Vol. 9, No. 10, 2015

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Bounds for Serre's open image theorem for elliptic curves over number fields

Davide Lombardo

Vol. 9 (2015), No. 10, 2347–2395
Abstract

For an elliptic curve $E∕K$ without potential complex multiplication we bound the index of the image of Gal$\left(\overline{K}∕K\right)$ in GL${}_{2}\left(\stackrel{̂}{ℤ}\right)$, the representation being given by the action on the Tate modules of $E$ at the various primes. The bound is explicit and only depends on $\left[K:ℚ\right]$ and on the stable Faltings height of $E$. We also prove a result relating the structure of closed subgroups of GL${}_{2}\left({ℤ}_{\ell }\right)$ to certain Lie algebras naturally attached to them.

Keywords
Galois representations, elliptic curves, Lie algebras, open image theorem
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 11F80, 14K15