Doubly isogenous curves of genus two with a rational action of $D_6$

Abstract: Let $C$ and $C'$ be curves over a finite field $K$, provided with embeddings $\iota$ and $\iota'$ into their Jacobian varieties. Let $D\to C$ and $D'\to C'$ be the pullbacks (via these embeddings) of the multiplication-by-$2$ maps on the Jacobians. We say that $(C,\iota)$ and $(C',\iota')$ are \emph{doubly isogenous} if $\mathrm{Jac}(C)$ and $\mathrm{Jac}(C')$ are isogenous over $K$ and $\mathrm{Jac}(D)$ and $\mathrm{Jac}(D')$ are isogenous over~$K$. When we restrict attention to the case where $C$ and $C'$ are curves of genus $2$ whose groups of $K$-rational automorphisms are isomorphic to the dihedral group $D_6$ of order $12$, we find many more doubly isogenous pairs than one would expect from reasonable heuristics. Our analysis of this overabundance of doubly isogenous curves over finite fields leads to the construction of a pair of doubly isogenous curves over a number field. That such a global example exists seems extremely surprising. We show that the Zilber--Pink conjecture implies that there can only be finitely many such examples. When we exclude reductions of this pair of global curves in our counts, we find that the data for the remaining curves is consistent with our original heuristic. Computationally, we find that doubly isogenous curves in our family of $D_6$ curves can be distinguished from one another by considering the isogeny classes of the Prym varieties of certain unramified covers of exponent $3$ and $4$. We discuss how our family of curves can be potentially be used to obtain a deterministic polynomial-time algorithm to factor univariate polynomials over finite fields via an argument of Kayal and Poonen.