Model theory of the Frobenius on the Witt vectors

Abstract. We show that the theory of the non-standard Frobenius automorphism, acting on an algebraically closed valued field of equal characteristic 0, is NTP 2 . More generally, in the contractive as well as in the isometric case, we prove that a σ-henselian valued difference field of equicharacteristic 0 is NTP 2 , provided both the residue difference field and the value group (as an ordered difference group) are NTP 2 .

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“…• We prove a similar result in the isometric case, where an Ax-Kochen-Ershov principle holds as well [Sca03,BMS07,AvdD11].…”

Section: Introductionsupporting

confidence: 63%

Valued difference fields and NTP2

Chernikov

Hils

2014

Isr. J. Math.

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“…Note that, if k L has characteristic zero, then one has L n = L 1 and ac n = ac 1 for all n 1. We also write ac for ac 1 .…”

Section: Non-archimedean Yomdin-gromov Parametrizations With Taylor Amentioning

confidence: 99%

“…From now on in Section 3, definable sets and functions will be so for the language L. Note that the study of definable sets was initiated in the works of Macintyre [25] and Denef and van den Dries [19] in the p-adic case, and was generalized later to this and other settings in for example [1,12,14,36].…”

Section: Non-archimedean Yomdin-gromov Parametrizations With Taylor Amentioning

confidence: 99%

“…This has two uses: to obtain that composition with well-chosen power maps makes the C r -norm less than or equal to 1, and to obtain that C r -norm bounded by 1 implies T r on each maximal ball included in the domain. Moreover, by using the results of the previous section on T 1 , we reduce to the situation where one has globally T 1 and locally (on maximal balls) T r . By composing with power maps once more, the previous T 1 condition on far-away points implies T r , which then follows globally on the pieces.…”

Section: Non-archimedean Yomdin-gromov Parametrizations With Taylor Amentioning

confidence: 99%

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Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

2015

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We prove an analog of the Yomdin-Gromov lemma for p-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of p-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over C((t)), in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

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“…[23] or [3]). On the face of it, Theorem 2.6 is not first-order expressible, but we shall show that a failure of Theorem 4.1 may be converted to a failure of Theorem 2.6.…”

Section: 1mentioning

confidence: 99%

Local André-Oort conjecture for the universal abelian variety

Scanlon

2005

Invent. math.

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Abstract. We prove a p-adic analogue of the André-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let g and n be integers with n ≥ 3 and p a prime number not dividing n. Let R be a finite extension of W [F alg p ], the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space A = Ag,1,n of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety π : X → A over A may be defined over R. We call a point ξ ∈ X (R) R-special if X π(ξ) is a canonical lift and ξ is a torsion point of its fibre. We show that an irreducible subvariety of XR containing a dense set of p-special points must be a special subvariety in the sense of mixed Shimura varieties. Our proof employs the model theory of difference fields.

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Model theory of the Frobenius on the Witt vectors

Cited by 26 publications

References 20 publications

Valued difference fields and NTP2

Valued difference fields and NTP2

Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

Local André-Oort conjecture for the universal abelian variety

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