Abelian varieties over cyclic fields

Im, Boe-Hae; Larsen, Michael

doi:10.1353/ajm.0.0020

Cited by 8 publications

(1 citation statement)

References 13 publications

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“…In the same spirit, the condition of linear disjointness is included in Question 1 to exclude certain trivial answers; it is, however, furthermore useful for theoretical purposes, since it guarantees that the rank of the respective elliptic curve becomes infinite over the compositum of these extensions. See, for example, [1,13,15] for consideration about infinite rank over such "large" fields.…”

Section: Introductionmentioning

confidence: 99%

Rank gain of Jacobians over number field extensions with prescribed Galois groups

König

2023

Mathematische Nachrichten

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We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has a Galois group permutation-isomorphic to a prescribed group 𝐺 (in short, "𝐺-extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups 𝐺, we show that most elliptic curves over (for example) ℚ gain rank over infinitely many 𝐺-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that "many" elliptic curves gain rank over infinitely many 𝐺-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group 𝐺 and certain local properties.

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Section: Introductionmentioning

confidence: 99%

Rank gain of Jacobians over number field extensions with prescribed Galois groups

König

2023

Mathematische Nachrichten

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Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

Arias-de-Reyna

Gajda

Petersen

2013

Mathematische Nachrichten

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In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on ℓ-torsion points, for almost all primes ℓ, contains the full symplectic group.

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Artin-Schreier extensions in NIP and simple fields

2011

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Abelian varieties over cyclic fields

Abstract: Abstract. Let K be a field of characteristic = 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. If K is not locally finite, the rank of A over K is infinite.

Cited by 8 publications

References 13 publications

Rank gain of Jacobians over number field extensions with prescribed Galois groups

Rank gain of Jacobians over number field extensions with prescribed Galois groups

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

Artin-Schreier extensions in NIP and simple fields

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