Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. We denote the Jacobian of Fermat curve by $F(p)$.
In Gross and Rohrlich's 1978 Invent paper "Some Results on the Mordell-Weil Group of the Jacobian of the Fermat Curve", the introduction says:
Faddeev $[7]$ proved that when $p<7$ the Mordell-Weil group of $J(p)$ over $\mathbb Q$ is finite. Here we show that this group contains points of infinite order whenever $p > 7$; for example, $J(11)$ has rank $6$ over $\mathbb Q$. More precisely, following Faddeev we consider an isogeny of $J(p)$ onto the product of $p-2$ abelian varieties, and we show that when $p \geq 7$ all but possibly $3$ of the factors in this product have infinite Mordell-Weil group. It seems likely that all $p-2$ factors have rational points of infinite order when $p > 11$.
Besides, they can determine solution of fermat equation for small $p$ over small degree number field. More precisely, let $p=3, 5, 7,$ or $11$, if $[K: \mathbb Q] < (p-1)/2$ then $F(p)(K)$ only contains those trivial points possibly along with $(w,w^{-1},1)$ and $(w^{-1},w,1)$ where $w$ is a primitive sixth root of unity).
For the purpose to apply Chabauty-Coleman method to generalize the result, I have a question: when is the rank of $J(p)(\mathbb Q)$ less than $\frac{(p-1)(p-2)}{2}$ i.e the genus of $F(p)$ ?
I suspect the answer is always true, how about the rank of $J(p)(K)$ when $K$ is a quadratic field?
What is the growth of $\text{rank}J(p)(\mathbb Q)$ with respect to $p$? It's at least $O(p)$ by above paper's result.
