$. Then Mordell's conjecture, now a theorem of Faltings, asserts that $C(K) \cap J(K)$ is finite.

One point of view, presented in J.S. Milne's notes on abelian varieties (Heuristic arguments, page 131), is that "since there is no reason to expect any relation between $C(\mathbb{C})$ and $J(K)$ as subsets of $J(\mathbb{C})$", finiteness ought to follow for dimension reasons (since when $g \geq 2$ we have $1 = \dim_{\mathbb{C}} C < \dim_{\mathbb{C}} J = g$). He also mentions that to date (the notes seem to have been updated last in 2008) there is no proof of Faltings' theorem that follows this strategy.

My question is to ask whether a refinement of Mordell's conjecture, following the heuristic above, is expected to hold. Indeed one can view $J(K)$ as a finitely generated subgroup, or $\mathbb{Z}$-module, inside $J(\mathbb{C})$ that in some sense is independent of $C(\mathbb{C})$. Thus let $\Gamma$ be a finitely generated subgroup of $J(\mathbb{C})$. Is it then expected that $\Gamma$ will intersect $C(\mathbb{C})$ inside $J(\mathbb{C})$ finitely many times? In other words, is Mordell's conjecture really about the geometry of finitely generated subgroups inside $J(\mathbb{C})$? Or is there some subtle arithmetic significance to the group of rational points over a number field?

Answer 1

In fact, you don't need $C$ to be defined over a number field: if $C$ is curve of genus $\geq 2$ defined over $\mathbb{C}$, and $\Gamma$ is a finitely generated subgroup of $J(\mathbb{C})$, then $C \cap \Gamma$ is finite. In fact, you can even take a subgroup $\Gamma$ of finite rank, i.e. $\Gamma \otimes \mathbb{Q}$ is finite dimensional over $\mathbb{Q}$. So for example, you can throw in $\Gamma$ all the torsion points of $J$. There is also a generalization for an algebraic variety $X$ sitting inside a semi-abelian variety $G$. The conclusion is that $X \cap \Gamma$ is finite unless $X$ contains a translate of an algebraic subgroup of $G$ of dimension $\geq 1$.

The history of this theorem is long, see for example Hindry's survey and Silverman's review of Zannier's book Some problems of unlikely intersections in arithmetic and geometry. I'm not in the domain, but there is currently much research about unlikely intersections.