NSF PAR Search | NSF Public Access Repository

https://doi.org/10.1090/mcom/4110

Balakrishnan, Jennifer; Çiperiani, Mirela; Mazur, Barry; Rubin, Karl (July 2025, Mathematics of Computation)

Let

E

be an elliptic curve over

Q

with Mordell–Weil rank

2

and

p

be an odd prime of good ordinary reduction. For every imaginary quadratic field

K

satisfying the Heegner hypothesis, there is (subject to the Shafarevich–Tate conjecture) a line, i.e., a free

Z_{p}

-submodule of rank

1

, in

E (K) \otimes<#comment/> Z_{p}

given by universal norms coming from the Mordell–Weil groups of subfields of the anticyclotomic

Z_{p}

-extension of

K

; we call it theshadow line. When the twist of

E

K

has analytic rank

1

, the shadow line is conjectured to lie in

E (Q) \otimes<#comment/> Z_{p}

; we verify this computationally in all our examples. We study the distribution of shadow lines in

E (Q) \otimes<#comment/> Z_{p}

K

varies, framing conjectures based on the computations we have made.

Free, publicly-accessible full text available July 31, 2026

Abstract Let

E / Q

\mathrm{E}/\mathbb{Q}be an elliptic curve and 𝑝 a prime of supersingular reduction for

E

\mathrm{E}.Consider a quadratic extension

L / Q_{p}

L/\mathbb{Q}_{p}and the corresponding anticyclotomic

Z_{p}

\mathbb{Z}_{p}-extension

L_{\infty} / L

L_{\infty}/L.We analyze the structure of the points

E (L_{\infty})

\mathrm{E}(L_{\infty})and describe two global implications of our results.

Search for: All records