Generalised Fermat Equations

• Generalised Fermat equations are Diophantine equations of the form Ax^a + By^b + Cz^c = 0, where the search for primitive solutions drives key research questions.
• They are examined particularly in the hyperbolic case (1/a + 1/b + 1/c < 1), linking the equations to modular forms through Frey varieties and modularity theorems.
• Researchers employ methods such as descent theory, level lowering, and Galois representation analysis to uncover the asymptotic behavior and finiteness of solutions.

A generalized Fermat equation is a Diophantine equation of the form

$A x^a + B y^b + C z^c = 0,$

where $A,B,C \in \mathbb{Z} \setminus \{0\}$ and $a,b,c \geq 2$ are fixed exponents. The study of such equations encompasses questions about the existence, quantity, and nature of primitive (i.e., $\gcd(x, y, z) = 1$) integral solutions, and leads to rich interconnections between Diophantine geometry, arithmetic of elliptic curves and abelian varieties, modularity theorems, descent theory, and explicit computation of modular forms.

1. Forms and Signatures of Generalized Fermat Equations

A generalized Fermat equation (GFE) can take a variety of forms, the most classical being the symmetric exponent version $x^p + y^p = z^p$. The signature is the triple $(a, b, c)$ of exponents. The literature primarily investigates signatures with $1/a + 1/b + 1/c < 1$ (the "hyperbolic" case), a setting conjectured by Darmon–Granville to admit only finitely many primitive solutions for fixed coefficients and exponents (Ratcliffe et al., 2024).

The principal families of interest, up to variable permutation, are:

• $(p,p,p)$: classical Fermat equations,
• $(p,p,r)$ and $(r,r,p)$: two exponents fixed, one varying,
• $(q,r,p)$: mixed exponents.

For each, research has focused on both existence (finiteness and explicit determination of all primitive solutions) and asymptotic behavior (for varying exponents or coefficients) (Sahoo, 4 Dec 2025, Siksek et al., 2013, Billerey et al., 2017).

2. Methodologies: Modularity and Descent

There are two main, highly developed methodologies:

Modularity and the "Modular Method"

For signatures $(p, p, r)$ and $(r, r, p)$ with large exponents, the modular method is fundamental. A putative primitive solution gives rise to a Frey curve or a higher-dimensional GL$_2$-type abelian variety (the "Frey variety"), whose $p$-torsion Galois representation is studied.

The steps are (Chen et al., 20 Jul 2025, Sahoo, 4 Dec 2025, Azon, 19 Mar 2025):

1. Frey Variety: Attach an elliptic curve or Jacobian of a hyperelliptic curve over a suitable number field $K$.
2. Modularity: Prove modularity (often via base change and modern automorphy lifting theorems) so the Galois representation comes from a (Hilbert or Bianchi) modular form/newform.
3. Irreducibility: Prove the residual representation is irreducible for large $p$.
4. Level Lowering: Relate the mod $p$ Galois representation to an explicit space of modular forms using a level-lowering theorem.
5. Elimination: Show there is no compatible newform, typically via trace-of-Frobenius/inertia calculations at auxiliary primes.

A crucial innovation is the use of *Frey hyperelliptic curves