Frey–Mazur Conjecture in Arithmetic Geometry

• The Frey–Mazur Conjecture asserts that for primes p ≥ 17, an isomorphism of the mod p Galois representations of elliptic curves over Q forces the curves to be isogenous.
• The modular method uses the conjecture to connect Frey curves with modular forms, thereby providing explicit bounds for solutions to Diophantine equations like those in Lucas sequences.
• Geometric analogs extend the conjecture to abelian surfaces with quaternionic multiplication, showing that isomorphic p-torsion on low-genus curves implies an isogeny for sufficiently large p.

The Frey–Mazur Conjecture posits that for elliptic curves over $\mathbb{Q}$, the isogeny class is determined by the isomorphism class of its mod $p$ Galois representation for sufficiently large primes $p$, specifically $p \geq 17$. This conjecture has profound implications for arithmetic geometry, Diophantine equations, and the modular method in number theory. Its classical statement and geometric analogs inform the understanding of the relationship between torsion representations and isogeny classes in various contexts, including elliptic curves and abelian surfaces with quaternionic multiplication ("fake elliptic curves").

1. Statement and Classical Context of the Frey–Mazur Conjecture

The classical Frey–Mazur Conjecture asserts that if $E_1$ and $E_2$ are elliptic curves defined over $\mathbb{Q}$ and $p \geq 17$, then an isomorphism of their mod $p$ Galois representations,

$\rho_{E_1,p} \simeq \rho_{E_2,p}$

forces $E_1$ and $E_2$ to be isogenous. Concretely, for $p > 17$, the Galois representation

$$\rho_{E,p}: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\mathbb{F}_p)$$

encodes sufficient arithmetic information to distinguish the isogeny class of $E$. This conjecture leverages profound results in the theory of modularity (the modularity theorem), and Ribet’s level-lowering theorem, which demonstrate that $\rho_{E,p}$ is modular and associated to a weight 2 newform of level independent of $E$.

2. Modular Method and its Arithmetic Implications

In the setting of Diophantine equations, particularly for perfect powers in Lucas sequences, the “modular method” attaches to each putative solution an auxiliary elliptic curve, called the Frey curve, whose $\mathbb{Q}$-arithmetic invariants (minimal discriminant, conductor) are tightly controlled by the solution parameters. Specifically, for a nondegenerate Lucas sequence defined by $U_{n+2} = b U_{n+1} + c U_n$, $U_0 = 0$, $U_1 = 1$, the problem of finding all $U_n = y^p$ is most effectively attacked by this method. The Frey–Mazur Conjecture then enters when considering the isomorphism between the mod $p$ Galois representation of the Frey curve and that attached to the corresponding newform:

$\rho_{E,p} \simeq \rho_{F,p}$

Imposing the conjecture implies an isogeny between $E$ and $F$ and, crucially, equates their conductors.

3. Explicit Conductor and Bounding Admissible Solutions

The conductor of the Frey curve attached to a solution $(n, y, p)$ is of the form

$N_E = 2^\alpha \cdot \mathrm{rad}(c(b^2 + 4c)y),$

or the general bound for admissible solutions,

$$N = 28 \cdot \mathrm{rad}'(c) \cdot \mathrm{rad}'(b^2 + 4c),$$

where $\mathrm{rad}'(m)$ is the product of odd primes dividing $m$. The isogeny implied by the Frey–Mazur Conjecture forces any contribution from the variable $y$ in the conductor to disappear:

$\mathrm{rad}(y) \mid \mathrm{rad}(2c(b^2 + 4c)).$

This result restricts the growth and “smoothness” of $y$, and subsequently constrains the index $n$. Applying classical results on the distribution of smooth numbers in Lucas sequences leads to explicit bounds on $n$, and thus on $p$ via

$p \leq 4 n \log|a|,$

where $a$ is the dominant root of $z^2 - bz - c$. The main theorem (Theorem 4.1) consolidates these estimates:

$$p \leq \max \left\{17,\ \Psi(N) + \Psi(N)/12 + 1,\ 4 \log|a| \cdot \max\{30, (N+1)\} \right\}$$

with $$\Psi(N) = N \prod_{p|N} \left(1 + \frac{1}{p}\right)$$ the Dedekind psi function.

4. Geometric Analog: Fake Elliptic Curves and Low Genus Phenomena

A geometric extension of the Frey–Mazur Conjecture is developed for abelian surfaces with quaternionic multiplication (“fake elliptic curves”) over the function fields of complex curves of low genus. For any $k > 0$, there exists $N > 0$ so that for any smooth complex curve $B$ of genus $g < k$, families $A_1$, $A_2$ with $\mathcal{O}_D$-action and isomorphic $p$-torsion local systems ($A_i[p]$ as $\mathcal{O}_D$-modules), $A_1$ and $A_2$ are $\mathcal{O}_D$-isogenous for $p > N$.

This injectivity result is reformulated in the context of Shimura curves and modular surfaces:

• The moduli space $Z^D(p)$ parametrizes pairs $(A_1, A_2)$ with level $p$ structure and isomorphism of $p$-torsion.
• The detection of isogeny classes is linked to the behavior of curves on $Z^D(p)$; any non-Hecke curve must lie on a Hecke divisor if $p$ is large compared to the genus.

Techniques employed include uniformization of Shimura curves, Riemann–Hurwitz calculations, multiplicity estimates at CM points, geometric “repulsion” phenomena among CM points, and volume bounds in hyperbolic geometry. These collectively force the injectivity of the map from isogeny classes to $p$-torsion Galois representations for abelian surfaces over low-genus base curves (Bakker et al., 2013).

5. Interplay of Galois Representations, Conductors, and Isogeny

The critical connection in the arithmetic setting is the identification of mod $p$ Galois representations and its consequences:

• If $\rho_{E,p} \simeq \rho_{F,p}$ for $p > 17$, then $E \sim F$ (isogenous).
• This imposes $N_E = N_F$ for conductors, restricting the prime divisors contributed by “exceptional” variables like $y$ in Lucas sequences.
• In the geometric context, isomorphism of $p$-torsion local systems for abelian surfaces with quaternionic multiplication forces isogeny as long as $p$ is large compared to the geometric complexity of the base.

A plausible implication is that, under the conjecture, the modular method extends systematically from Fermat’s Last Theorem to a wider array of Diophantine problems governed by recurrence relations and moduli spaces. The repulsion and volume estimates in the geometric approach suggest a robust framework capable of handling higher-dimensional generalizations.

6. Broader Mathematical Implications and Associated Conjectures

The linkage of the Frey–Mazur Conjecture to explicit bounds for Diophantine equations, control over perfect powers in recurrence sequences, and identification of isogeny classes is a significant application of deep results from the arithmetic of elliptic curves. The approach demonstrates how:

• The modular method, enriched by level-lowering and representation-theoretic insights, produces uniform results across different number-theoretic settings.
• The geometric analog in low-genus curve contexts reveals how much information about the isogeny class is captured in torsion representations, generalizing the classical arithmetic phenomenon.
• The methods align with broader goals in arithmetic geometry, including understanding rational points on modular and Shimura curves and connections to the Bombieri–Lang conjecture.

This suggests ongoing relevance for mod $p$ Galois representations and their geometric counterparts not only in classifying isogeny classes but also in limiting the arithmetic complexity of solutions to recurrence-based equations (1307.50781309.6568).