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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

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Clarification of argument in "Elliptic curves over $\mathbb{Q}_{\infty}$ are modular"

In https://arxiv.org/abs/1505.04769 in the proof of Theorem 5 it is asserted that since $\rho_{E, l}:G_\mathbb{Q}\to\mathrm{GL}_2(\mathbb{Z}_l)$ is surjective then $E_{\mathbb{Q}_\infty}[l^\infty]=0$. …