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Heegner Points on Cartan Non-split Curves

Published online by Cambridge University Press:  20 November 2018

Daniel Kohen  and
Ariel Pacetti
Daniel Kohen
Affiliation:
IMAS-CONICET, Buenos Aires, Argentina e-mail: kohendaniel@gmail.com
Ariel Pacetti
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, CONICET, Argentina e-mail: apacetti@dm.uba.ar
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Abstract

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Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ , and let $K$ be an imaginary quadratic field such that the root number of $E/K$ is −1. Let $O$ be an order in $K$ and assume that there exists an odd prime $p$ such that ${{p}^{2}}\,\parallel \,N$ , and $p$ is inert in $O$ . Although there are no Heegner points on ${{X}_{0}}(N)$ attached to $O$ , in this article we construct such points on Cartan non-split curves. In order to do that, we give a method to compute Fourier expansions for forms on Cartan non-split curves, and prove that the constructed points form a Heegner system as in the classical case.

Keywords

11G0511F30Cartan curvesHeegner points
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 2 , 01 April 2016 , pp. 422 - 444
Copyright
Copyright © Canadian Mathematical Society 2016

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