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Twists of Shimura Curves

  Published:2013-09-04
 Printed: Aug 2014
  • James Stankewicz,
    Mathematics and Computer Science, Wesleyan University, Middletown, CT, USA
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Abstract

Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livné on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.
Keywords: Shimura curves, complex multiplication, modular curves, elliptic curves Shimura curves, complex multiplication, modular curves, elliptic curves
MSC Classifications: 11G18, 14G35, 11G15, 11G10 show english descriptions Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
Complex multiplication and moduli of abelian varieties [See also 14K22]
Abelian varieties of dimension $> 1$ [See also 14Kxx]
11G18 - Arithmetic aspects of modular and Shimura varieties [See also 14G35]
14G35 - Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
11G15 - Complex multiplication and moduli of abelian varieties [See also 14K22]
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
 

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