Prime Hasse principles via Diophantine second moments

Abstract

We show that almost all primes $p\not\equiv \pm 4\mathrm{mod}9$ are sums of three cubes, assuming a conjecture due to Hooley, Manin, et al. on cubic fourfolds. This conjecture is approachable under standard statistical hypotheses on geometric families of $L$-functions.

22 pages; accepted version