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Mathematics of Computation

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On $ L$-functions of quadratic $ \mathbb{Q}$-curves

Authors: Peter Bruin and Andrea Ferraguti
Journal: Math. Comp.
MSC (2010): Primary 11G05, 11G40, 11F30
Published electronically: June 13, 2017
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Abstract: Let $ K$ be a quadratic number field and let $ E$ be a $ \mathbb{Q}$-curve without CM completely defined over $ K$ and not isogenous to an elliptic curve over $ \mathbb{Q}$. In this setting, it is known that there exists a weight $ 2$ newform of suitable level and character, such that $ L(E,s)=L(f,s)L({}^\sigma \! f,s)$, where $ {}^\sigma \! f$ is the unique Galois conjugate of $ f$. In this paper, we first describe an algorithm to compute the level, the character and the Fourier coefficients of $ f$. Next, we show that given an invariant differential $ \omega _E$ on $ E$, there exists a positive integer $ Q=Q(E,\omega _E)$ such that $ L(E,1)/P(E/K)\cdot Q$ is an integer, where $ P(E/K)$ is the period of $ E$. Assuming a generalization of Manin's conjecture, the integer $ Q$ is made effective. As an application, we verify the weak BSD conjecture for some curves of rank two, we compute the $ L$-ratio of a curve of rank zero and we produce relevant examples of newforms of large level.

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Additional Information

Peter Bruin
Affiliation: Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, Netherlands

Andrea Ferraguti
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
Address at time of publication: DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Keywords: Number fields, $\mathbb{Q}$-curves, $L$-functions, newforms, BSD
Received by editor(s): March 21, 2016
Received by editor(s) in revised form: September 11, 2016
Published electronically: June 13, 2017
Additional Notes: The first author was partially supported by the Swiss National Science Foundation through grants 124737 and 137928, and by the Netherlands Organisation for Scientific Research (NWO) through Veni grant 639.031.346.
The second author was partially supported by Swiss National Science Foundation grant 168459
Article copyright: © Copyright 2017 American Mathematical Society