## Abstract

We show that for certain non-CM elliptic curves E/_{Q} such that 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E_{ψ} of E have Mordell–Weil rank one and the 3-adic height pairing on E_{ψ}(Q) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p-adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero.

Original language | American English |
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Pages (from-to) | 13-32 |

Number of pages | 20 |

Journal | Proceedings of the American Mathematical Society, Series B |

Volume | 10 |

DOIs | |

State | Published - 2023 |

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Algebra and Number Theory