On the solutions of the generalized Fermat equation over totally real number fields
Satyabrat Sahoo
TL;DR
This work extends the modular approach for generalized Fermat-type equations to totally real number fields with even $ABC$, proving nonexistence of asymptotic solutions in both $W_K$ and, under further hypotheses, in $K^3$ for the equation $Ax^p+By^p+Cz^p=0$. Central to the method are Frey curves attached to potential solutions, modularity results for large primes $p$, irreducibility of mod-$p$ Galois representations, and level-lowering arguments; the analysis crucially uses $S_K'$-unit bounds and connections to $S_K'$-unit equations. The paper also provides several purely local criteria on $K$ guaranteeing no asymptotic solutions and shows a density-1 result for real quadratic fields, meaning that almost all such fields satisfy the no-solution conclusions for a broad class of coefficients. Together, these results advance the understanding of asymptotic Diophantine behavior over number fields and yield practical criteria and density statements that complement existing results over $\\mathbb{Q}$. The methods bridge modular forms, Galois representations, and unit equations to establish strong nonexistence results in a broad arithmetic setting.
Abstract
Let $K$ be a totally real number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$ with prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even. For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2|abc$. Then, under some assumptions on $A,B,C$, we also prove that $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field.