We prove that $164\, 634\, 913$ is the smallest positive integer that is a sum of two rational sixth powers, but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$ , we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell–Weil sieve, to rule out the existence of rational points on $C_{k}$ for various k.

Let $C$ denote the Fermat curve over $\mathbb{Q}$ of prime exponent $l$ . The Jacobian $\text{Jac(}C\text{)}$ of $C$ splits over $\mathbb{Q}$ as the product of Jacobians $\text{Jac(}{{C}_{k}})$ , $1\,\le \,k\,\le \,\ell \,-\text{2}$ , where ${{C}_{k}}$ are curves obtained as quotients of $C$ by certain subgroups of automorphisms of $C$ . It is well known that $\text{Jac(}{{C}_{k}}\text{)}$ is the power of an absolutely simple abelian variety ${{B}_{k}}$ with complex multiplication. We call degenerate those pairs $(l,\,k)$ for which ${{B}_{k}}$ has degenerate $\text{CM}$ type. For a non-degenerate pair $(l,\,k)$ , we compute the Sato–Tate group of $\text{Jac(}{{C}_{k}}\text{)}$ , prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the moments and measures of the involved distributions. Regardless of whether $(l,\,k)$ is degenerate, we also obtain Frobenius equidistribution results for primes of certain residue degrees in the $l$ -th cyclotomic field. Key to our results is a detailed study of the rank of certain generalized Demjanenko matrices.

In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

Let X be the Fermat curve of degree q+1 over the field k of q$^2$ elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.