Ranks of elliptic curves over $$\mathbb{Z}_p^2$$-extensions

Abstract

Let E be an elliptic curve with good reduction at a fixed odd prime p and K an imaginary quadratic field where p splits. We give a growth estimate for the Mordell-Weil rank of E over finite extensions inside the Z 2 pextension of K.

Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p

“…When E has good ordinary reduction at all primes above p, the above conjecture is a natural extension of Mazur's conjecture (see [6,7,9,10,32]). When E has supersingular reduction with F = Q and F an imaginary quadratic field where p splits, this was studied in [18,22,23].…”

“…When F = F = Q, Kobayashi [20] defined the plus and minus Selmer groups of E over Q cyc by constructing the plus and minus norm groups E ± (Q cyc p ), which are subgroups of the formal group of E at p. He was able to describe the algebraic structure of these plus and minus norm groups precisely and show that the plus and minus Selmer groups are cotorsion over Z p [[Γ]]. These Selmer groups have been extended to different settings by various authors (see [1,2,3,15,16,17,18,19,21,23,24,25,31]). When E is defined over Q and F is a number field where p is unramified, Kim [15,16] studied the structure of plus and minus Selmer groups over F cyc .…”

Chromatic Selmer groups and arithmetic invariants of elliptic curves

“…In our proof, we do this for a slightly larger module Y n . (See [20,Section 5.3] for the precise definitions. We have the inequality rk Zp Y n rk Zp Y n .…”

Asymptotic growth of Mordell–Weil ranks of elliptic curves in noncommutative towers

“…In this setting, it is the extension contained in 𝐹 ∞ such that Gal(𝐹 (𝑛) /𝐹) = (Z/𝑝 𝑛 Z) 2 . For elliptic curves 𝐸 /𝐹 , asymptotic formulas for the growth of the rank of 𝐸 (𝐹 (𝑛) ) as 𝑛 → ∞ have been proven by A. Lei and F. Sprung in [19]. More Mordell-Weil ranks in noncommutative towers 3 recently, such growth questions are studied in admissible uniform pro-𝑝 extensions of number fields by D. Delbourgo and A. Lei in [7], and by P. C. Hung and M. F. Lim in [14].…”