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Local solubility of generalised Fermat equations

Peter Koymans, Ross Paterson, Tim Santens, Alec Shute

TL;DR

The paper determines the asymptotic count of coefficient triples $(a,b,c)$ with bounded size for which the diagonal generalised Fermat equation $a x^n + b y^n + c z^n = 0$ is soluble everywhere locally. It develops a robust framework of local solubility criteria, unfolds the local indicator into a character-sum structure, and isolates the main term using a bilinear-sieve approach combined with Siegel–Walfisz-type bounds, ultimately producing an explicit leading constant $C_n$ and exponent $\alpha_n$. The results verify the Loughran–Smeets and Loughran–Rome–Sofos conjectures for this diagonal family, and yield corollaries such as sharp upper bounds for equations with rational solutions. The leading constant is expressed via local and real densities and Tamagawa measures, connecting arithmetic statistics with Manin–Peyre-type predictions and confirming the conjectural framework in this natural Diophantine setting.

Abstract

For every $n \geq 2$ we determine the asymptotic formula for the number of integer triples $(a,b,c)$ of bounded absolute value such that the generalised Fermat equation given by $ax^n+by^n+cz^n=0$ is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations.

Local solubility of generalised Fermat equations

TL;DR

The paper determines the asymptotic count of coefficient triples with bounded size for which the diagonal generalised Fermat equation is soluble everywhere locally. It develops a robust framework of local solubility criteria, unfolds the local indicator into a character-sum structure, and isolates the main term using a bilinear-sieve approach combined with Siegel–Walfisz-type bounds, ultimately producing an explicit leading constant and exponent . The results verify the Loughran–Smeets and Loughran–Rome–Sofos conjectures for this diagonal family, and yield corollaries such as sharp upper bounds for equations with rational solutions. The leading constant is expressed via local and real densities and Tamagawa measures, connecting arithmetic statistics with Manin–Peyre-type predictions and confirming the conjectural framework in this natural Diophantine setting.

Abstract

For every we determine the asymptotic formula for the number of integer triples of bounded absolute value such that the generalised Fermat equation given by is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations.

Paper Structure

This paper contains 34 sections, 43 theorems, 208 equations, 1 table.

Table of Contents

  1. Introduction
  2. Local solubility
  3. Definitions and notation
  4. Hensel's Lemma
  5. The character sum
  6. Unfolding the indicator function
  7. Change of variables
  8. The auxiliary function f
  9. The solubility criteria
  10. Rewriting the sum
  11. Isolating the main term
  12. Analytic preliminaries
  13. Elementary bounds
  14. Statement of the bilinear sieve
  15. Statement of Siegel--Walfisz
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $n > 1$ be an integer. Then as $B\to\infty$ where, writing $\phi$ for Euler's phi function, and $C_n > 0$ is an explicit constant which is independent of $B$.

Theorems & Definitions (88)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Example 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Hensel's lemma
  • ...and 78 more