Prime Hasse principles via Diophantine second moments

TL;DR

The paper addresses the Hasse-principle question for representations by $F_0(y)=y_1^3+y_2^3+y_3^3$ through a Diophantine second-moment framework. It develops a six-variable variance approach that splits into non-archimedean and archimedean analyses, introduces truncated singular-series models $s_a(K)$, and proves that these secondary terms are typically large, enabling a probabilistic control of the exceptional set. Conditional on Hooley–Manin-type conjectures for cubic fourfolds and related $L$-function hypotheses, the authors deduce that almost all primes $p$ with $p\not\equiv \pm 4 \bmod 9$ are sums of three cubes; they also obtain (under these hypotheses) analogous results for integer values with a careful choice of weights that amplify archimedean contributions. The work further develops the machinery to handle nonnegative-cube representations and poses natural open questions about extensions and unconditional barriers, highlighting the potential of a second-moment, density-based approach in Diophantine problems tied to higher-dimensional geometry.

Abstract

We show that almost all primes $p\not\equiv \pm 4 \bmod{9}$ 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.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof