Ibukiyama-Katsura-Oort Matrix

• The Ibukiyama-Katsura-Oort matrix is a Hermitian 2×2 matrix defined over the maximal order of a definite rational quaternion algebra that uniquely encodes the principal polarization of a superspecial abelian surface.
• It is computed algorithmically by resolving the endomorphism ring structure and solving the principal-ideal problem, with key steps relying on quaternion conjugation and isogeny frameworks under GRH assumptions.
• Its study is critical for explicit moduli description, isomorphism classification, and advances in isogeny-based cryptography, bridging theoretical insights with practical computations.

The Ibukiyama–Katsura–Oort (IKO) matrix is a Hermitian $2 \times 2$ matrix with entries in the maximal order $O_0$ of the definite rational quaternion algebra $B_{p, \infty}$ (ramified only at $p$ and $\infty$), which uniquely encodes the principal polarization of a principally polarized superspecial abelian surface (PPSAS) over the finite field $k = \mathbb{F}_{p^2}$. This matrix is central to the explicit description of moduli, computation of endomorphism rings, isomorphism classification, and explicit isogeny constructions for abelian surfaces, especially in the context of isogeny-based cryptography and the study of superspecial jacobians and products of elliptic curves (Montessinos, 26 Jan 2026).

1. Formal Definition and Fundamental Structure

Given a fixed base supersingular elliptic curve $E_0/k$ with endomorphism ring $O_0$ and its self-product $A_0 = E_0 \times E_0$, the endomorphism ring of $A_0$ is $M_2(O_0)$. For any principal polarization $\lambda$ on $A_0$, the associated IKO matrix is defined as

$$\mu(\lambda) = \lambda_0^{-1} \circ \lambda \in M_2(O_0),$$

where $\lambda_0$ is the product polarization. The principal theorem of Ibukiyama–Katsura–Oort establishes that the map

$$\{\text{principal polarizations on }A_0\}/\sim \longrightarrow M_2(O_0),\qquad \lambda \mapsto \mu(\lambda)$$

is injective, with image exactly the set of positive-definite Hermitian matrices

$\Mat(A_0) = \left\{ g = \begin{pmatrix} s & r \ \bar{r} & t \end{pmatrix} : s, t \in \mathbb{Z}_{\ge 1},\, r \in O_0,\, st - r\bar{r} = 1 \right\}.$

The equivalence on polarizations, and thus on matrices, is determined up to conjugation by $GL_2(O_0)$: $g \sim \gamma^* g \gamma$ for $\gamma \in GL_2(O_0)$.

2. The IKO Matrix and the Endomorphism Structure

For any PPSAS $(A, \lambda_A)$ over $k$, superspeciality provides an isomorphism to $(A_0, \lambda)$, and the IKO matrix $\mu(A)$ is $\mu(\lambda)$ up to $GL_2(O_0)$ conjugation. The Rosati involution $r_\lambda$ on $\End(A)$, induced by the principal polarization $\lambda$, is connected to the IKO matrix via

$$r_\lambda(\gamma) = \mu(\lambda)^{-1}\, \gamma^*\, \mu(\lambda), \qquad \gamma \in M_2(O_0),$$

where $^*$ denotes quaternionic conjugate transpose. This correspondence is bidirectional: knowing $r_\lambda$ on a $\mathbb{Z}$-basis of $M_2(O_0)$ determines $\mu(\lambda)$ by solving the linear system

$g\, r_\lambda(\gamma) = \gamma^*\, g,\quad \forall\ \gamma \in M_2(O_0),\ g \in \Mat(A_0).$

Thus, the IKO matrix provides a canonical representative of the principal polarization (or equivalently, the Rosati involution) in the endomorphism ring of the abelian surface.

