Modularity and effective Mordell I

Abstract

We give an effective proof of Faltings' theorem for curves mapping to Hilbert modular stacks over odd-degree totally real fields.We do this by giving an effective proof of the Shafarevich conjecture for abelian varieties of GL2-type over an odd-degree totally real field.We deduce for example an effective height bound for K-points on the curves Ca : x 6 + 4y 3 = a 2 (a ∈ K × ) when K is odd-degree totally real.(Over Q all hyperbolic hyperelliptic curves admit an étale cover dominating C1.) Introduction.Faltings' theorem is one of the classic ineffective results in mathematics. His (first) method of proof is roughly to realize, following Parshin, a given curve inside a moduli space, and then to prove a finiteness result for integral points on that moduli space (compactness provides integrality). He is forced to work with the entire moduli space of principally polarized abelian varieties because of a construction of Kodaira that he invokes.This forces him to consider quite general Galois representations. We will deal only with Galois representations valued in GL 2 by working only with curves mapping to Hilbert modular varieties 1 (or, slightly more generally, curves over which there is a non-isotrivial family of GL 2 -type abelian varieties). We will further put ourselves in a situation where it is known that there are motives attached to the conjecturally corresponding automorphic forms by working only over odd-degree totally real fields.In that situation we will prove that all relevant abelian varieties become modular over a computable 2 finite list of odd-degree totally real extensions, and then deduce a height bound on all such abelian varieties using the usual construction 1 We will use this terminology for the fine moduli spaces, which are, without level structures imposed, not varieties but stacks.2 We remind the reader that to say that a quantity is computable, or, equivalently, effectively computable, is to say that there is a Turing machine that terminates on all inputs and which, on input the relevant parameters -in this case the base number field and bounds on the dimensions and conductors of the relevant abelian varieties -outputs said quantity.

Un peu d'effectivité pour les variétés modulaires de Hilbert-Blumenthal

“…4 Before writing [1] it was clear to us that this lemma must be known at least with ineffective implied constant (indeed it is a special case of a theorem of Zarhin which relies on Faltings' finiteness of isogeny classes). However in the context of [2] the lemma can also be phrased as a statement about congruences of Hilbert modular forms, whence it was natural to look in the literature on Hilbert modular forms for such a statement. And exactly such a statement was proven by Mladen Dimitrov [6] via a technique that we effectivized with a trick.…”

“…(For example, all solvable covers of P 1 over a number field are şirin, via e.g. the family given in Section 6 of [2] and the main theorem of Poonen's [13]. More generally, all curves C/K admitting a diagram C ϕ ← − C f − → P 1 over a finite L/K with ϕ étale and f Belyi with all ramification indices above 0, 1, ∞ respectively divisible by a, b, c with 1 a + 1 b + 1 c < 1 are also şirin, by pulling back a suitable hypergeometric family of abelian varieties.…”

“…The crux of Faltings' Great [8] is his proof of the finiteness of isogeny classes of abelian varieties over number fields, from which he deduces a finiteness conjecture of Shafarevich. His proof does not 1 give an estimate on the size #|A g (o K,S )| of the relevant finite set, and nor do the subsequent effectivizations and improvements of Raynaud, Masser-Wüstholz, Bost, David, Gaudron-Rémond, and others 2 . This is because all known 3 proofs of his isogeny theorem give bounds depending on the height of a representative in the relevant isogeny class, a quantity over which one still has no a priori effective control.…”

“…There are now of course numerous other ways to proceed. 2 Let us also note work of the Chudnovsky brothers [5] and Koshikawa [11]. Our list certainly omits important work, and we ask the reader to forgive our ignorance of the literature.…”

“…3 We exclude the case of elliptic curves because it is trivialized by Baker's explicit height bounds (since the relevant moduli space is so simple). key point in the argument will be provided by a lemma which greatly simplified the argument in [2]. 4 Such an isogeny estimate easily gives an a priori estimate on the number of such abelian varieties of bounded dimension, conductor, and field of definition (and their endomorphism rings, and e.g.…”