The Arithmetic of Elliptic Pairs and An $\nu +1$-variable Artin Conjecture

The theory of elliptic pairs, as investigated in a paper by Castravet, Laface, Tevelev, and Ugaglia, provides useful conditions to determine polyhedrality of the pseudo-effective cone, which give rise to interesting arithmetic questions when reducing the variety modulo $p$. In this paper, we examine one such case, namely the blow-up $X$ of 9 points in ${\mathbb{P}}^2$ lying on the nodal cubic, and study the density of primes $p$ for which the pseudo-effective cone of the reduction of $X$ modulo $p$ is polyhedral. This problem reduces to an analogue of Artin’s Conjecture on primitive roots like that investigated by Stephens and then Moree and Stevenhagen. As a result, we find that the density of such “polyhedral primes” hover around a higher analogue of the Stephens’ Constant under the assumption of the Generalized Riemann Hypothesis. Finally, in order to determine a precise value for the density of polyhedral primes, we look at the containment of rank 8 root sublattices of ${\mathbb{E}}_8$.