A p‐converse theorem for real quadratic fields
Abstract
Abstract Let be an elliptic curve defined over a real quadratic field . Let be a rational prime that is inert in and assume that has split multiplicative reduction at the prime of dividing . Let denote the Tate–Shfarevich group of over and be the Hasse–Weil complex ‐function of over . In this setting, we establish a rank 1 ‐converse theorem to Zhang's generalisation of Gross–Zagier and Kolyvagin theorem. More precisely, under some technical assumptions, we show that image We also indicate an application to a ‐converse to Gross–Zagier and Kolyvagin theorem in a parallel setting over .
Fields of Study & Subjects
Algebraic Geometry and Number Theory
Analytic Number Theory Research
Cryptography and Residue Arithmetic
Publication details
- Journal / source
- Journal of the London Mathematical Society
- Volume
- 114
- Issue
- 1
- Publication date
- 1 July 2026
- Language
- EN