Open problems: Descending cohomology, geometrically

Mazur, Barry

doi:10.4310/iccm.2014.v2.n1.a7

Cited by 5 publications

(5 citation statements)

References 17 publications

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“…For example, for a prime number different from the characteristic of , it is well known that the first -adic cohomology group can be modeled by the Albanese variety in the sense that it is isomorphic as a -representation to . Our primary motivation is a special case of a problem posed by Mazur [Maz14] at the birthday conference for Joe Harris in 2011. To fix ideas, suppose that is a smooth projective variety over a number field .…”

Section: Introductionmentioning

confidence: 99%

On descending cohomology geometrically

Achter¹,

Casalaina-Martin²,

Vial

2017

Compositio Math.

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Abstract. In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur's question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel-Jacobi map admits a distinguished model over the rationals.

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Section: Introductionmentioning

confidence: 99%

On descending cohomology geometrically

Achter¹,

Casalaina-Martin²,

Vial

2017

Compositio Math.

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“…Such an abelian variety, if it exists, is determined up to isogeny ; this is called the phantom isogeny class for V ℓ . Mazur asks the following question [Maz14,p.38] : Let X be a smooth projective variety over a field K ⊆ C, and let n be a nonnegative integer. If H 2n+1 (X C , Q) has Hodge coniveau n (i.e., H 2n+1 (X C , C) = H n,n+1 (X) ⊕ H n+1,n (X)), does there exist a phantom abelian variety for H 2n+1 (X K , Q ℓ (n))?…”

Section: As a First Application Of Theorem A We Recover A Results Of mentioning

confidence: 99%

Distinguished Models of Intermediate Jacobians

Achter

Casalaina-Martin

Vial

2018

J. Inst. Math. Jussieu

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We show that the image of the Abel-Jacobi map admits functorially a model over the field of definition, with the property that the Abel-Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level one.

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“…The integral Selmer set S 0 (K, A g ), defined in the previous section, corresponds to the set of Galois representations that are almost everywhere unramified and, locally, come from abelian varieties (which thus are of good reduction for almost all places of K) and we will also consider a few variants of the question of surjectivity of σ Ag /K to S 0 (K, A g ) by different local hypotheses and discuss what we can and cannot prove. A version of this kind of question has also been considered by B. Mazur [29].…”

Section: Moduli Of Abelian Varietiesmentioning

confidence: 99%

Anabelian geometry and descent obstructions on moduli spaces

Patrikis¹,

Voloch²,

Zarhin³

2016

Algebra Number Theory

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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.

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Open problems: Descending cohomology, geometrically

Cited by 5 publications

References 17 publications

On descending cohomology geometrically

On descending cohomology geometrically

Distinguished Models of Intermediate Jacobians

Anabelian geometry and descent obstructions on moduli spaces

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