1 Addendum to lecture 3: strong uniformity

1.1 Theorem (Caporaso-Harris-Mazur)

Assume that Lang’s “strong” conjecture holds true. Then there is a number N(g), such that, for any number field extension K, the number of curves of genus g > 1 defined over K which have more than N(g) rational points over L is finite.

Start with family over Hilbert scheme. Apply:

1.2 Theorem (CHM)

Let g > 1 be an integer, and fix a subvariety X_{0} ⊂ M_{g}. Let X_{n} ⊂ M_{g,n} be the inverse image. Then for large enough n, the variety X_{n} is of general type.

and proof of:

1.3 Lemma (CHM)

Suppose X → B is a family of curves and P ⊂ X is n-correlated in X → B. Then there is an open Zariski - dense set U ⊂ B such that the number of points of P_{b} on all fibers of X → B over b ∈ U is uniformly bounded.

this time using the Lang exceptional set Z. Get

1. a bound N_{0}(g) which is valid outside a finite set of geometric isomorphism classes.

2. a finite number of K-rational points P_{i} on M_{g,n_{i}} giving rise to the exceptions. May assume n_{i} > 1.

Let N_{1}(g) = maximum number of fixed points of automorphisms on a curve of genus g. Claim: N(g) = N_{0}(g) + N_{1}(g) works.

P = P_{i} lies over [C] ∈ M_{g}, the fiber is an open set in C^{n}/AutC. Assume DK̄ = CK̄ and there is a rational point Q ∈ D^{n} whose image is P. Also assume Q not fixed by AutD. Denote π : D^{n} → C^{n}/AutC. Q = (x_{1}, … x_{n}). At least one x_{i} not fixed by automorphisms, say x_{1}. Therefore (x_{1}, … x_{n−1}) not fixed. By the exercise, the inverse image of π(x_{1}, … x_{n−1}) is isomorphic to D. Thus D is determined by P.