Finite descent obstruction for Hilbert modular varieties

Baldi, Gregorio; Grossi, Giada

doi:10.4153/s0008439520000569

Cited by 2 publications

(5 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This observation, namely that there are complete curves in these moduli spaces, and thus curves whose rational points are (unconditionally!) controlled by explicit automorphic forms, was the origin of this work -note, though, that the former point is also observed in recent independent work of Baldi-Grossi [5] about obstructions to rational points on these moduli spaces.…”

Section: Remarksmentioning

confidence: 52%

“…As in Remark 12 of Archinard's [4], we see that the holomorphic differentials on X λ are given by: 5 . Now let us discuss how to compute the periods of X λ .…”

mentioning

confidence: 67%

“…4 It follows that, at least for K/Q totally real of odd degree, we may compute H o (o K,S ) in finite time. 5 Definition 1.2.…”

Section: Main Theoremsmentioning

confidence: 99%

“…4 We are free to increase S to avoid subtleties with the latter. 5 We are implicitly invoking the polarization estimates of Gaudron-Rémond (Theorem 1.1 of their [13]) and the endomorphism estimates of Masser-Wüstholz (the main theorem of their [17], though an explicit such estimate can be extracted from Lemma 9.2 of Gaudron-Rémond's [13]) in order to deal with the o-linear polarizations that are hidden by the notation. 6 (Add level structure to produce curves rather than just stacky curves from this argument.…”

Section: Main Theoremsmentioning

confidence: 99%

“…We will project the period lattice to C 2 via π : C 3 ։ C 2 via (x, y, z) → (x, z) since it suffices to work with the period lattice of an isogenous abelian surface. We find the following generators for the relevant period lattice: π( µ), π( ν), diag(ζ 6 , ζ 5 6 )• π( µ), diag(ζ 6 , ζ 5 6 ) • π( ν). Via the Euler identity…”

mentioning

confidence: 99%

See 4 more Smart Citations

Modularity and effective Mordell I

Alpoge¹

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Remarksmentioning

confidence: 52%

“…As in Remark 12 of Archinard's [4], we see that the holomorphic differentials on X λ are given by: 5 . Now let us discuss how to compute the periods of X λ .…”

mentioning

confidence: 67%

“…4 It follows that, at least for K/Q totally real of odd degree, we may compute H o (o K,S ) in finite time. 5 Definition 1.2.…”

Section: Main Theoremsmentioning

confidence: 99%