An explicit expression of the Richelot isogeny through Kleinian hyperellyptic functions

1. Problem Statement

The paper addresses the problem of explicitly relating Kleinian hyperelliptic functions of weight 2 associated with a genus 2 curve $C$ to those associated with a Richelot isogenous curve $\hat{C}$.

• Context: The author aims to develop an algorithm (analogous to the classical Landen algorithm for elliptic curves) to compute Kleinian functions for genus 2 curves. This involves constructing a sequence of isogenous curves where the functions of one curve can be expressed in terms of the next.
• Specific Challenge: While the algebraic construction of the Richelot isogeny (mapping Jacobians of genus 2 curves) is well-known, finding an explicit transcendental formula that connects the specific Kleinian functions ($\wp$-functions of weight 2) of the source curve to the target curve has been an open problem. Previous algebraic approaches (e.g., in [3]) did not directly yield relations for these transcendental functions.
• Goal: To derive an explicit formula expressing the vector of Kleinian functions $S_f(z)$ for a curve $X_f$ in terms of the vector $S_{\hat{f}}(z)$ for the isogenous curve $X_{\hat{f}}$, utilizing the geometry of Kummer surfaces.

2. Methodology

The author employs a combination of transcendental methods (theta functions, period lattices) and algebraic geometry (Kummer surfaces, tropes, nodes).

• Kleinian Functions of Weight 2: Instead of classical $\sigma$-functions (which require a Weierstrass point at infinity), the paper uses the space $S_f$ of holomorphic functions on $\mathbb{C}^2$ satisfying specific quasi-periodicity conditions. These functions embed the Kummer surface $K(\text{Jac}(X_f))$ into $\mathbb{CP}^3$.
• Richelot Construction: The paper utilizes the standard Richelot construction where a genus 2 curve defined by $y^2 = f(x)$ (with $f = pqr$ for quadratic polynomials $p,q,r$) is mapped to a new curve defined by $y^2 = \hat{f}(x)$, where $\hat{f}$ is derived via resultants and discriminants of $p,q,r$.
• Kummer Surface Geometry:
  • The Kummer surface $K_f$ has 16 nodes (images of 2-torsion points) and 16 tropes (hyperplanes containing 6 nodes).
  • The Richelot isogeny induces a map between the Kummer surfaces $K_{\hat{f}} \to K_f$.
  • The author proves that this induced map is a quadratic map (degree 2 homogeneous polynomial map) on $\mathbb{CP}^3$.
• Matrix Formulation:
  • The map is represented by a quadruple of symmetric $4 \times 4$ matrices $(A_1, A_2, A_3, A_4)$.
  • The author constructs two specific matrices, $B$ and $\hat{B}$, which relate the basis of theta functions to the coordinates of the Kummer surface.
  • By analyzing the symmetries of the Kummer surface (specifically the action of 2-torsion points on tropes and nodes), the author determines the structure of these matrices up to scalar factors.
• Taylor Expansion Matching: The author uses the Taylor expansions of the Kleinian functions near the origin (order 2) to fix the scalar constants and the specific form of the matrices, ensuring the quasi-periodicity conditions (Legendre's identities) are satisfied.

3. Key Contributions

1. Explicit Quadratic Map: The paper derives an explicit formula (Theorem 4.4) for the Richelot isogeny as a map $R: \mathbb{CP}^3 \to \mathbb{CP}^3$. This map is given by a specific composition of matrices involving the coefficients of the polynomials $p, q, r$ and their derived forms $\hat{p}, \hat{q}, \hat{r}$.
2. Transcendental Relation: The main result (Theorem 4.6) provides the explicit relation between the Kleinian function vectors:
   $$S_f(z) = -\frac{\exp(z^T H_{p,q,r} z)}{32\Delta(p,q,r)^3} \begin{pmatrix} S_{\hat{f}}(z)^T A_{p,q,r} S_{\hat{f}}(z) \\ S_{\hat{f}}(z)^T A_{p,q,r}^{(22)} S_{\hat{f}}(z) \\ S_{\hat{f}}(z)^T A_{p,q,r}^{(12)} S_{\hat{f}}(z) \\ S_{\hat{f}}(z)^T A_{p,q,r}^{(11)} S_{\hat{f}}(z) \end{pmatrix}$$
   where $H_{p,q,r}$ is a symmetric matrix derived from the period matrices and $A$ matrices are explicitly defined symmetric matrices.
3. Matrix $H_{p,q,r}$: The paper explicitly calculates the matrix $H_{p,q,r}$ (Lemma 3.3 and Theorem 4.6), which accounts for the change in the exponential factor (the "characteristic") when moving between the isogenous curves. This matrix is crucial for preserving the quasi-periodicity of the functions.
4. Geometric Construction of Matrices: The author provides a constructive method to find the matrices $B$ and $\hat{B}$ by identifying the intersection of specific tropes (hyperplanes) on the Kummer surface, resolving the indeterminacy of row permutations and scalings.

4. Key Results

• Theorem 4.4: Establishes that the Richelot isogeny on the Kummer surface is a homogeneous quadratic map defined by the matrix product $D_{p,q,r} T_{p,q,r} (C_{p,q,r}^{-1} a)$.
• Theorem 4.6: The final explicit expression for the Kleinian functions. It shows that the functions of the original curve are quadratic forms of the functions of the isogenous curve, multiplied by an exponential factor involving a quadratic form $z^T H z$.
• Verification: The complex algebraic formulas derived in the paper were verified using Maple (code available on GitHub), ensuring the correctness of the cumbersome polynomial coefficients.

5. Significance

• Algorithmic Implementation: This work provides the necessary "step" for an algorithm to compute Kleinian functions recursively. By expressing functions of a curve in terms of an isogenous curve, one can potentially reduce the problem to a curve where the functions are easier to approximate (e.g., via degeneration or numerical methods).
• Bridge Between Algebra and Transcendence: The paper successfully bridges the gap between the purely algebraic description of the Richelot isogeny (polynomials and resultants) and the transcendental description (Kleinian functions). This is essential for applications in cryptography (isogeny-based crypto) and numerical analysis of Abelian varieties.
• Generalization: The approach relies on weight 2 Kleinian functions, which are defined for general hyperelliptic curves without requiring a point at infinity, making the results more general than those relying on classical $\sigma$-functions.
• Foundation for Future Work: The author notes that the choice of the specific isogenous curve in a sequence (to ensure convergence or specific properties) is a subject for future work, but this paper solves the fundamental problem of expressing the functions explicitly.

In summary, Matvey Smirnov provides the missing explicit link between the geometry of Richelot isogenies and the analytic theory of genus 2 Kleinian functions, enabling the potential development of efficient algorithms for computing these special functions.