A note on rank one quadratic twists of elliptic curves and the non-degeneracy of p-adic regulators at eisenstein primes

Ashay A. Burungale; Christopher Skinner

doi:10.1090/bproc/144

A note on rank one quadratic twists of elliptic curves and the non-degeneracy of p-adic regulators at eisenstein primes

Ashay A. Burungale, Christopher Skinner

Mathematics

Research output: Contribution to journal › Article › peer-review

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	English (US)
Pages (from-to)	13-32
Number of pages	20
Journal	Proceedings of the American Mathematical Society, Series B
Volume	10
DOIs	https://doi.org/10.1090/bproc/144
State	Published - 2023

All Science Journal Classification (ASJC) codes

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

Access to Document

10.1090/bproc/144

Cite this

@article{bf1c4d2e7c6a40acb63eaab2a5742084,

title = "A note on rank one quadratic twists of elliptic curves and the non-degeneracy of p-adic regulators at eisenstein primes",

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 Eψ of E have Mordell–Weil rank one and the 3-adic height pairing on Eψ(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.",

author = "Burungale, {Ashay A.} and Christopher Skinner",

note = "Publisher Copyright: {\textcopyright} 2023 by the author(s) under Creative Commons Attribution-NonCommercial 3.0 License.",

year = "2023",

doi = "10.1090/bproc/144",

language = "English (US)",

volume = "10",

pages = "13--32",

journal = "Proceedings of the American Mathematical Society, Series B",

issn = "2330-1511",

publisher = "American Mathematical Society",

}

TY - JOUR

T1 - A note on rank one quadratic twists of elliptic curves and the non-degeneracy of p-adic regulators at eisenstein primes

AU - Burungale, Ashay A.

AU - Skinner, Christopher

PY - 2023

Y1 - 2023

N2 - 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 Eψ of E have Mordell–Weil rank one and the 3-adic height pairing on Eψ(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.

AB - 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 Eψ of E have Mordell–Weil rank one and the 3-adic height pairing on Eψ(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.

UR - http://www.scopus.com/inward/record.url?scp=85150977633&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85150977633&partnerID=8YFLogxK

U2 - 10.1090/bproc/144

DO - 10.1090/bproc/144

M3 - Article

AN - SCOPUS:85150977633

SN - 2330-1511

VL - 10

SP - 13

EP - 32

JO - Proceedings of the American Mathematical Society, Series B

JF - Proceedings of the American Mathematical Society, Series B

ER -

A note on rank one quadratic twists of elliptic curves and the non-degeneracy of p-adic regulators at eisenstein primes

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this