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.

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    $\begingroup$ If you want to apply Chabauty's method, it is usually better to work with one of the abelian varieties occurring as a factor of $J(p)$, for example the Jacobian of $y^2 = 4 x^p + 1$ (the curve is a quotient of $F(p)$, so knowing its rational points is enough). It is possible to use a variant of Chabauty's method without actually knowing the rank exactly, see Section 7 in mathe2.uni-bayreuth.de/stoll/schrift.html#AG54. This approach should generalize to symmetric powers of the curve, which would give information on higher-degree points. $\endgroup$ – Michael Stoll Feb 14 '19 at 17:58

The rank can be estimated and Chabauty's method applied assuming a conjecture about cyclotomic fields which is still open:

McCallum, William G. On the method of Coleman and Chabauty. Math. Ann. 299 (1994), no. 3, 565–596.

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    $\begingroup$ Thank you, I didn't notice that paper, it seems for regular primes the bound is true. $\endgroup$ – sawdada Feb 15 '19 at 0:08

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