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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

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    $\begingroup$ I think this question would be improved if you tried to narrow down your enquiry to something more specific, rather than a general request for people to explain a topic $\endgroup$
    – Yemon Choi
    Commented Jan 21, 2015 at 21:28

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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.

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