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A rank zero p-converse to a theorem of Gross-Zagier, Kolyvagin and Rubin
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DOI：10.4007/annals.2026.203.1.1.png)
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Abstract
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Let E be a CM elliptic curve defined over Q and p a prime. We show that corankZp Selp infinity (E/Q) = 0 =double right arrow ords=1L(s, E/Q) = 0 for the p infinity- Selmer group Selp infinity(E/Q) and the complex L-function L(s, E/Q). Along with Smith's work on the distribution of 2 infinity-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 ords=1L(s, E/Q(n)) = 0, where E(n) : ny2 = x3-x is a quadratic twist of the congruent number elliptic curve E:y2=x3-x.
Keywords:
The Birch and Swinnerton-Dyer conjecture
The Goldfeld conjecture
CM elliptic curves
Iwasawa theory
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