Behaviors of the Tate–Shafarevich Group of Elliptic Curves under Quadratic Field Extensions
Abstract
Let E/Q be an elliptic curve. We study the behavior of the Tate–Shafarevich group of E under quadratic extensions Q(D)/Q. By analyzing the cokernel of the restriction map, without assuming the finiteness of the Tate–Shafarevich group, we prove that the ratio #Ш(E/Q(D))[4]#Ш(ED/Q)[2] and #Ш(ED/Q)[2] can, under some conditions on E/Q, grow arbitrarily large simultaneously, where ED denotes the quadratic twist of E by D. For elliptic curves of the form E:y2=x3+px with p≡1mod4 being an odd prime, assuming the finiteness of the relevant Tate–Shafarevich groups, we prove that #Ш(E/Q(D))[2]≤4 and Ш(ED/Q)[2]=0 for infinitely many square-free integers D with −D being a prime number. Additionally, Ш(E/Q(−D))[2]≠0 for all D when p=257.
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Advanced Algebra and Geometry
Analytic Number Theory Research
Publication details
- Journal / source
- Tokyo Journal of Mathematics
- Publisher
- Tokyo Journal of Mathematics
- Volume
- 49
- Issue
- 1
- Pages
- 199–224
- Publication date
- 1 June 2026
- Language
- EN