# rank of Jacobian of Fermat curve and Chabauty-Coleman method

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 $$$$ proved that when $$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.

• 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. – Michael Stoll Feb 14 '19 at 17:58