3. Algorithmic Computation of $\mu(A)$

The computation of the IKO matrix from an explicit "good" representation of $\End(A)$ is algorithmically well-structured:

1. Compute structure constants for $\End^0(A) = \End(A) \otimes \mathbb{Q}$ and obtain an explicit isomorphism $\tau: \End^0(A) \xrightarrow{\sim} M_2(B_{p, \infty})$ such that $\tau(\End(A)) = R$ for some order $R$.
2. Solve the principal-ideal problem (PIP) in $M_2(O_0)$ to find a generator $\gamma \in M_2(B_{p, \infty})^\times$ realizing $R\, M_2(O_0) = \gamma M_2(O_0)$.
3. Adjust $\gamma$ by the required outer automorphism to orient the isomorphism as per the determinant constraints (Mestre–Oesterlé–Voight orientation).
4. Transport the Rosati involution through $\tau$ and $\gamma$ to obtain $$\sigma: M_2(B_{p, \infty}) \rightarrow M_2(B_{p, \infty})$$, defined by $$\sigma(\alpha) = \gamma^{-1}\, (\tau \circ r_A \circ \tau^{-1})(\gamma\, \alpha\, \gamma^{-1})\, \gamma$$.
5. Solve $\sigma(\alpha) = g^{-1}\, \alpha^*\, g$ in $M_2(O_0)$ for $g \in \Mat(A_0)$, yielding $\mu(A)$.

The complexity of this process is polynomial in $\log p$ under the Generalized Riemann Hypothesis (GRH) for the central simple algebra (CSA) isomorphism and PIP steps, and the standard assumptions for $\text{KLPT}^2$ algorithms, though the CSA isomorphism carries substantial constants (Montessinos, 26 Jan 2026).

4. Key Formulas and Isogeny Conditions

The formalism centralizes several explicit formulas:

• Hermitian condition (IKO matrices):

$$g = \begin{pmatrix} s & r \ \bar{r} & t \end{pmatrix},\quad s, t \in \mathbb{Z}_{\ge 1},\ r \in O_0,\quad st - r\bar{r}=1.$$

• Rosati involution on $A_0$:

$$r_\lambda(\gamma) = \mu(\lambda)^{-1}\, \gamma^*\, \mu(\lambda).$$

• Polarized isogeny:

$$\gamma: (A_0, \lambda) \to (A_0, \lambda')\ \text{is polarized} \iff \gamma^* \mu(\lambda') \gamma = N \mu(\lambda).$$

• Trace pairing on endomorphisms:

$$\langle f, g \rangle = \mathrm{Tr}(f \circ r_\lambda(g)) \in \mathbb{Z}_{>0}.$$

These relations characterize the behaviour of polarizations, isogenies, and endomorphism structure within and between superspecial abelian surfaces.

5. Computational Equivalences and Reductions

Computation of the IKO matrix is, under GRH and $\text{KLPT}^2$ hypotheses, polynomial-time equivalent to:

• Computing a “good” representation of $\End(A)$.
• Computing an “effective” ($4 \times 4$ quaternion-order) $\mathbb{Z}$-basis of $\End(A)$.
• Computing an unpolarized isomorphism $A \simeq A'$ to a PPSAS of known IKO matrix.

In the product-of-elliptics case, all problems are polynomial-time equivalent. In the Jacobian case, effective endomorphism ring computation reduces to isomorphism, which reduces to IKO computation, which in turn reduces to good endomorphism representation, potentially leaving only the “effective” versus “good” distinction unresolved in genus two (Montessinos, 26 Jan 2026).

6. Illustrative Examples

• For $A = E_0^2$, the trivial case, $\mu(A) = I_2$.
• For $A = \mathrm{Jac}(C)$, a superspecial genus-2 Jacobian, one uses a composition of Richelot $(2,2)$-isogenies (via $\text{KLPT}^2$), adjusting kernels at each step, to recover an explicit unpolarized isomorphism $E_3 \times E_4 \simeq A$ with known elliptic factors and endomorphism rings. The product-curve machinery of Gaudry–Spaenlehauer–Soumier is then used to invert and compose isogenies (Montessinos, 26 Jan 2026).

7. Challenges and Research Directions

Outstanding issues include:

• Removing the GRH assumption for PIP and CSA isomorphism steps, toward fully unconditional $\text{KLPT}^2$ algorithms.
• Representing genus-2 endomorphism rings more compactly or natively, avoiding the need to pass through “good” endomorphism representations.
• Extending the IKO matrix formalism to non-principal polarizations and to higher-dimensional superspecial abelian varieties.
• Practical optimization of the central simple algebra isomorphism stage to minimize computational overhead.

The Ibukiyama–Katsura–Oort matrix thus constitutes the canonical Hermitian matrix invariant of a PPSAS, encapsulating the data of the principal polarization and providing the framework for algorithmic reduction between central computational tasks—endomorphism ring computation, isomorphism testing, and explicit isogeny construction—in abelian surface arithmetic (Montessinos, 26 Jan 2026).