Naslov
The growth of the rank of Abelian varieties upon extensions

Autori
Bruin, Peter ; Najman, Filip

Izvornik
Ramanujan journal (1382-4090) 39 (2016), 2; 259-269

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Elliptic curves ; Rank growth

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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