#### Vol. 9, No. 10, 2015

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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