Open Access
April, 2008 A moduli approach to quadratic $\mathbb{Q}$-curves realizing projective mod $p$ Galois representations
Julio Fernández
Rev. Mat. Iberoamericana 24(1): 1-30 (April, 2008).


For a fixed odd prime $p$ and a representation $\varrho$ of the absolute Galois group of $\mathbb{Q}$ into the projective group ${\rm PGL}_2(\mathbb{F}_p)$, we provide the twisted modular curves whose rational points supply the quadratic $\mathbb{Q}$-curves of degree $N$ prime to $p$ that realize $\varrho$ through the Galois action on their $p$-torsion modules. The modular curve to twist is either the fiber product of $X_0(N)$ and $X(p)$ or a certain quotient of Atkin-Lehner type, depending on the value of $N$ mod $p$. For our purposes, a special care must be taken in fixing rational models for these modular curves and in studying their automorphisms. By performing some genus computations, we obtain as a by-product some finiteness results on the number of quadratic $\mathbb{Q}$-curves of a given degree $N$ realizing $\varrho$.


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Julio Fernández . "A moduli approach to quadratic $\mathbb{Q}$-curves realizing projective mod $p$ Galois representations." Rev. Mat. Iberoamericana 24 (1) 1 - 30, April, 2008.


Published: April, 2008
First available in Project Euclid: 16 July 2008

zbMATH: 1206.11077
MathSciNet: MR2435964

Primary: 11F80 , 11G05 , 11G15 , 14G35 , 14H10 , 14H52 , 14K10

Keywords: $p$-torsion points , Elliptic curves , mod $p$ Galois representations , moduli problem , quadratic $\mathbb{Q}$-curves , twisted modular curves

Rights: Copyright © 2008 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.24 • No. 1 • April, 2008
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