On superspecial hyperelliptic curves of Rosenhain forms

Technical Summary: Enumeration of Superspecial Hyperelliptic Curves in Rosenhain Forms

Problem Statement
The paper addresses the fundamental problem of determining the number of isomorphism classes of superspecial genus-$g$ hyperelliptic curves in a given characteristic $p$. While the case for genus $g \le 3$ has been largely resolved, and the existence of such curves for $g \ge 4$ remains open in general, specific counts for higher genera in small characteristics have been limited. Previous work by Kudo and Harashita utilized Gröbner basis computations to solve this for genus 4 in characteristics up to 19. This paper aims to extend these results to genera 4, 5, and 6 for characteristics $p$ beyond the previously covered ranges, specifically up to $p=41$.

Methodology
The authors propose a new algorithm (Algorithm 3.1) for enumerating these curves, which relies on a specific necessary condition derived for curves in Rosenhain form.

1. Rosenhain Form and Necessary Conditions:
   The paper focuses on hyperelliptic curves defined by the Rosenhain equation:
   $$C: y^2 = x(x-1) \prod_{i=1}^{2g-1} (x - \lambda_i)$$
   The core theoretical contribution is Theorem 1.3, which establishes a necessary condition for such a curve to be superspecial in characteristic $p > 2$. The theorem states that if $C$ is superspecial, then for all $i, j \in \{1, \dots, 2g-1\}$, the elements $\lambda_i$, $1-\lambda_i$, and $\lambda_i - \lambda_j$ must be squares in the finite field $\mathbb{F}_{p^2}$.

   • Proof Strategy: The proof utilizes the properties of the Jacobian $J$ of the curve. It demonstrates that for a superspecial curve, the 4-torsion subgroup $J[4]$ is defined over $\mathbb{F}_{p^2}$. By analyzing the Mumford representations of 4-torsion points and their half-points, the authors show that specific cross-ratios of the branch points must be squares in $\mathbb{F}_{p^2}$.

2. Enumeration Algorithm:
   Based on Theorem 1.3, the algorithm proceeds as follows:

   • Filtering: It constructs a set $S \subset \mathbb{F}_{p^2}$ containing elements $\lambda$ where both $\lambda$ and $1-\lambda$ are squares.
   • Candidate Generation: It recursively selects distinct sets $\{\lambda_1, \dots, \lambda_{2g-1}\} \subset S$ such that the difference $\lambda_i - \lambda_j$ is also a square for all pairs.
   • Isomorphism Handling: To avoid counting isomorphic curves multiple times, the algorithm generates all Rosenhain forms equivalent to a candidate set under the action of the symmetric group (via Möbius transformations permuting the branch points $\{0, 1, \infty, \lambda_1, \dots, \lambda_{2g-1}\}$) and stores canonical representatives.
   • Superspeciality Check: For each valid candidate set, the algorithm computes the Cartier-Manin matrix of the curve. The curve is superspecial if and only if this matrix is the zero matrix. This involves checking if specific coefficients in the expansion of $f(x)^{(p-1)/2}$ vanish.

Key Contributions and Results
The primary contribution is the successful application of this algorithm to compute the exact number of isomorphism classes for genera 4, 5, and 6 in various small characteristics, filling gaps in the existing literature. The results are summarized as follows:

• Genus 4: The paper determines the counts for characteristics $p \in \{23, 29, 31, 37, 41\}$.

  • $p=23$: 4 classes
  • $p=29$: 8 classes
  • $p=31$: 10 classes
  • $p=37$: 23 classes
  • $p=41$: 34 classes
  • Note: Results for $p \le 19$ match previous findings by Kudo and Harashita.

• Genus 5: The paper determines the counts for characteristics $p \in \{13, 17, 19, 23, 29, 31, 37, 41\}$.

  • No curves exist for $p=13, 17$.
  • $p=19, 29$: 1 class each.
  • $p=23$: 2 classes.
  • $p=31$: 6 classes.
  • $p=37$: 5 classes.
  • $p=41$: 3 classes.

• Genus 6: The paper determines the counts for characteristics $p \in \{17, 19, 23, 29, 31\}$.

  • No curves exist for $p=17, 19, 29$.
  • $p=23, 31$: 1 class each.

Significance and Claims
The paper claims that its approach offers a distinct advantage over previous methods (specifically those using Gröbner bases) by avoiding complex symbolic computations. Instead, it leverages the arithmetic properties of the Rosenhain parameters to filter candidates efficiently. The authors state that for small characteristics, the number of candidate sets satisfying the square conditions is sufficiently small to make the algorithm computationally feasible.

By implementing this algorithm in Magma, the authors successfully extended the known classification of superspecial hyperelliptic curves to higher genera and larger characteristics than previously documented. The paper explicitly notes that for genera $g \in \{5, 6\}$, the number of isomorphism classes had not been determined for any characteristic $p > 2g+1$ prior to this work. The results provide concrete data points for the fundamental problem of enumerating superspecial curves, which is relevant to applications in isogeny-based cryptography and algebraic geometry codes.