Darmon's Program: Modular Approach for Fermat Equations

• Darmon's Program is a modular method extending classical techniques by replacing Frey elliptic curves with higher-dimensional abelian varieties in generalized Fermat equations.
• The strategy attaches a Frey object to a candidate solution and employs modularity, residual irreducibility, and level lowering to eliminate non-trivial cases.
• Recent advances have unified approaches for signatures (p,p,r) and (r,r,p), enabling effective computational elimination and deeper insights into GL2-type representations.

Searching arXiv for papers on Darmon’s Program and generalized Fermat equations. Darmon’s Program is a modular strategy for families of generalized Fermat equations with one varying prime exponent. In its modern form, it treats equations such as

$$x^p+y^p=z^r,\qquad x^r+y^r=z^p,\qquad A x^r+B y^q=C z^p,$$

by attaching to a putative primitive non-trivial solution a Frey object whose residual $2$-dimensional Galois representations can be studied by modularity, irreducibility, level lowering, and explicit elimination. Its distinctive feature is that, beyond the classical modular method, the natural Frey object is often not an elliptic curve but a higher-dimensional abelian variety of $\mathrm{GL}_2$-type over a totally real field, typically a Jacobian of a hyperelliptic curve (Chen et al., 20 Jul 2025).

1. Historical formulation and scope

Darmon formulated the program in 2000 as an extension of the Wiles-style modular method from Frey elliptic curves over $\mathbf Q$ to abelian varieties of $\mathrm{GL}_2$-type over totally real fields. The motivating problem is the generalized Fermat equation

$x^p+y^q=z^r,$

especially in one-parameter families where one exponent varies and the others are fixed. After permuting variables, the principal signatures are $(p,p,r)$, $(r,r,p)$, and $(q,r,p)$. The guiding philosophy is unchanged from the classical case: construct a Frey object from a putative primitive non-trivial solution, prove modularity of its $p$-adic representation, show residual irreducibility, lower the level, and derive a contradiction from the scarcity of Hilbert newforms at the resulting Serre level (Billerey et al., 2022).

What makes the program genuinely different from the classical modular method is the geometry of the natural Frey objects. For signatures such as $2$0 with $2$1, Darmon predicted that one must leave the elliptic setting and work with Jacobians of hyperelliptic curves. The 2025 survey emphasizes that the general framework was developed first for $2$2 and $2$3, while later work extends the same philosophy systematically to signature $2$4 by means of a Frey hyperelliptic curve whose Jacobian is of $2$5-type over $2$6 (Chen et al., 20 Jul 2025, García et al., 4 May 2026).

2. Frey abelian varieties and Frey representations

A central formal notion is that of a Frey representation. For odd primes $2$7, points $2$8, and $2$9, a Frey representation of signature $\mathrm{GL}_2$0 with respect to $\mathrm{GL}_2$1 is a representation

$\mathrm{GL}_2$2

whose projectivization is unramified outside $\mathrm{GL}_2$3 and whose inertia images at those points have orders $\mathrm{GL}_2$4. In the signatures relevant to Darmon’s Program, these representations arise from Jacobians $\mathrm{GL}_2$5 of hyperelliptic curves with real multiplication, so that for each $\mathrm{GL}_2$6 one has a $\mathrm{GL}_2$7-dimensional representation

$\mathrm{GL}_2$8

The determinant is cyclotomic: $\mathrm{GL}_2$9 which is decisive for modularity and for lowering to Hilbert newforms of trivial character (Chen et al., 20 Jul 2025).

For signature $\mathbf Q$0, Darmon’s hyperelliptic families are built from

$\mathbf Q$1

and

$\mathbf Q$2

The associated curves are

$\mathbf Q$3

with Jacobians $\mathbf Q$4. For signature $\mathbf Q$5, one also has Kraus’ hyperelliptic curves $\mathbf Q$6 and Freitas’ Frey elliptic curves over $\mathbf Q$7. A major conceptual advance of the effective 2025 framework is that the Frey hyperelliptic curves for both $\mathbf Q$8 and $\mathbf Q$9 can be viewed as quadratic twists of specializations of a single family $\mathrm{GL}_2$0, yielding a uniform treatment of both signatures (Azon, 19 Mar 2025).

