[Paper Review] Modularity and the Fermat Equation over Totally Real Fields

Nuno Freitas, Samir Siksek|arXiv (Cornell University)|Jul 11, 2013

History and Theory of Mathematics4 citations

TL;DR

This paper establishes a criterion for the asymptotic Fermat's Last Theorem over totally real fields using modularity, level lowering, and image of inertia comparisons. It proves the theorem holds for $\mathbb{Q}(\sqrt{d})$ with squarefree $d$ of density $5/6$, and density 1 under the Eichler-Shimura conjecture, advancing the number field analog of the classical Fermat equation.

ABSTRACT

Let $K$ be a totally real field. By the asymptotic Fermat's Last Theorem over $K$ we mean the statement that there is a constant $B_K$ such that for prime exponents $p>B_K$ the only solutions to the Fermat equation $a^p + b^p + c^p = 0$ with $a$, $b$, $c$ in $K$ are the trivial ones satisfying $abc = 0$. With the help of modularity, level lowering and image of inertia comparisons we give an algorithmically testable criterion which if satisfied by $K$ implies the asymptotic Fermat's Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $K = \mathbb{Q}(\sqrt{d})$ for a subset of $d$ having density $5/6$ among the squarefree positive integers. We can improve this to density 1 if we assume a standard Eichler-Shimura conjecture.

Motivation & Objective

• To establish a testable criterion for the asymptotic Fermat's Last Theorem over totally real fields using modular forms and Galois representations.
• To determine the density of totally real quadratic fields $\mathbb{Q}(\sqrt{d})$ for which the asymptotic Fermat equation has only trivial solutions.
• To extend the modularity approach from $\mathbb{Q}$ to arbitrary totally real fields, leveraging level lowering and image of inertia arguments.
• To use analytic number theory techniques to verify the criterion for a positive density of quadratic fields.
• To improve the density result under the assumption of the Eichler-Shimura conjecture.

Proposed method

• Apply modularity theorems to associate Galois representations to modular forms over totally real fields.
• Use level lowering techniques to reduce the conductor of the associated Galois representation to a form compatible with the Fermat equation.
• Compare the image of inertia groups at primes above $p$ in the Galois representation to constrain possible solutions to $a^p + b^p + c^p = 0$.
• Construct an algorithmically testable criterion based on the image of inertia and modularity properties.
• Apply analytic number theory to estimate the density of squarefree $d$ for which the criterion holds.
• Assume the Eichler-Shimura conjecture to strengthen the density bound from $5/6$ to $1$.

Experimental results

Research questions

• RQ1 For which totally real fields $K$ does the asymptotic Fermat's Last Theorem hold, i.e., are there only trivial solutions to $a^p + b^p + c^p = 0$ for large prime $p$?
• RQ2 Can a criterion based on modularity and Galois image properties be formulated and tested algorithmically for such fields?
• RQ3 What is the natural density of squarefree $d$ for which the asymptotic Fermat equation holds over $\mathbb{Q}(\sqrt{d})$?
• RQ4 How does the Eichler-Shimura conjecture affect the density of fields satisfying the asymptotic Fermat condition?
• RQ5 Can the modularity method be extended from $\mathbb{Q}$ to general totally real fields to resolve the Fermat equation?

Key findings

• The paper establishes a testable criterion for the asymptotic Fermat's Last Theorem over any totally real field using modularity and Galois representation techniques.
• For $K = \mathbb{Q}(\sqrt{d})$ with $d$ squarefree and positive, the criterion holds for a subset of $d$ of natural density $5/6$.
• Under the assumption of the Eichler-Shimura conjecture, the density of such $d$ for which the asymptotic Fermat's Last Theorem holds increases to $1$.
• The method relies on comparing the image of inertia in Galois representations to rule out nontrivial solutions to the Fermat equation.
• The algorithmic nature of the criterion allows for computational verification in specific cases.
• The results extend the modularity approach beyond $\mathbb{Q}$ to arbitrary totally real fields, providing a framework for future generalizations.