- Title
- On the soft p-converse to a theorem of Gross-Zagier and Kolyvagin
- KIAS Author
- Kim, Chan-Ho,Kim, Chan-Ho
- Journal
- MATHEMATISCHE ANNALEN, 2023
- Archive
- https://arxiv.org/abs/2109.12344
- Abstract
- 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 the equivalence rk(Z)E(Q) = 1, #III(E/Q) < infinity double left right arrow ord(s=1)L(E, S) = 1 for every elliptic curve E over Q.