On $p$-adic solubility of $A{x}^{\ell }+B{y}^m+C{z}^n=0$

Abstract

We study $p$-adic solubility of generalized Fermat equations $A{x}^{\ell }+B{y}^m+C{z}^n=0$ for positive integers $\ell ,m,n$. For all but finitely many primes $p$, the probability of having a $p$-adic solution is described by a rational function in $p$ depending only on $\gcd (p-1,\ell ,m)$, $\gcd (p-1,\ell ,n)$, and $\gcd (p-1,m,n)$. When $\ell ,m,n$ are pairwise coprime, we deduce that the proportion of these equations which are everywhere locally soluble is positive, given by a product of these local probabilities; when $\ell ,m,n$ are not pairwise coprime, the proportion is 0\%. We then give several detailed examples demonstrating the explicit nature of the results.

Main article by Keyes and Kobin; joint appendix with Arango-Piñeros