The $5$-divisible integer group determinants for the elementary abelian group of order 25

Technical Summary: The 5-Divisible Integer Group Determinants for the Elementary Abelian Group of Order 25

Problem Statement
This paper addresses the Taussky–Todd integer group determinant problem for the elementary abelian group of order 25, denoted as $G_5 = C_5 \times C_5$. The problem seeks to characterize the set $S(G)$, defined as the set of all integer values taken by the group determinant $\Theta_G((x_g)_{g \in G})$ when the variables $x_g$ are restricted to integers.

While the values of $S(G)$ coprime to the prime $p=5$ were previously determined by DeSilva and Pinner, the structure of the subset of values divisible by 5 remained incomplete. Specifically, Mossinghoff and Pinner established a necessary condition that any 5-divisible value must be divisible by $5^8$. However, it was unknown whether every multiple of $5^8$ is actually attainable as a group determinant value. This paper resolves that gap by proving the converse inclusion.

Methodology
The proof relies on a combination of algebraic factorization, a general "shift identity," and the explicit construction of specific polynomial "seeds."

1. Factorization and Setup:
   For $G_p = C_p \times C_p$, the group determinant $\Theta_{G_p}$ is expressed via Dedekind's factorization as a product over characters. This is reformulated using a bivariate polynomial $F(x, y) \in \mathbb{Z}[x, y]$ such that the determinant value is:
   $$M_p(F) = \prod_{i,j=0}^{p-1} F(\zeta_p^i, \zeta_p^j)$$
   where $\zeta_p = e^{2\pi i/p}$.

2. The Shift Identity:
   The core technical tool is a shift identity (Lemma 2.1) involving the polynomial $H_p(x, y) = \Phi_p(x)\Phi_p(y)$, where $\Phi_p$ is the $p$-th cyclotomic polynomial. The identity states that for any $F \in \mathbb{Z}[x, y]$ and integer $c$:
   $$M_p(F + cH_p) = \frac{F(1, 1) + p^2 c}{F(1, 1)} M_p(F)$$
   (assuming $F(1,1) \neq 0$). This identity allows the generation of new determinant values from existing ones by shifting the polynomial, effectively scaling the original value $M_p(F)$ by the factor $\frac{F(1, 1) + p^2 c}{F(1, 1)}$ while preserving the product structure over non-trivial characters.

3. The Seed Criterion:
   Building on the shift identity, the paper proposes a "Seed Criterion" (Proposition 2.2). To prove that $p^{p+3}\mathbb{Z} \subseteq S(G_p)$, it suffices to find a set of polynomials $\{F_a\}$ indexed by a set $A \subset \mathbb{Z} \setminus \{0\}$ that maps surjectively onto $\mathbb{Z}/p\mathbb{Z}$, such that for each $a \in A$:

   • $F_a(1, 1) = p^a$
   • $M_p(F_a) = a \cdot p^{p+3}$

   If such seeds exist, the shift identity can generate all integer multiples of $p^{p+3}$.

Key Contributions and Results
The primary contribution is the explicit construction of the required seeds for the case $p=5$. The author defines three specific polynomials in $\mathbb{Z}[x, y]$:

• $F_1(x, y) = 1 + y + y^2 + x + x^4y$
• $F_2(x, y) = 1 + y + y^2 + y^3 + x + xy + xy^2 + x^4 + x^4y + x^4y^4$
• $F_5(x, y) = H_5(x, y) - 1 - y + x + x^4$

Through direct reduction using properties of the cyclotomic field $\mathbb{Q}(\zeta_5)$ and norm calculations, the paper proves (Proposition 3.1) that these polynomials satisfy the seed conditions for $a = 1, 2, 5$:

• $M_5(F_1) = 5^8$
• $M_5(F_2) = 2 \cdot 5^8$
• $M_5(F_5) = 5^9$ (which corresponds to $a=5$ in the seed criterion context, as $5 \cdot 5^8 = 5^9$)

Using the symmetry property $M_p(-F) = -M_p(F)$, seeds for $a = -1$ and $a = -2$ are also obtained. Since $\{1, -1, 2, -2, 5\}$ covers all residue classes modulo 5, the Seed Criterion is satisfied.

Main Theorem and Corollary
The paper establishes the following results:

• Theorem 1.1: For $G_5 = C_5 \times C_5$, the set of 5-divisible group determinants is exactly the set of multiples of $5^8$:
  $$S(G_5) \cap 5\mathbb{Z} = 5^8\mathbb{Z}$$
• Corollary 1.2: Combining this with the known classification of values coprime to 5, the complete set of integer group determinants for $G_5$ is:
  $$S(G_5) = \{m \in \mathbb{Z} : m \equiv \pm 1 \text{ or } \pm 7 \pmod{25}\} \cup 5^8\mathbb{Z}$$

Significance and Scope
The paper claims that this result, together with the previously determined $S(C_{25})$ by Mossinghoff and Pinner, completes the Taussky–Todd integer group determinant problem for all groups of order 25.

The author notes that while the shift identity and seed criterion are valid for any odd prime $p$, the existence of the specific seeds satisfying the required conditions is a separate, difficult problem. The argument presented is specific to $p=5$. The paper does not assert that the inclusion $p^{p+3}\mathbb{Z} \subseteq S(C_p \times C_p)$ holds for primes $p \geq 7$, suggesting that the clean formulas found for $p=3$ and $p=5$ may reflect exceptional behavior rather than a general rule.