Approximation Theory and the Rank of Abelian Varieties Over Large Algebraic Fields

Frey, Gerhard; Jarden, Moshe

doi:10.1112/plms/s3-28.1.112

Cited by 69 publications

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“…This is Theorem 2.2 of the paper of G. Frey and M. Jarden [2]. The statement of the theorem is that EiQ) has infinite rank but they remark directly after the proof that £(Í2) already has infinite rank.…”

Section: Propositionmentioning

confidence: 91%

Elliptic curves and Dedekind domains

Rosen¹

1976

Proc. Amer. Math. Soc.

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Abstract.Some results are obtained on the group of rational points on elliptic curves over infinite algebraic number fields. A certain naturally defined class of Dedekind domains, elliptic Dedekind domains, are described and it is shown that every countable abelian group can be realized as the class group of an elliptic Dedekind domain.Introduction. Let E be an elliptic curve defined over a field K. Let S be a set of K rational points on E and RS(E) the ring of K rational functions on E having all their poles in S. Then RS(E) is a Dedekind domain. We call such a ring an elliptic Dedekind domain. What abelian groups arise as class groups of such rings? In [6] it is shown that every finitely generated abelian group arises in this way. In this paper we extend this result as follows.

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Section: Propositionmentioning

confidence: 91%

Elliptic curves and Dedekind domains

Rosen¹

1976

Proc. Amer. Math. Soc.

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“…Suppose that every Abelian variety over Q has infinite rank over Q(2); this is a variant of [5,page 127,Problem]. Then by taking the Weil restriction, we obtain a positive answer to Question 11.…”

Section: Q-curves and Ranks Of Twistsmentioning

confidence: 99%

The growth of the rank of Abelian varieties upon extensions

Bruin

Najman

2014

Ramanujan J

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Abstract. We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields.First, we show that if L/K is a finite Galois extension of number fields such that Gal(L/K) does not have an index 2 subgroup and A/K is an Abelian variety, then rk A(L) − rk A(K) can never be 1. We obtain more precise results when Gal(L/K) is of odd order, alternating, SL 2 (Fp) or PSL 2 (Fp). This implies a restriction on rk E(K(E[p])) − rk E(K(ζp)) when E/K is an elliptic curve whose mod p Galois representation is surjective. Similar results are obtained for the growth of the rank in certain non-Galois extensions.Second, we show that for every n ≥ 2 there exists an elliptic curve E over a number field K such that Q ⊗ End Q Res K/Q E contains a number field of degree 2 n . We ask whether every elliptic curve E/K has infinite rank over KQ(2), where Q(2) is the compositum of all quadratic extensions of Q. We show that if the answer is yes, then for any n ≥ 2, there exists an elliptic curve E/K admitting infinitely many quadratic twists whose rank is a positive multiple of 2 n .

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“…Interesting problems arise if one studies the rank in other infinite algebraic extensions of K. For elliptic curves E|Q Frey and Jarden showed that E(Ω) is of infinite rank where Ω denotes the maximal Kummer extension of Q of exponent 2. In the light of these facts Frey and Jarden asked in their paper [2]:…”

Section: Introductionmentioning

confidence: 98%

On a question of Frey and Jarden about the rank of Abelian varieties

Petersen

2006

Journal of Number Theory

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This paper deals with a classical question of Frey and Jarden, who asked in their 1974 paper if any nonzero Abelian variety over a number field K acquires infinite rank over the maximal Abelian extension K ab of the ground field. We generalize recent results of Rosen and Wong on the subject. However, the original question in full generality remains open. Some further results on the rank in certain other infinite extensions are included.

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Approximation Theory and the Rank of Abelian Varieties Over Large Algebraic Fields

Cited by 69 publications

References 0 publications

Elliptic curves and Dedekind domains

Elliptic curves and Dedekind domains

The growth of the rank of Abelian varieties upon extensions

On a question of Frey and Jarden about the rank of Abelian varieties

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