Let $X$ be a smooth projective curve over $\mathbb{Q}$. I heard (if I did not misunderstood) that the geometry of the complex points $X(\mathbb{C})$ (flat, hyperbolic case) dicts the shape (group structure, number) of rational points $X(\mathbb{Q})$. Can some one explain this relationship ? Thanks



rational point of a curve
Not sure if this is really MathOverflow level, but:


(A) genus X is 0 <==> X(C) is a sphere <==> X(Q) is either empty or looks like P^1(Q)


(B) genus X is 1 <==> X(C) is a 1-holed torus <==> X(Q) is either empty or a finitely generated abelian group (Mordell's theorem)


(C) genus X is g > 1 <==> X(C) is a g-holed torus <==> X(Q) is finite (Faltings' theorem, originally the Mordell conjecture, with alternative proof via Diophantine approximation methods due to Vojta)


Statement (B) is true for abelian varieties of all dimensions and over all number fields, which is due to Weil. An analogue of (C) is a conjecture of Bombieri and Lang which states that if X is of general type, then X(Q) is not Zariski dense in X. A quantitative version of this is given by very deep conjectures of Vojta.


