A rank zero p-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin

2025/06/04 by Ashay Burungale, Ye Tian, Burungale, Ashay A. +1
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2506.03465

Abstract

Let E be a CM elliptic curve defined over ℚ and p a prime. We show that \mathrm corank_ℤ_p \mathrm Sel_p^∞(E_/ℚ)=0 ⇒ \mathrm ord_s=1L(s,E_/ℚ)=0 for the p^∞-Selmer group \mathrm Sel_p^∞(E_/ℚ) and the complex L-function L(s,E_/ℚ). Along with Smith's work on the distribution of 2^∞-Selmer groups, this leads to the first instance of the even parity Goldfeld conjecture: For 50% of the positive square-free integers n, we have \mathrm ord_s=1L(s,E⁽ⁿ⁾_/ℚ)=0, where E⁽ⁿ⁾: ny²=x³-x is a quadratic twist of the congruent number elliptic curve E: y²=x³-x.

Citations

• Genus Periods, Genus Points and Congruent Number Problem
• A converse to a theorem of Gross, Zagier, and Kolyvagin
• On the 2-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication
• Congruent Numbers and Heegner Points
• Disparity in Selmer ranks of quadratic twists of elliptic curves
• The average analytic rank of elliptic curves

Related

• The even parity Goldfeld conjecture: congruent number elliptic curves
• p-Selmer ranks of CM abelian varieties
• Invariance of the parity conjecture for p-Selmer groups of elliptic curves in a D_2pⁿ-extension
• p-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes
• On the p-converse of the Kolyvagin-Gross-Zagier theorem