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The growth of the rank of Abelian varieties upon extensions (CROSBI ID 235373)

Prilog u časopisu | izvorni znanstveni rad | međunarodna recenzija

Bruin, Peter ; Najman, Filip The growth of the rank of Abelian varieties upon extensions // Ramanujan journal, 39 (2016), 2; 259-269. doi: 10.1007/s11139-014-9627-y

Podaci o odgovornosti

Bruin, Peter ; Najman, Filip

engleski

The growth of the rank of Abelian varieties upon extensions

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 A is an Abelian variety over a number field K and L/K is a finite Galois extension such that Gal(L/K) does not have an index 2 subgroup, then rkA(L)−rkA(K) can never be 1. We show that rkA(L)−rkA(K) is either 0 or ≥p−1, where p is the smallest prime divisor of #Gal(L/K), and we obtain more precise results when Gal(L/K) is alternating, SL2(Fp) or PSL2(Fp) for p>2. This implies a restriction on rkE(K(E[p]))−rkE(K(ζp)) when E/K is an elliptic curve whose mod p Galois representation is surjective. We obtain similar results for the growth of the rank over certain non-Galois extensions. Second, we show that for every n≥2 there exists an elliptic curve En over a number field Kn such that Q⊗EndQResKn/QEn contains a number field of degree 2n. 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 En over a number field Kn admitting infinitely many quadratic twists whose rank is a positive multiple of 2n.

Elliptic curves ; Rank growth

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Podaci o izdanju

39 (2)

2016.

259-269

objavljeno

1382-4090

1572-9303

10.1007/s11139-014-9627-y

Povezanost rada

Matematika

Poveznice
Indeksiranost