The $\mathrm{GL}_2$1-type condition is the mechanism that keeps the modular method two-dimensional despite the higher dimension of the Frey variety. For the Jacobians $\mathrm{GL}_2$2 and related families, one has real multiplication by the maximal totally real subfield $\mathrm{GL}_2$3, and the Tate module decomposes into $\mathrm{GL}_2$4-dimensional $\mathrm{GL}_2$5-adic pieces. This is why higher-dimensional Jacobians can still be compared with Hilbert modular forms of parallel weight $\mathrm{GL}_2$6, exactly as elliptic curves are compared with classical modular forms (Azon, 19 Mar 2025).

3. Modular-method architecture

The program is usually organized into five steps. First, one constructs a Frey curve or Frey abelian variety $\mathrm{GL}_2$7 from a putative primitive non-trivial solution. Second, one proves modularity of $\mathrm{GL}_2$8. Third, one proves absolute irreducibility of the residual representation

$\mathrm{GL}_2$9

Fourth, one lowers the level to obtain

$x^p+y^q=z^r,$0

for a Hilbert newform $x^p+y^q=z^r,$1 of parallel weight $x^p+y^q=z^r,$2, trivial character, and controlled level. Fifth, one eliminates all such $x^p+y^q=z^r,$3 by comparing Frobenius traces, local conductor exponents, inertial types, or coefficient-field constraints (Billerey et al., 2022).

The modularity step is no longer purely formal, but modern modularity lifting results make it viable in many of the cases targeted by the program. The survey highlights a theorem of Khare–Thorne: if

$x^p+y^q=z^r,$4

is unramified almost everywhere, de Rham with Hodge–Tate weights $x^p+y^q=z^r,$5 at all $x^p+y^q=z^r,$6, and has residual representation modular and irreducible on $x^p+y^q=z^r,$7, then $x^p+y^q=z^r,$8 is modular and arises from a Hilbert modular form of weight $x^p+y^q=z^r,$9. This is the form of modularity used for Frey Jacobians over totally real fields (Chen et al., 20 Jul 2025).

Level lowering over totally real fields uses the theorems of Fujiwara, Jarvis, and Rajaei. To apply them, one needs precise conductor control and finite-flatness or unramifiedness at primes above $(p,p,r)$0. For hyperelliptic Frey Jacobians this requires explicit analysis of discriminants, semistable reduction, multiplicative or potentially good reduction, and local Weil–Deligne types. In the effective framework of 2025, the lowered level in the generalized Fermat settings depends only on the fixed coefficients $(p,p,r)$1 and the exponent $(p,p,r)$2, not on the hypothetical solution itself, which is what makes large-scale elimination possible (Azon, 19 Mar 2025).

4. Technical bottlenecks and conjectural inputs

The hardest obstruction in the early stages of the program was irreducibility. For elliptic curves over totally real fields there are many irreducibility criteria, but Darmon’s Program requires analogous control for residual $(p,p,r)$3-dimensional pieces cut out of higher-dimensional $(p,p,r)$4-type abelian varieties. The 2016 paper on

$(p,p,r)$5

provided a decisive new ingredient: an irreducibility criterion for mod-$(p,p,r)$6 representations attached to certain abelian varieties of $(p,p,r)$7-type over totally real fields. That paper also gave the first unconditional Diophantine result in the program whose proof genuinely uses a higher-dimensional Frey abelian variety, rather than an elliptic curve in disguise (Billerey et al., 2016).

A second bottleneck is the “trivial solution obstruction.” In the $(p,p,r)$8 setting, the trivial primitive solution $(p,p,r)$9 yields a nonsingular Frey hyperelliptic curve $(r,r,p)$0 whose Jacobian has CM by $(r,r,p)$1. Because level lowering may land in the same residual modular world as such a CM object, the modular method can lose its separating power. Recent work repeatedly isolates this obstruction: in the $(r,r,p)$2 case the treated congruence classes are exactly those in which the trivial-solution obstruction can be avoided by choosing the appropriate Frey curve $(r,r,p)$3 or $(r,r,p)$4, and in the $(r,r,p)$5 case the remaining unresolved part is reduced to the Cartan-normalizer behavior forced by the CM Jacobian attached to the trivial solution (Chen et al., 2022, Billerey et al., 2023).

The third persistent issue is big image. Darmon’s big image conjecture predicts that for large $(r,r,p)$6, the image of $(r,r,p)$7 should be large, excluding reducible, exceptional, and Cartan-normalizer cases except in special geometric situations such as CM. Modern work has sharpened the conjectural remainder. For $(r,r,p)$8 and $(r,r,p)$9, recent results reduce the unresolved part essentially to the Cartan case of Darmon’s conjecture, eliminating the Borel/reducible alternative by a combination of higher-dimensional and multi-Frey arguments (Billerey et al., 2022, Billerey et al., 2023).

