Arithmetic dynamics and Generalized Fermat's conjecture

TL;DR

The paper develops an arithmetic-dynamics framework over function fields by introducing a commuting system of endomorphisms $F$ compatible with an ample line bundle $L$, and a height $h_F$ on $X(K)$ that scales under each $f_N$ via $h_F(f_N(x))=d_N\,h_F(x)$. For individual endomorphisms $f$ compatible with $L$, it constructs a canonical height $h_f$ with $h_f(f(x))=d(f)h_f(x)$ and proves a key lemma connecting finite point sets to a zero-height condition, underpinning finiteness phenomena. The central contribution is the Generalized Fermat's conjecture, linking finite sets $Y_N(K)$ to the eventual Fermat-property of preimages $Y_N=f_N^{-1}(Y)$, with precise refinements for additive and multiplicative systems and density results in the multiplicative case. The work unifies Fermat-type Diophantine finiteness with dynamical heights, providing a pathway to extend finiteness results (à la Faltings) to arithmetic dynamics and to characterize when preimage varieties exhibit height-zero behavior for large $N$.

Abstract

We propose generalized Fermat's conjecture in the framework of arithmetic dynamics, and give evidences.

Theorems & Definitions (16)

• Conjecture 1.1: Generalized Fermat's conjecture
• Theorem 1.2
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 3.1
• proof
• Example 3.2
• Example 3.3