2021/09/25 by Chan-Ho Kim, Kim, Chan-Ho
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2109.12344
We give a proof of a soft version of the p-converse to a theorem of Gross--Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at p >3 under the irreducibility assumption on the residual representation. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain that the Mordell-Weil rank is one and the Tate-Shafarevich group is finite if and only if the analytic rank is one for every elliptic curve over the rationals.