Factorization of Additive Polynomials and van der Geer--van der Vlugt curves in characteristic 2

Technical Summary: Factorization of Additive Polynomials and van der Geer–van der Vlugt Curves in Characteristic 2

Problem Statement
The paper addresses the explicit computation of Frobenius eigenvalues for van der Geer–van der Vlugt curves defined over finite fields of characteristic 2. These curves are given by the equation $C_R: y^p - y = xR(x)$, where $R(x)$ is an $\mathbb{F}_p$-linearized polynomial. While formulas for these eigenvalues exist for odd characteristics (expressed via quadratic Gauss sums), the case $p_0=2$ has remained more elusive. Previous work by the authors (specifically reference [7]) provided a formula for characteristic 2, but it relied on the choice of a maximal abelian subgroup of a Heisenberg group associated with the curve. This dependence on arbitrary choices rendered the formula complicated and difficult to apply for explicit computations. The primary goal of this work is to derive a simpler, choice-independent formula and to utilize it to characterize maximal and minimal curves within this family.

Methodology
The authors adopt a novel approach centered on the factorization of additive polynomials rather than the group-theoretic analysis of Heisenberg groups used in prior work.

1. Adjoint Polynomials and Factorization: The core insight involves the adjoint operator $F \mapsto F^*$ for $\mathbb{F}_p$-linearized polynomials. The authors consider a pair $(F, t)$, where $F$ is an $\mathbb{F}_p$-linearized polynomial satisfying specific conditions (separability, $F^*(1)=0$, and kernel constraints) and $t \in \mathbb{F}_q$. They demonstrate that for a given polynomial $R$, if it can be expressed in the form $R = R_{F,t}$ derived from such a pair, the associated curve $C_R$ is isomorphic to a curve $D_{F,t}$ defined by $y^p - y = x R_{F,t}(x)$.
2. Cohomological Computation: The authors compute the $\ell$-adic cohomology of these curves by relating them to Lang torsors over the Witt vector scheme $W_2$ of length two. They utilize the Lang morphism $L_p: W_2 \to W_2$ and associated sheaves to decompose the cohomology groups.
3. Explicit Eigenvalue Derivation: By analyzing the action of the geometric Frobenius on these sheaves, they derive an explicit formula for the eigenvalues. This formula relies on a character sum involving a specific function $Q_q$ (an even-characteristic analogue of a quadratic form) evaluated over the kernel of the adjoint polynomial $F^*$.
4. Linear-Algebraic Characterization: The paper establishes a correspondence between the existence of such a factorization and the geometric properties of the curve, specifically the structure of the kernel of $R + R^*$.

Key Contributions and Results

• A New Explicit Formula (Theorem 1.1): The authors provide a simplified formula for the $L$-polynomial of the curve $D_{F,t}$. The eigenvalues are given by $\tau_v = Q_q(t+v)^{-1}(-1-\sqrt{-1})^{[F_q:F_2]}$ for $v$ in the kernel of $F^*$. This formula is independent of the auxiliary choices required in previous literature.
• Characterization of Extremal Curves (Theorem 1.2 & 1.3): The paper provides a necessary and sufficient condition for a van der Geer–van der Vlugt curve to be $\mathbb{F}_q$-maximal or $\mathbb{F}_q$-minimal.
  • It proves that every $\mathbb{F}_q$-extremal curve in this family arises from a specific linear-algebraic construction involving a subspace $W \subset \mathbb{F}_q$ and an element $t$.
  • This yields a constructive recipe: one can generate all maximal/minimal curves by choosing a subspace $W$ and an element $t$ satisfying specific trace and quadratic form conditions.
• Construction of Maximal Curves over $\mathbb{F}_{q^2}$ (Theorem 1.4): The authors give an explicit condition under which the curve $D_{F,t}$ is maximal over $\mathbb{F}_{q^2}$, specifically relating to the trace of $t$.
• Periods and Parities (Section 1.4 & 6.5): The paper investigates the "period" (the smallest extension degree where the curve becomes extremal) and "parity" (whether it is maximal or minimal at that degree).
  • It is shown that the period $\mu$ is always even.
  • The paper classifies which pairs $(\mu, \delta)$ are realizable. Notably, it proves that for $p=2$, the pair $(4, 1)$ (period 4, minimal) is impossible.
  • It establishes that if $\mathbb{F}_p$ contains a primitive $m$-th root of unity for an odd integer $m$, then the pair $(2m, 1)$ is never realized.

Significance
The paper claims that its primary significance lies in the simplicity and utility of the new formula. By removing the dependency on the choice of a maximal abelian subgroup of a Heisenberg group, the new approach allows for more direct and explicit computations of Frobenius eigenvalues.

Furthermore, the work provides a complete classification of the maximal and minimal curves within the family of van der Geer–van der Vlugt curves in characteristic 2. It demonstrates that these extremal curves are not isolated phenomena but form a subclass that can be systematically constructed from linear-algebraic data (subspaces and trace conditions). This bridges the gap between the abstract existence of such curves and their concrete construction, offering a "recipe" for generating them. The results also refine the understanding of the periods and parities of these curves, ruling out specific combinations of period and parity that were previously uncertain.