2026/07/01 by Muskan Bansal, Somnath Jha, Aprameyo Pal +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic
paper · doi:10.1112/jlms.70629
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 .