Quartic reductions and elliptic obstructions for perfect Euler bricks

1. Problem Statement

The paper addresses the Perfect Euler Brick (or Perfect Cuboid) problem, a longstanding open question in number theory dating back to the 18th century.

• Definition: A perfect Euler brick is a rectangular parallelepiped with integer edge lengths $(e_1, e_2, e_3)$ such that all three face diagonals ($d_{ij}$) and the space diagonal ($D$) are also integers.
• Equations:
  $$e_i^2 + e_j^2 = d_{ij}^2 \quad (1 \le i < j \le 3)$$
  $$e_1^2 + e_2^2 + e_3^2 = D^2$$
• Current Status: While "Euler bricks" (satisfying only face-diagonal conditions) exist, no perfect Euler brick has ever been found. The paper does not claim to prove non-existence but establishes a rigorous framework of obstructions and reductions.

2. Methodology and Reduction Framework

The author transforms the geometric problem into an algebraic problem involving quartic equations and elliptic curves.

A. Parametrization via Euclid's Formula

Using Euclid's formula for Pythagorean triples, the edges are parametrized by two coprime pairs of integers $(a, b)$ and $(m, n)$ (with opposite parity). This reduces the brick conditions to two simultaneous square conditions on a rational parameter $\lambda = m/n$:

1. Master Condition: $f_2 = W_1^2(\lambda^2 - 1)^2 + 4V_1^2\lambda^2 = \square$
2. H-total Condition: $f_1 = W_1^2(\lambda^2 - 1)^2 + 4U_1^2\lambda^2 = \square$
   Where $U_1, V_1, W_1$ are derived from the pair $(a, b)$.

B. Normalization to Quartic Pair

By setting $s = a/b$ and normalizing, the problem is reduced to finding coprime integers $a, b$ and a rational $\lambda$ such that:
$$r^2 = \lambda^4 + 2c\lambda^2 + 1$$
$$x^2 = \lambda^4 - 2c\lambda^2 + 1$$
Here, $c = \frac{s^4 - 6s^2 + 1}{(1+s^2)^2}$. The product of these expressions yields a single octic equation:
$$f_1 f_2 = \lambda^8 + A\lambda^4 + 1 = \square$$
where $A = 2 - 4c^2$.

C. Genus-3 Curve and Elliptic Quotients

The equation $w^2 = \lambda^8 + A\lambda^4 + 1$ defines a genus-3 hyperelliptic curve $C_A$.

• Key Insight: A perfect Euler brick exists if and only if $C_A$ possesses a non-degenerate rational point (where $\lambda \notin \{0, \pm 1\}$).
• Jacobian Decomposition: The Jacobian of $C_A$ decomposes (up to isogeny) into a product of three elliptic curves: $J(C_A) \sim E_A \times E'_A \times E''_A$.
• The Bridge: The problem is mapped to the elliptic quotient $E_A: y^2 = (x+A)(x-2)(x+2)$. A rational point on $C_A$ corresponds to a rational point $P \in E_A(\mathbb{Q})$ such that a specific rational function $f(P)$ is a perfect square in $\mathbb{Q}^*$.

3. Key Contributions

I. The Kummer Character Obstruction

The paper introduces a Kummer character $\chi_f$ associated with the function $f(P)$ on the elliptic curve $E_A$.

• Definition: $\chi_f(T) = [f(\cdot + T)/f(\cdot)] \in \mathbb{Q}(E_A)^*/\mathbb{Q}(E_A)^{*2}$.
• Main Result (Theorem 5.4): The character is non-trivial on the 4-torsion subgroup of $E_A$. Specifically, $\chi_f(T_4) \neq 1$ for any rational point $T_4$ of order 4.
• Implication: This structural obstruction implies that for any rational point $P$, it is generally impossible for both $f(P)$ and $f(P+T_4)$ to be squares simultaneously, severely restricting the search space for solutions.

II. 2-Descent and Algebraic Obstructions

Using 2-descent techniques on $E_A$, the author derives conditions on the "descent classes" $(\delta_1, \delta_2, \delta_3)$ of points on the curve.

• Theorem 6.2:
  • If the third component $\delta_3 = 1$, then $f(P)$ is congruent to $2 \pmod{\mathbb{Q}^{*2}}$, meaning $f(P)$ cannot be a square.
  • If an odd prime $p$ divides $\delta_3$, it must also divide $\delta_1$ or $\delta_2$.
• Significance: These results algebraically exclude large families of potential solutions based on the arithmetic properties of the descent classes.

III. Computational Verification

The author performed extensive computations using PARI/GP and SageMath:

• Verified that for all parameters $a, b, m, n$ up to $10^3$, the product $f_1 f_2$ is never a perfect square.
• Identified that for every tested tuple, there exists a "blocker" prime $p$ (either $p=2$ or $p \equiv 1 \pmod 4$) such that the valuation $v_p(f_1 f_2)$ is odd, preventing the product from being a square.
• No prime $p \equiv 3 \pmod 4$ ever acted as a blocker, consistent with Gaussian integer norm properties.

4. Results and Significance

Unconditional Results

1. Reduction: The existence of a perfect Euler brick is equivalent to the existence of a non-degenerate rational point on the genus-3 curve $C_A$ where a specific function $f(P)$ is a square.
2. Torsion Obstruction: The Kummer character is non-trivial on 4-torsion, creating a fundamental structural barrier.
3. Descent Constraints: Points with specific descent classes (e.g., $\delta_3=1$) are provably non-solutions.

Limitations and Open Gap

• The results do not unconditionally prove non-existence.
• The Gap: The current obstructions do not cover all possible descent classes. Specifically, cases where $\delta_3 \neq 1$ and the prime factors of $\delta_3$ divide $\delta_1$ (but not $\delta_2$) remain unruled out by the current algebraic arguments.
• Computational Limit: The verification is finite (up to $10^3$) and cannot cover the infinite parameter space.

Future Directions Proposed

1. Genus-5 Covering: The non-triviality of $\chi_f$ on 4-torsion implies a double cover of $E_A$ is a genus-5 curve. By Faltings' Theorem, this curve has finitely many rational points. If the rank of its Jacobian is low enough, Chabauty-Coleman methods could prove finiteness effectively.
2. Generalizing the Character: Proving $\chi_f(G) \neq 1$ for all non-torsion points $G$, not just torsion points.
3. Brauer-Manin Obstruction: Investigating geometric obstructions on the surface defined by the quartic pair.
4. Connection to $\mathbb{Q}(\sqrt{2})$: Exploring the factorization of the parameter $c(s)$ over $\mathbb{Q}(\sqrt{2})$ to link with previous work on cuboid polynomials.

Conclusion

René Peschmann's paper provides a significant advancement in the study of the Perfect Euler Brick by shifting the problem from a search for integer solutions to a deep analysis of the arithmetic geometry of genus-3 curves and their elliptic quotients. While it stops short of a final proof of non-existence, it establishes powerful new obstructions (Kummer characters on torsion, 2-descent constraints) and outlines a clear path toward a complete solution via higher-genus covering techniques and Chabauty methods.