On Larsen's conjecture on the ranks of Elliptic Curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=⟨{\sigma }_1,\dots ,{\sigma }_n⟩$ be a finitely generated subgroup of $\mathrm{Gal}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})$. Larsen’s conjecture claims that the rank of the Mordell-Weil group $E({\bar{\mathbb{Q}}}^G)$ is infinite where ${\bar{\mathbb{Q}}}^G$ is the $G$-fixed sub-field of $\bar{\mathbb{Q}}$. In this paper we prove the conjecture for the case in which ${\sigma }_i$ for each $i=1,\dots ,n$ is an element of some infinite families of elements of $\mathrm{Gal}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})$.