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On an elliptic curve defined over \(\mathbb{Q}(\sqrt -23)\). (English. Russian original) Zbl 1335.11042

St. Petersbg. Math. J. 24, No. 4, 575-589 (2013); translation from Algebra Anal. 24, No. 4, 64-83 (2012).
Let \(K\) be a number field and \(\mathbb A\) the adele ring of \(K\). An elliptic curve \(E\) defined over \(K\) is modular if there exists an automorphic cusp eigenform \(f\) of \(\mathrm{GL}_2(\mathbb A)\) such that the \(\mathfrak p\)-component of \(L\)-function of \(E\) is equal to that of \(L\)-function of \(f\) for any prime ideal \(\mathfrak p\) of \(K\). In the cases that \(K=\mathbb Q\) or \(K\) is a totally real number field, the modularity of elliptic curves over \(K\) have been proved by A. Wiles and others. However, in the case that \(K\) is an imaginary quadratic field, the modularity of elliptic curves without complex multiplication has been shown only for three elliptic curves defined over \(\mathbb Q(\sqrt{-23})\) in [the first author et al., Math. Comput. 79, No. 270, 1145–1170 (2010; Zbl 1227.11073)].
In this article, the authors show that the elliptic curve \(E\) defined by the equation \(y^2+(\omega+1)xy+y=x^3+(\omega+1)x^2-7x+(5-3\omega)\), (\(\omega=(1+\sqrt{-23})/2\)) is modular, by using the algorithm to check the modularity in the article cited above. The automorphic cusp eigenform \(f\) associated with \(E\) is \(f_{44}\) in M. Lingham’s list [Modular forms and elliptic curves over imaginary quadratic fields. Nottingham: University of Nottingham (PhD Thesis) (2005)]. In particular, \(E\) satisfies Hasse-Weil conjecture.

MSC:

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Citations:

Zbl 1227.11073

Software:

PARI/GP
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Full Text: DOI

References:

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