A p-Converse theorem for Real Quadratic Fields

2025/04/30 by Bansal, Muskan, Jha, Somnath, Pal, Aprameyo +1
#11G05 #11G40 #11R23 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.21799

Abstract

Let E be an elliptic curve defined over a real quadratic field F. Let p > 5 be a rational prime that is inert in F and assume that E has split multiplicative reduction at the prime \mathfrakp of F dividing p. Let \underlineIII(E/F) denote the Tate-Shafarevich group of E over F and L(E/F,s) be the Hasse-Weil complex L-function of E over F. Under some technical assumptions, we show that when rank_ℤ \hspace0.01mm \hspace1mm E(F) = 1 and #(\underlineIII(E/F)_ p^∞) < ∞, then ord_s=1 L(E/F,s) = 1. Further, we give an applictaion to a p-converse theorem over ℚ.

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