Construction of Generically Ordinary Families of Hyperelliptic Curves

Technical Summary: Construction of Generically Ordinary Families of Hyperelliptic Curves

Problem Statement
The paper addresses a conjecture posed by Katz (2018) regarding the generic ordinarity of specific one-parameter families of hyperelliptic curves over the $t$-line. A smooth projective curve $C$ of genus $g$ over a field of characteristic $p > 0$ is defined as ordinary if the middle coefficient of the numerator of its zeta function is prime to $p$. Equivalently, the Jacobian of $C$ is an ordinary abelian variety, characterized by a Newton polygon with only slopes 0 and 1. This condition is detected by the nonvanishing of the determinant of the Hasse–Witt matrix (or Cartier–Manin matrix).

Katz conjectured that the family of curves defined by $K: y^2 = x^d - dx + t$ (where $d \in \{2g+1, 2g+2\}$ and $g \ge 2$) is generically ordinary for all sufficiently large primes $p$. While the family possesses "large monodromy" (finite index in $Sp_{2g}(\mathbb{Z})$), it was known to fail generic ordinarity for small primes. The specific problem is to establish a quadratic bound $P^+(d)$ such that for all $p > P^+(d)$, the family is generically ordinary, and to generalize this result to families of the form $C_\alpha: y^2 = x^d + \alpha x + t$ for any nonzero algebraic integer $\alpha$.

Methodology
The proof strategy relies on analyzing the Hasse–Witt matrix associated with the family $C_\alpha$ and demonstrating that its determinant (the Hasse–Witt polynomial) does not vanish identically modulo $p$ for sufficiently large $p$.

1. Hasse–Witt Polynomial Construction:
   The authors define a coefficient matrix $(M_{ij}(X, Y))_{1 \le i,j \le g}$ over $\mathbb{Z}[X, Y]$, where $M_{ij}$ is the coefficient of $x^{pi-j}$ in the expansion of $(x^d + Xx + Y)^{(p-1)/2}$. The Hasse–Witt determinant polynomial is defined as $H_{p,\alpha}(Y) = \det(M_{ij}(\alpha, Y))$. The generic fiber is ordinary if and only if this polynomial is nonzero modulo $p$.

2. Leading Term Analysis:
   To prove $H_{p,\alpha}(Y) \not\equiv 0 \pmod p$, the authors analyze the leading term of the polynomial. They identify a specific degree $E$ and show that the leading coefficient is of the form $\alpha^m u_p \Delta_p$, where:

   • $m$ is a non-negative integer.
   • $u_p$ is a product of binomial coefficients coprime to $p$.
   • $\Delta_p$ is a determinant involving binomial coefficients $\binom{(p-1)/2 - Q_i}{s_i - j}$, where $s_i$ are residues of $pi \pmod d$.

3. Reduction to a Binomial Determinant:
   The core technical innovation is the reduction of $\Delta_p$ modulo $p$ to a constant rational determinant $\Delta(s_1, \dots, s_g)$ independent of $p$. Using the congruence $(p-1)/2 - Q_i \equiv \beta_i \pmod p$ (where $\beta_i = s_i \frac{d-1}{2}$), they show $\Delta_p \equiv \Delta(s_1, \dots, s_g) \pmod p$.

4. Product Formula and Prime Factor Bounds:
   The authors derive an explicit product formula for the determinant $\Delta(s_1, \dots, s_g)$ by reducing it to a Vandermonde-type determinant. This formula involves factors of the form $(k - \lambda_r)$ and binomial coefficients.

   • Lemma 2.1 & 2.2: Establish that certain rational parameters $\lambda_r$ are non-integers, ensuring the determinant is well-defined and nonzero.
   • Lemma 2.6: Crucially, the authors prove that every prime factor of the numerator and denominator of the product formula for $\Delta$ is bounded by $P^+(d)$. This is achieved by analyzing the function $h(x, y) = (2d-2)x - 2dy - d$ over specific lattice regions $U$ and $V$.

5. Nonvanishing Modulo $p$:
   For any prime $p > P^+(d)$, the determinant $\Delta$ is a $p$-adic unit (nonzero modulo $p$). Consequently, the leading coefficient of $H_{p,\alpha}(Y)$ is nonzero modulo $p$ (provided $\alpha \not\equiv 0 \pmod p$), ensuring the polynomial itself is nonzero.

Key Contributions and Results

• Theorem 1.3: The paper proves that for every genus $g \ge 2$ and every nonzero algebraic integer $\alpha$, the family $C_\alpha: y^2 = x^d + \alpha x + t$ (with $d \in \{2g+1, 2g+2\}$) is generically ordinary at every prime $p > P^+(d)$, provided $p$ does not divide the norm of $\alpha$.
  • The bound is defined as:
    $$P^+(d) = \begin{cases} d^2 - 4d + 2 & \text{if } d \text{ is odd} \\ (d^2 - 3d + 2)/2 & \text{if } d \text{ is even} \end{cases}$$
• Resolution of Katz's Conjecture: As a direct corollary (Corollary 4.2), the paper confirms Katz's 2018 conjecture for the specific family $K$ (where $\alpha = -d$) for all $p > P^+(d)$.
• Generalization to Two-Parameter Families: Corollary 4.3 extends the result to the two-parameter family $y^2 = x^d + sx + t$ over the affine plane, showing it is generically ordinary for $p > P^+(d)$.
• Refinement for $p \equiv 1 \pmod d$: Corollary 4.5 shows that if $p \equiv 1 \pmod d$, the family is generically ordinary for all $p$ (without the $P^+(d)$ bound), as the determinant $\Delta_p$ becomes 1 in this case.

Significance and Claims
The paper claims to settle Katz's conjecture for all $g \ge 2$. The significance lies in moving beyond previous results (such as Miller's families where the coefficient matrix is monomial) to handle families where the Hasse–Witt matrix is generally non-monomial and contains many nonzero entries. The authors note that their approach requires a "different and new approach" because the determinant does not reduce to a single product.

The work establishes a concrete, quadratic bound $P^+(d)$ for generic ordinarity. The authors note that this bound is sharp in certain cases; data provided by Katz shows that generic ordinarity fails for $p = P^+(d)$ in specific instances (e.g., $(d, p) = (21, 359)$), indicating the bound cannot be weakened to $p \ge P^+(d)$ in general.

The paper situates its results within the tradition of Miller (1972) and the study of large monodromy in hyperelliptic loci, confirming that the ordinary locus is non-empty and dense for these families at large primes.