## Heights of points on elliptic curves over $\mathbb {Q}$

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- by Michael Griffin, Ken Ono and Wei-Lun Tsai PDF
- Proc. Amer. Math. Soc.
**149**(2021), 5093-5100 Request permission

## Abstract:

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb {Q})$ by making use of suitable elliptic curve ideal class pairings \begin{equation*} \Psi _{E,-D}: \ E(\mathbb {Q})\times E_{-D}(\mathbb {Q})\mapsto \mathrm {CL}(-D). \end{equation*} In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove \begin{equation*} \widehat {h}(P)> \frac {|E_{\mathrm {tor}}(\mathbb {Q})|^2}{\left ( H(-D)+ |E_{\mathrm {tor}}(\mathbb {Q})|\right )^2}\cdot T_E(-D). \end{equation*}## References

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## Additional Information

**Michael Griffin**- Affiliation: Department of Mathematics, 275 TMCB, Brigham Young University, Provo, Utah 84602
- MR Author ID: 943260
- ORCID: 0000-0002-9014-3210
- Email: mjgriffin@math.byu.edu
**Ken Ono**- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 342109
- Email: ken.ono691@virginia.edu
**Wei-Lun Tsai**- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 1305416
- ORCID: 0000-0002-8747-5230
- Email: tsaiwlun@gmail.com
- Received by editor(s): July 18, 2020
- Received by editor(s) in revised form: January 15, 2021, and March 28, 2021
- Published electronically: September 24, 2021
- Additional Notes: The second author was supported by the NSF (DMS-2002265 and DMS-2055118) and the UVa Thomas Jefferson fund
- Communicated by: Matthew A. Papanikolas
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 5093-5100 - MSC (2020): Primary 11G05, 11G50
- DOI: https://doi.org/10.1090/proc/15605
- MathSciNet review: 4327417