5. Major advances and representative applications

Recent progress has turned the program from a largely conjectural framework into a sequence of concrete Diophantine applications.

Paper	Representative result	Significance
(Billerey et al., 2016)	For a fixed regular prime $(q,r,p)$0, there exists $(q,r,p)$1 such that for every prime $(q,r,p)$2, the equation $(q,r,p)$3 has no non-trivial proper solutions with $(q,r,p)$4 and $(q,r,p)$5.	First unconditional Diophantine result in the program that genuinely requires a higher-dimensional Frey abelian variety.
(Billerey et al., 2022)	For all integers $(q,r,p)$6, there are no integer solutions to $(q,r,p)$7 with $(q,r,p)$8, $(q,r,p)$9, and $p$0 or $p$1.	Carries out all but the fifth step for signature $p$2 in almost full generality and reduces $p$3 to the Cartan case of Darmon’s big image conjecture.
(Chen et al., 2022)	For $p$4, there are no non-trivial primitive solutions to $p$5 in either case $p$6 or $p$7.	First full proof-of-concept for signature $p$8 with genus-$p$9 Frey Jacobians, including modularity, irreducibility, conductor analysis, level lowering, and elimination.
(Billerey et al., 2023)	For all integers $2$00, there are no non-trivial primitive solutions to $2$01.	Demonstrates that higher-dimensional Frey varieties can make the proof more efficient than an elliptic-only strategy; for $2$02, only the Cartan case remains.
(Azon, 19 Mar 2025)	Gives an effective, uniform framework for $2$03 and $2$04, together with a Magma package, and solves several families for $2$05.	Makes the program algorithmic: the lowered level depends only on the fixed coefficients, and elimination can be carried out systematically.
(García et al., 4 May 2026)	Develops the program for $2$06 and proves, for $2$07, $2$08, that $2$09 has no primitive solutions with $2$10 and $2$11.	Extends the framework to signature $2$12 and applies it to a conjecture of Laradji–Mignotte–Tzanakis.

These papers also clarify the internal structure of the program. In some signatures the natural proof is genuinely higher-dimensional; in others a multi-Frey method is more effective, combining Frey elliptic curves over $2$13 or totally real fields with Frey hyperelliptic Jacobians. The resulting synthesis is especially visible in the resolution of $2$14, where the Jacobian $2$15 contributes stronger coefficient-field restrictions and Galois symmetries than the elliptic Frey curves alone (Billerey et al., 2023).

6. Current status and open directions

The current state of Darmon’s Program is markedly stronger than its original formulation. There are now unconditional Diophantine applications in which the Frey object is genuinely higher-dimensional, effective conductor and level-lowering frameworks for the main one-parameter signatures, explicit elimination procedures, and large-scale computations of Hilbert newforms. The 2025 survey presents the program as a working research methodology rather than a speculative blueprint (Chen et al., 20 Jul 2025).

At the same time, the survey also identifies the remaining structural difficulties. Step $2$16, the elimination of all Hilbert newforms at the Serre level, is still often the hardest part. Trivial primitive solutions yield CM Frey varieties and obstruct naive elimination. Big-image statements for $2$17-type abelian varieties remain open in general. Conductor computations for hyperelliptic Jacobians, especially at $2$18, are delicate. Even after level lowering, Hilbert modular spaces can be very large. These issues explain why many of the strongest current theorems are either asymptotic in the varying exponent or conditional only on the Cartan case of Darmon’s big image conjecture (Chen et al., 20 Jul 2025).

Two recent developments indicate the likely direction of the subject. First, the effective 2025 framework shows that the program can be made algorithmic, with a common formalism for $2$19 and $2$20, explicit local analysis of Néron models, and a Magma implementation of the elimination step (Azon, 19 Mar 2025). Second, the 2026 treatment of $2$21 shows that the hyperelliptic, $2$22-type strategy is not confined to the original two signatures; it extends to a broader hypergeometric setting in which elliptic Frey curves are unavailable (García et al., 4 May 2026).

In its mature form, Darmon’s Program is therefore best understood as a higher-dimensional modular method for generalized Fermat equations. Its defining insight is that the modular method does not depend on elliptic curves as such, but on access to $2$23-dimensional Galois representations with tightly controlled local behavior. Frey Jacobians with real multiplication provide exactly that structure, and the recent literature shows that, once modularity, irreducibility, level lowering, and elimination can be organized around them, the program becomes a practical mechanism for resolving infinite Diophantine families far beyond the classical Fermat equation.