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\(\mathbb{Q}\)-curves, Hecke characters, and some Diophantine equations. II. (English) Zbl 1527.11027

In the article [Part I, Math. Comput. 91, No. 338, 2817–2865 (2022; Zbl 1503.11072)], the authors presented a general procedure to study solutions of the equations \(x^4 - dy^2 = z^p\), where \(d\) is a negative integer. The novelty was to use the theory of Hecke characters over imaginary quadratic fields.
The purpose of the present article is to extend their previous results to positive values of \(d\). The authors prove the existence of Hecke characters over real quadratic fields with prescribed local behaviour in section 2 (Theorem 2.1). In the fourth section, they give an explicit version of Ellenberg’s large image result (regarding images of Galois representations coming from \(\mathbb Q\)-curves over imaginary quadratic fields [J. S. Ellenberg, Am. J. Math. 126, No. 4, 763–787 (2004; Zbl 1059.11041)]) for real quadratic fields where the prime \(2\) splits. The last section contains the examples for small values of \(d\) (\(1\leq d \leq 20\) and \(d=129\)); see Theorems 5.1 to 5.5.

MSC:

11D41 Higher degree equations; Fermat’s equation
11F80 Galois representations

Software:

PARI/GP; Magma
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References:

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