Composite images of Galois for elliptic curves over \(\mathbb {Q}\) and entanglement fields. (English) Zbl 1470.11154
Summary: Let \( E\) be an elliptic curve defined over \( \mathbb {Q}\) without complex multiplication. For each prime \( \ell \), there is a representation \( \rho _{E,\ell }\colon \mathrm{Gal}(\overline {\mathbb {Q}}/\mathbb {Q}) \rightarrow \mathrm{GL}_2(\mathbb {Z}/\ell \mathbb {Z})\) that describes the Galois action on the \( \ell \)-torsion points of \( E\). Building on recent work of J. Rouse and D. Zureick-Brown [Res. Number Theory 1, Paper No. 12, 34 p. (2015; Zbl 1397.11095)] and D. Zywina [Bull. Lond. Math. Soc. 42, No. 5, 811–826 (2010; Zbl 1221.11136)], we find models for composite level modular curves whose rational points classify elliptic curves over \( \mathbb {Q}\) with simultaneously non-surjective, composite images of Galois. We also provably determine the rational points on almost all of these curves. Finally, we give an application of our results to the study of entanglement fields.
MSC:
| 11G05 | Elliptic curves over global fields |
| 11D45 | Counting solutions of Diophantine equations |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11F80 | Galois representations |
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