p -Selmer groups and rational points on CM elliptic curves

Ashay Burungale, Francisco Castella Cabello, Christopher Skinner, Ye Tian

Research output: Contribution to journalArticlepeer-review

Abstract

Let E/ Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Selp∞(E/Q) has Zp-corank one, then E(Q) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ords=1L(E,s)=1, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p> 3 , this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].

Original languageEnglish (US)
JournalAnnales Mathematiques du Quebec
DOIs
StateAccepted/In press - 2022

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Elliptic waves
  • Heegen points
  • L-functions
  • P-adic
  • Selmen groups

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