Rational Points of Abelian Varieties over Function Fields

Lang, Serge; Neron, André

doi:10.1007/978-1-4612-2118-0_19

Cited by 41 publications

(59 citation statements)

References 11 publications

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“…This curve uniquely extends to a minimal regular proper elliptic fibration E → P 1 K . The group E (K(T )) is finitely generated, by the Lang-Néron theorem [18,Theorem 1]. (See [4, §6] for a proof of the Lang-Néron theorem using the language of schemes.)…”

Section: Introductionmentioning

confidence: 99%

Root numbers and ranks in positive characteristic

Conrad

Helfgott

2005

Advances in Mathematics

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For a global field K and an elliptic curve E over K(T ), Silverman's specialization theorem implies rank(E (K(T ))) rank(E t (K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve E is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T ) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = (u) over any finite field with characteristic = 2, we construct an explicit 2-parameter family E c,d of non-isotrivial elliptic curves over K(T ) (depending on arbitrary c, d ∈ × ) such that, under the parity conjecture, each E c,d has elevated rank.

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Section: Introductionmentioning

confidence: 99%

Root numbers and ranks in positive characteristic

Conrad

Helfgott

2005

Advances in Mathematics

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“…But this contradicts the fact that (A η ) 0 (k(S)) is a finitely generated abelian group by the Lang-Néron theorem [20]. Thus, M 0 = {0}, as desired.…”

Section: Lemma 24 For Any Finite Extensionmentioning

confidence: 60%

“…When k is finitely generated over its prime field, the trivial character [20] and the p-adic cyclotomic character [3,Lemma 2.6] are typical examples of non-Tate characters.…”

Section: Non-tate Charactersmentioning

confidence: 99%

Uniform boundedness of p-primary torsion of abelian schemes

Cadoret

Tamagawa

2011

Invent. math.

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Let k be a field finitely generated over Q and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d = 1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d = 1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E → S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E s , s ∈ S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.More precisely, we prove the following. Denote by k the absolute Galois group of k. For an abelian variety A over k and a character χ : k → Z * p , define A[p ∞ ](χ ) to be the module of p-primary torsion of A(k) on which k acts as χ -multiplication. Assume that χ does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p ∞ ](χ ) is always finite, but the exponent of A[p ∞ ](χ ) may de-

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“…From the Mordell-Weil theorem (proved by Lang and Néron [6] in this case) it follows that MW(X/C) is finitely generated. We call the rank of MW(X/C) the Mordell-Weil rank.…”

Section: Review Of Mordell-weil Lattices and Necessary Definitionsmentioning

confidence: 99%