Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois. (English) Zbl 1397.11095
In this highly important paper the authors give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over \(\mathbb {Q}\). To this end, they compute the ‘arithmetically maximal’ tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.
Central is the following:
Theorem 1.1. Let \(H\in \mathrm{GL}_2(\mathbb Z_2)\) be a subgroup, and \(E\) be an elliptic curve whose 2-adic image is contained in \(H\). Then one of the following holds:
– The modular curve \(X_H\) has infinitely many rational points.
– The curve \(E\) has complex multiplication.
– The \(j\)-invariant of \(E\) appears in the table 1 of exceptional j-invariants given in the paper.
Further we state their
Corollary 1.3. Let \(E\) be an elliptic curve over \(\mathbb Q\) without complex multiplication. Then the index of \(\rho_{E,2^\infty}(G_{\mathbb Q})\) divides 64 or 96; all such indices occur. Moreover, the image of \(\rho_{E,2^\infty}(G_{\mathbb Q})\) is the inverse image in \( \mathrm{GL}_2(\mathbb Z_2)\) of the image of \(\rho_{E,32}(G_{\mathbb Q})\). For non-CM elliptic curves \(E/\mathbb Q\), there are precisely 1208 possible images for \(\rho_{E,2^\infty}\).
Several interesting remarks show relations with other papers and some are of interest in their own.
Central is the following:
Theorem 1.1. Let \(H\in \mathrm{GL}_2(\mathbb Z_2)\) be a subgroup, and \(E\) be an elliptic curve whose 2-adic image is contained in \(H\). Then one of the following holds:
– The modular curve \(X_H\) has infinitely many rational points.
– The curve \(E\) has complex multiplication.
– The \(j\)-invariant of \(E\) appears in the table 1 of exceptional j-invariants given in the paper.
Further we state their
Corollary 1.3. Let \(E\) be an elliptic curve over \(\mathbb Q\) without complex multiplication. Then the index of \(\rho_{E,2^\infty}(G_{\mathbb Q})\) divides 64 or 96; all such indices occur. Moreover, the image of \(\rho_{E,2^\infty}(G_{\mathbb Q})\) is the inverse image in \( \mathrm{GL}_2(\mathbb Z_2)\) of the image of \(\rho_{E,32}(G_{\mathbb Q})\). For non-CM elliptic curves \(E/\mathbb Q\), there are precisely 1208 possible images for \(\rho_{E,2^\infty}\).
Several interesting remarks show relations with other papers and some are of interest in their own.
Reviewer: Olaf Ninnemann (Uffing am Staffelsee)
MSC:
| 11G05 | Elliptic curves over global fields |
| 11F80 | Galois representations |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
Software:
ell-adic-galois-imagesReferences:
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