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Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a numberfield $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper weshow the existence of infinitely many linearly disjoint degree $d$-extensions$L/K$ over which the Jacobian of $\mathcal{X}$ gains rank. In the case where 0,1 and $\infty$ are the only branch points, and there is an automorphism$\sigma$ of $\mathcal{X}$ which cyclically permutes these branch points, weobtain the same result for the Jacobian of $\mathcal{X}/\sigma$. In particularif $\mathcal{X}$ is the Klein quartic, then the construction provides anelliptic curve which gains rank over infinitely many degree $7$-extensions of$\mathbb{Q}$. As an application, we show the existence of infinitely manyelliptic curves that gain rank over infinitely many cyclic cubic extensions of$\mathbb{Q}$.

Rank gain of Jacobians over finite Galois extensions