A p-adic nonabelian criterion for good reduction of curves

Andreatta, Fabrizio; Iovită, Adrian; Kim, Minhyong

doi:10.1215/00127094-3146817

Cited by 17 publications

(57 citation statements)

References 17 publications

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“…If

C / K

is a curve (smooth, projective and geometrically connected), then results of Oda [, Theorem 3.2] show that

C

has good reduction if and only if the outer

G_{K}

‐action on its

Q_{ℓ}

‐unipotent fundamental group

π_{1}^{\overset{e}{true}} t {(C_{\bar{K}})}_{Q_{ℓ}}

is unramified for one (respectively, all) primes

ℓ \neq p

. The

p

‐adic version of this result was proven by Andreatta, Iovita and Kim .…”

Section: Introductionmentioning

confidence: 92%

A Néron–Ogg–Shafarevich criterion for K3 surfaces

Chiarellotto

Lazda

Liedtke

2019

Proc. London Math. Soc.

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The naive analogue of the Néron–Ogg–Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified ℓ‐adic étale cohomology groups, but which do not admit good reduction over K. Assuming potential semi‐stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if Hnormalét2false(XK¯,Qℓfalse) is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain ‘canonical reduction’ of X. We also prove the corresponding results for p‐adic étale cohomology.

show abstract

“…If

C / K

is a curve (smooth, projective and geometrically connected), then results of Oda [, Theorem 3.2] show that

C

has good reduction if and only if the outer

G_{K}

‐action on its

Q_{ℓ}

‐unipotent fundamental group

π_{1}^{\overset{e}{true}} t {(C_{\bar{K}})}_{Q_{ℓ}}

is unramified for one (respectively, all) primes

ℓ \neq p

. The

p

‐adic version of this result was proven by Andreatta, Iovita and Kim .…”

Section: Introductionmentioning

confidence: 92%

A Néron–Ogg–Shafarevich criterion for K3 surfaces

Chiarellotto

Lazda

Liedtke

2019

Proc. London Math. Soc.

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“…The deformation theory of such curves is unobstructed, so this special fibre can be deformed to a stable curve in mixed characteristic. By comparing suitable monodromy operators I can then in fact deduce Theorem 7.5 from the mixed characteristic result in [AIK15].…”

Section: Good Reduction Criteriamentioning

confidence: 98%

“…In the rest of this article, I will show how to use the 'non-abelian' information contained in π rig 1 (X/E † K , x) to give a criterion for a smooth, semistable curve over F to have good reduction, analogous to that provided by Andreatta, Iovita and Kim in [AIK15]. To formulate this criterion, I will first need to discuss the correct analogue of being 'crystalline' for (ϕ, ∇)-modules over E † K .…”

Section: Good Reduction Criteriamentioning

confidence: 99%

Fundamental groups and good reduction criteria for curves over positive characteristic local fields

Lazda¹

2017

J. Théor. Nombres Bordeaux

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In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian (ϕ,∇)-module over the bounded Robba ring E † K , whose underlying unipotent group (after base changing to the Amice ring E K ) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, p-adic analogue of Oda's theorem that a semistable curve over a p-adic field has good reduction iff the Galois action on its ℓ-adic unipotent fundamental group is unramified.

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“…Given two different choices of splitting of the Hodge filtration s (1) and s (2) , we obtain two different pre-heights h (1) p and h…”

Section: Global Height: Definition and Basic Properties Definementioning

confidence: 99%

“…First suppose ρ f (A)d + d(d−1)e(A)/2−1 > d(r−dim(A)). By Lemma 3.3, we have a Galois-stable quotient of U 2 which is an extension 2 , then we use Proposition 5.1. We take R, as above, to be Mat d (Q), acting trivially on B and in the obvious way on A d .…”

Section: Proof Of Theoremmentioning

confidence: 99%