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The $L$-polynomials of van der Geer--van der Vlugt curves in characteristic $2$

Tetsushi Ito, Daichi Takeuchi, Takahiro Tsushima

TL;DR

This work extends the explicit description of L-polynomials for van der Geer–van der Vlugt curves to characteristic $2$ by developing a Witt-vector Lang torsor framework. The authors decompose the cohomology via Heisenberg group quotients, compute Frobenius eigenvalues $ au_{R,oldsymbol \xi,q}$ in terms of characters of $W_2({ m F}_q)$, and assemble the $L$-polynomial as a product over isotypic components. They further show how to pass to certain quotients, construct ${ m F}_{q^2}$-maximal curves via twists, and realize supersingular elliptic curves as quotients, illustrating rich interplays between abelian covers, Lang torsors, and Frobenius actions in characteristic $2$. The results yield explicit maximal curve examples and provide a versatile toolkit for studying L-polynomials and maximality in this setting.

Abstract

The van der Geer--van der Vlugt curves form a class of Artin--Schreier coverings of the projective line over finite fields. We provide an explicit formula for their $L$-polynomials in characteristic $2$, expressed in terms of characters of maximal abelian subgroups of associated Heisenberg groups. For this purpose, we develop new methods specific to characteristic $2$ that exploit the structure of the Heisenberg groups and the geometry of Lang torsors for $W_2$. As an application, we construct examples of curves in this family attaining the Hasse--Weil bound.

The $L$-polynomials of van der Geer--van der Vlugt curves in characteristic $2$

TL;DR

This work extends the explicit description of L-polynomials for van der Geer–van der Vlugt curves to characteristic by developing a Witt-vector Lang torsor framework. The authors decompose the cohomology via Heisenberg group quotients, compute Frobenius eigenvalues in terms of characters of , and assemble the -polynomial as a product over isotypic components. They further show how to pass to certain quotients, construct -maximal curves via twists, and realize supersingular elliptic curves as quotients, illustrating rich interplays between abelian covers, Lang torsors, and Frobenius actions in characteristic . The results yield explicit maximal curve examples and provide a versatile toolkit for studying L-polynomials and maximality in this setting.

Abstract

The van der Geer--van der Vlugt curves form a class of Artin--Schreier coverings of the projective line over finite fields. We provide an explicit formula for their -polynomials in characteristic , expressed in terms of characters of maximal abelian subgroups of associated Heisenberg groups. For this purpose, we develop new methods specific to characteristic that exploit the structure of the Heisenberg groups and the geometry of Lang torsors for . As an application, we construct examples of curves in this family attaining the Hasse--Weil bound.

Paper Structure

This paper contains 24 sections, 52 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Review of van der Geer--van der Vlugt curves
  3. The van der Geer--van der Vlugt curves associated with ${\mathbb F}_p$-linearized polynomials
  4. Heisenberg groups and their maximal abelian subgroups
  5. Decomposition of the $\ell$-adic cohomology
  6. Geometry of the curve $y^p + y = x^{p+1} + a_0 x^2$
  7. Witt vectors of length $2$
  8. The Heisenberg group $H_S$
  9. The Lang torsor over $W_2$
  10. Identification of $\mathcal{Q}_\xi$
  11. Calculation of Frobenius eigenvalues
  12. Quotients of van der Geer--van der Vlugt curves
  13. Maps between van der Geer--van der Vlugt curves
  14. The case where $R_2(x)=x^p+s_0x$
  15. Proof of the main theorem
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $a_{W_2},b_{W_2}$ be elements of ${\mathbb F}_p$ defined below. Define Then we have where we regard $(c_{R,\xi},0) \in W_2({\mathbb F}_q)$.

Theorems & Definitions (102)

  • Theorem 1.1: see Theorem \ref{['qppc']}
  • Theorem 1.2: see Theorem \ref{['qppt']}
  • Theorem 1.3: see Theorem \ref{['main']} (1)
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3: TT
  • Lemma 2.5: TT
  • Definition 2.6
  • Proposition 2.7
  • ...and 92 more