Fermat quotients with CM Jacobians and trivial automorphism group

Abstract

Let $F_n$ be the Fermat curve of degree $n$ over $\mathbb Q$, and let $D_n=F_n/S_3$ be its quotient by coordinate permutations. We prove that $D_n$ has trivial geometric automorphism group for every $n\ge 11$ with $\gcd(n,6)=1$. Its Jacobian has complex multiplication. For $n=11$, this Jacobian is absolutely simple of dimension five. We construct an isogeny over $\mathbb Q$ of degree $11^5$ from $\operatorname{Jac}(D_{11})$ to the Jacobian of $y^{11}=x(1-x)^2$. For primes $p\equiv 1\pmod 3$, we also consider the metacyclic quotients $C_p$ of $F_p$ over $\mathbb Q$. By a theorem of Carocca, Lange and Rodríguez, their Jacobians are absolutely simple and have complex multiplication. We prove that $C_p$ has trivial geometric automorphism group if and only if $p\ge 19$. Write $p=r^2+rs+s^2$ with $r,s$ positive integers. The natural degree-three cover $Y_p\to C_p$ satisfies $\operatorname{gon}(Y_p)=r+s$, so $\operatorname{gon}(C_p)\ge \lceil(r+s)/3\rceil$. Since $g(C_p)=(p-1)/6$, both the genus and the gonality of $C_p$ tend to infinity with $p$.