Semisimplicity of the Frobenius Action on π1

Betts, L. Alexander; Litt, Daniel

doi:10.1007/978-3-031-21550-6_2

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…]/I^n)\Bigr ) \,. \end{equation*}$$As in [7, section 3.1], we have a canonical isomorphism

$$\begin{equation} {\mathcal O}(\pi _1^{\mathrm{k\acute{e}t},{\mathbb {Q}}_p}(X_{{\mathbb {C}}_K},\bar{x}))^\vee \cong {\mathbb {Q}}_p[\! [\hat{\pi }_1^\mathrm{k\acute{e}t}(X_{{\mathbb {C}}_K},\bar{x})]\!

…”

Section: Comparison Between éTale and De Rham Fundamental Groupoidsmentioning

confidence: 99%

“…Remark We do not actually need the full strength of Theorem 1.4 for the results in this paper; in fact it just suffices to know that

{}_x{\mathbb {E}}^{\mathrm{\acute{e}t},{\rm an}}

is a pro‐de Rham local system. This can actually be proved rather quickly and indirectly, see, for example, [7, Remark 3.2]. However, Theorem 1.4 implies something stronger about the pro‐unipotent Kummer map

j_p

: not only is it locally constant modulo

\operatorname{H}^1_e

, but it is locally analytic and can be explicitly described in terms of iterated integrals on small discs, much as in the Chabauty–Kim method for curves.…”

Section: Introductionmentioning

confidence: 99%

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Local constancy of pro‐unipotent Kummer maps

Betts

2023

Proceedings of London Math Soc

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It is a theorem of Kim–Tamagawa that the ‐pro‐unipotent Kummer map associated to a smooth projective curve over a finite extension of is locally constant when . This paper establishes two generalisations of this result. First, we extend the Kim–Tamagawa theorem to the case that is a smooth variety of any dimension. Second, we formulate and prove the analogue of the Kim–Tamagawa theorem in the case , again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale–de Rham comparison theorem for pro‐unipotent fundamental groupoids using methods of Scholze and Diao–Lan–Liu–Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids.

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“…]/I^n)\Bigr ) \,. \end{equation*}$$As in [7, section 3.1], we have a canonical isomorphism

$$\begin{equation} {\mathcal O}(\pi _1^{\mathrm{k\acute{e}t},{\mathbb {Q}}_p}(X_{{\mathbb {C}}_K},\bar{x}))^\vee \cong {\mathbb {Q}}_p[\! [\hat{\pi }_1^\mathrm{k\acute{e}t}(X_{{\mathbb {C}}_K},\bar{x})]\!

…”

Section: Comparison Between éTale and De Rham Fundamental Groupoidsmentioning

confidence: 99%

“…Remark We do not actually need the full strength of Theorem 1.4 for the results in this paper; in fact it just suffices to know that

{}_x{\mathbb {E}}^{\mathrm{\acute{e}t},{\rm an}}

j_p

: not only is it locally constant modulo

\operatorname{H}^1_e

, but it is locally analytic and can be explicitly described in terms of iterated integrals on small discs, much as in the Chabauty–Kim method for curves.…”

Section: Introductionmentioning

confidence: 99%

Local constancy of pro‐unipotent Kummer maps

Betts

2023

Proceedings of London Math Soc

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The Ceresa Class: Tropical, Topological and Algebraic

Corey,

Ellenberg,

2023

J. Inst. Math. Jussieu

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The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for nonhyperelliptic curves is in general not easy. We present a ‘combinatorialization’ of this problem, explaining how to define a Ceresa class for a tropical algebraic curve and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over $\mathbb {C}(\!(t)\!)$ and show that the Ceresa class in each of these settings is torsion.

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The Motivic Anabelian Geometry of Local Heights on Abelian Varieties

Betts

2024

Memoirs of the AMS

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We study the problem of describing local components of Néron–Tate height functions on abelian varieties over characteristic 0 0 local fields as functions on spaces of torsors under various realisations of the 2 2 -step unipotent motivic fundamental group of the G m \mathbb G_m -torsor corresponding to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the Q ℓ \mathbb Q_\ell - and Q p \mathbb Q_p -pro-unipotent étale realisations when the base field is p p -adic, and in terms of the R \mathbb R -pro-unipotent Betti–de Rham realization when the base field is archimedean. In the course of proving the p p -adic instance of these theorems, we develop a new technique for studying local nonabelian Bloch–Kato Selmer sets, working with certain explicit cosimplicial group models for these sets and using methods from homotopical algebra. Among other uses, these models enable us to construct a pro-unipotent generalization of the Bloch–Kato exponential sequence under minimal assumptions.

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Semisimplicity of the Frobenius Action on π1

Cited by 3 publications

References 23 publications

Local constancy of pro‐unipotent Kummer maps

Local constancy of pro‐unipotent Kummer maps

The Ceresa Class: Tropical, Topological and Algebraic

The Motivic Anabelian Geometry of Local Heights on Abelian Varieties

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