Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gauss sums and Van der Geer--Van der Vlugt curves

Daichi Takeuchi, Takahiro Tsushima

Abstract

We study Van der Geer--Van der Vlugt curves in a ramification-theoretic view point. We give explicit formulae on L-polynomials of these curves. As a result, we show that these curves are supersingular and give sufficient conditions for these curves to be maximal or minimal.

Gauss sums and Van der Geer--Van der Vlugt curves

Abstract

We study Van der Geer--Van der Vlugt curves in a ramification-theoretic view point. We give explicit formulae on L-polynomials of these curves. As a result, we show that these curves are supersingular and give sufficient conditions for these curves to be maximal or minimal.
Paper Structure (9 sections, 23 theorems, 42 equations)

This paper contains 9 sections, 23 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Van der Geer--Van der Vlugt curves
  3. Evaluations of $\tau_\xi$
  4. Formulae for $\tau_\xi$ in terms of local epsilon factors
  5. Evaluation of Gauss sums
  6. Conclusion
  7. Another computation of $\tau_{\xi}$
  8. Taking quotients of $C_R$
  9. Value of $\tau_{\xi}$

Key Result

Theorem 1.1

We assume that $A_R \subset \mathbb{F}_q^2$. For each $\psi \in \mathbb{F}_p^{\vee} \setminus \{1\}$ and $\xi \in A_{\psi}^{\vee}$, there exists a certain number $\tau_{\xi}$ which is $q^{1/2}$ times a root of unity such that a formula holds. Consequently, $\overline{C}_R$ is supersingular.

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 36 more