Anabelian geometry and descent obstructions on moduli spaces

Abstract

Abstract.We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we show how the sufficiency of the finite descent obstruction implies the same for all hyperbolic curves.

Finite descent obstruction for Hilbert modular varieties

“…We believe that it is easier to make the results of this paper unconditional. We notice here that the absolute Hodge conjecture is not enogh for such papers: in [27,3] the Hodge conjecture is not only needed to descend complex abelian varieties over number fields. Finally the version of Serre's conjecture we are assuming here is always about GL 2 -coefficients, and so is certainly easier than the full Fontaine-Mazur conjecture [17].…”

Local to global principle for the moduli space of K3 surfaces

“…It is reasonable to expect a result of the same fashion for varieties whose geometry is well captured from cohomological invariants. For example the above theorem can not tell the difference between a curve of genus N > 1 and its Jacobian (see [PVZ16, Section 5] for a more detailed discussion about this). From this point of view K3 surfaces (and hyperkähler varieties) are very similar to abelian varieties.…”

“…Indeed the proof uses the To conclude the introduction we point out that Theorem 1.3 can be interpreted in the setting of anabelian geometry and the section conjecture for the moduli space classifying primitively polarized K3 surfaces of degree 2d. For more about this we refer the reader to the introduction of [PVZ16] and the second part of Section 1.1 in [Kle18].…”

Recognizing Galois representations of K3 surfaces

“…This paper grew out of an attempt to answer a question on the section conjecture for moduli spaces of K3 surfaces, inspired by recent work of Patrikis, Voloch and Zarhin [PVZ16]. In this paper, the authors study the section conjecture for the moduli space of principally polarized abelian varieties.…”

“…It is known that moduli spaces A g of abelian varieties should not be anabelian by results of Ihara and Nakamura [IN97]. However, theorem 1.1 of [PVZ16] shows that under the assumption of well known motivic conjectures, a large subset of sections S 0 (K, A g ) ⊂ H(K, A g ) is contained in the image of σ Ag , where the sections S 0 (K, A g ) are those coming from points…”