Models of Abelian varieties over valued fields, using model theory

Yatir Halevi

doi:10.1016/j.aim.2024.109938

Models of Abelian varieties over valued fields, using model theory

Yatir Halevi

Research output: Contribution to journal › Article › peer-review

Abstract

Given an elliptic curve E over a perfect defectless henselian valued field (F,val) with perfect residue field k_F and valuation ring O_F, there exists an integral separated smooth group scheme E over O_F with E×_{Spec O_F}Spec F≅E. If char(k_F)≠2,3 then one can be found over O_F^alg such that the definable group E(O) is the maximal generically stable subgroup of E. We also give some partial results on general Abelian varieties over F. The construction of E is by means of generating a birational group law over O_F by the aid of a generically stable generic type of a definable subgroup of E.

Original language	English
Article number	109938
Journal	Advances in Mathematics
Volume	457
DOIs	https://doi.org/10.1016/j.aim.2024.109938
State	Published - 1 Nov 2024
Externally published	Yes

Keywords

ACVF
Elliptic curves
Generically stable groups
Group schemes
Neron models
Valued fields

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2024.109938

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title = "Models of Abelian varieties over valued fields, using model theory",

abstract = "Given an elliptic curve E over a perfect defectless henselian valued field (F,val) with perfect residue field kF and valuation ring OF, there exists an integral separated smooth group scheme E over OF with E×Spec OFSpec F≅E. If char(kF)≠2,3 then one can be found over OFalg such that the definable group E(O) is the maximal generically stable subgroup of E. We also give some partial results on general Abelian varieties over F. The construction of E is by means of generating a birational group law over OF by the aid of a generically stable generic type of a definable subgroup of E.",

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N2 - Given an elliptic curve E over a perfect defectless henselian valued field (F,val) with perfect residue field kF and valuation ring OF, there exists an integral separated smooth group scheme E over OF with E×Spec OFSpec F≅E. If char(kF)≠2,3 then one can be found over OFalg such that the definable group E(O) is the maximal generically stable subgroup of E. We also give some partial results on general Abelian varieties over F. The construction of E is by means of generating a birational group law over OF by the aid of a generically stable generic type of a definable subgroup of E.

AB - Given an elliptic curve E over a perfect defectless henselian valued field (F,val) with perfect residue field kF and valuation ring OF, there exists an integral separated smooth group scheme E over OF with E×Spec OFSpec F≅E. If char(kF)≠2,3 then one can be found over OFalg such that the definable group E(O) is the maximal generically stable subgroup of E. We also give some partial results on general Abelian varieties over F. The construction of E is by means of generating a birational group law over OF by the aid of a generically stable generic type of a definable subgroup of E.

KW - ACVF

KW - Elliptic curves

KW - Generically stable groups

KW - Group schemes

KW - Neron models

KW - Valued fields

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DO - 10.1016/j.aim.2024.109938

M3 - Article

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SN - 0001-8708

VL - 457

JO - Advances in Mathematics

JF - Advances in Mathematics

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ER -

Models of Abelian varieties over valued fields, using model theory

Abstract

Keywords

ASJC Scopus subject areas

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