Infinitely p-Divisible Points on Abelian Varieties Defined over Function Fields of Characteristic p&gt;0

Rössler, Damian

doi:10.1215/00294527-2143943

Cited by 5 publications

(7 citation statements)

References 22 publications

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“…Then S(G ♯ )(K 1 ) = G 0 ♯ (K 1 ) + A G ♯ (K 1 ). By the result for abelian varieties ( [22]), A G ♯ (K 1 ) is contained in the torsion of A G . As G 0 is isogenous to some S 0 defined over K 0 , the field of constants of K 1 , it follows that G 0 ♯ (K 1 ) is definably isogenous to S ♯ 0 (K 1 ) = S 0 (K 0 ) = S 0 (F p alg ) which is exactly the group of torsion elements of S 0 .…”

Section: Characteristic P Case Quantifier Elimination For the Socle O...mentioning

confidence: 95%

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On function field Mordell–Lang: the semiabelian case and the socle theorem

Benoist

Bouscaren

Pillay

2017

Proc. London Math. Soc.

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In this paper we complete our 'second-generation' model-theoretic account of the function field Mordell-Lang conjecture, where we avoid appeals to dichotomy theorems for (generalized) Zariski geometries. In the current paper we reduce the semiabelian case to the abelian case using modeltheoretic tools. We also use our results from Benoist, Bouscaren and Pillay [J. Math. Log. 16 (2016)] to prove modularity of the 'Manin kernels' (over Fp(t) sep in positive characteristic and over C(t) alg in characteristic 0).

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Section: Characteristic P Case Quantifier Elimination For the Socle O...mentioning

confidence: 95%

“…Then every point in A ♯ (K) is a torsion point. Remark 6.4 The characterisic 0 case is proved in [3], and the characteristic p case in [22]. The trace 0 hypothesis is only mandatory in characteristic 0.…”

Section: Some Examplesmentioning

confidence: 96%

On function field Mordell–Lang: the semiabelian case and the socle theorem

Benoist

Bouscaren

Pillay

2017

Proc. London Math. Soc.

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“…The statements of the function field Mordell-Lang which we prove in this paper, as well as the main arguments we use, are specific to abelian varieties. They are also restricted to the function field case in one variable (C(t) alg in characteristic 0, and F p (t) sep in characteristic p) as the "Theorem of the Kernel" has been proved only in these cases, in [7] and in [29]. Basic background, as well as references, on abelian varieties, Mordell-Lang and Manin Mumford can be for example found in [14].…”

Section: Preliminariesmentioning

confidence: 99%

On function field Mordell–Lang and Manin–Mumford

2016

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We give a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the "Theorem of the Kernel", and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: "quantifier-elimination" for the corresponding A = p ∞ A(U ) where U is a saturated separably closed field.

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“…and applying a theorem of Zarhin (see [41,Th. 3.1] for a statement, explanations and further references), we conclude that in the sequence (4), there are only finitely many isomorphism classes of semiabelian schemes over S ′ .…”

Section: Proofs Of the Claims Made In Subsection 21mentioning

confidence: 99%

On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic

Rössler¹

2015

Comment. Math. Helv.

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Let A be an abelian variety over the function field K of a curve over a finite field. We describe several mild geometric conditions ensuring that the group A(K perf ) is finitely generated and that the p-primary torsion subgroup of A(K sep ) is finite. This gives partial answers to questions of Scanlon, Ghioca and Moosa, and Poonen and Voloch. We also describe a simple theory (used to prove our results) relating the Harder-Narasimhan filtration of vector bundles to the structure of finite flat group schemes of height one over projective curves over perfect fields. Finally, we use our results to give a complete proof of a conjecture of Esnault and Langer on Verschiebung divisibility of points in abelian varieties over function fields in the situation where the base field is the algebraic closure of a finite field.

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Infinitely p-Divisible Points on Abelian Varieties Defined over Function Fields of Characteristic p>0

Cited by 5 publications

References 22 publications

On function field Mordell–Lang: the semiabelian case and the socle theorem

On function field Mordell–Lang: the semiabelian case and the socle theorem

On function field Mordell–Lang and Manin–Mumford

On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic

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