Hilbert's tenth problem via additive combinatorics

TL;DR

The paper establishes an unconditional negative answer to Hilbert's tenth problem for every infinite ring $R$ finitely generated over $ Z$, by constructing elliptic curves $E$ with full rational $E[2]$-torsion and proving rank growth in carefully chosen quadratic extensions $L/K$. The authors introduce a novel blend of $2$-descent on curves with $E(K)[2] eq 0$ and additive combinatorics (via Kai's four-linear forms results) to prescribe twists $t$ that force the $2$-Selmer rank to stay small over $K$ while ensuring positive rank over $L$. A detailed transition-process framework tracks changes in the $2$-Selmer rank under controlled twists, and a constructive auxiliary twist $oldsymbol{\ kappa}$ combined with auxiliary primes achieves the required local-global data. This approach yields unconditional rank-growth in quadratic extensions, enabling Diophantine encodings that transfer undecidability from $ Z$ to a wide class of rings, and it links arithmetic statistics with Diophantine geometry in a new way with potential broader impact on decidability questions across number fields.

Abstract

For all infinite rings $R$ that are finitely generated over $\mathbb{Z}$, we show that Hilbert's tenth problem has a negative answer. This is accomplished by constructing elliptic curves $E$ without rank growth in certain quadratic extensions $L/K$. To achieve such a result unconditionally, our key innovation is to use elliptic curves $E$ with full rational $2$-torsion which allows us to combine techniques from additive combinatorics with $2$-descent.

Theorems & Definitions (86)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• Corollary 2.5