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Eisenstein integers and equilateral ideal triangles

Greg McShane

Abstract

We discuss the relationship between Penner's $λ$-length and the norms of Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that every prime $p$ of the form $3k + 1$ is the norm of an Eisenstein integer that is can be written as $a^2 - ab + b^2$ for some $a,b \in \mathbb{Z}$.

Eisenstein integers and equilateral ideal triangles

Abstract

We discuss the relationship between Penner's -length and the norms of Eisenstein integers. This leads to a geometric proof of the fact, attributed to Fermat, that every prime of the form is the norm of an Eisenstein integer that is can be written as for some .
Paper Structure (17 sections, 9 theorems, 25 equations, 6 figures)

This paper contains 17 sections, 9 theorems, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Eisenstein integers
  3. Farey diagram and $\lambda$-lengths
  4. Properly immersed ideal triangles
  5. Sketch of proof
  6. The case $p=3$
  7. Thanks
  8. Reciprocals of norms and arcs
  9. Ford circles, $\lambda$-lengths
  10. $\lambda$-length
  11. Gaussian integers
  12. Eisenstein integers
  13. The three punctured sphere
  14. Cusp regions
  15. Automorphism groups of $\mathbb{H}/\Gamma(2)$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $p$ be a prime then the equation has a solution in integers iff $p =2$ or $p$ is of the form $4k + 1$.

Figures (6)

  • Figure 1: Farey diagram.
  • Figure 2: Three punctured sphere with cusps labelled by $\Gamma(2)$-orbits and some arcs. Note that the dotted geodesic in the middle is the edge of an ideal triangle which is embedded but not properly immersed as two spikes meet at the cusp $\infty$.
  • Figure 3: Three intersecting lifts of an equilateral ideal triangle. The dotted lines intersect in two points the lower of which is the barycenter of $1/3, 2/3,\infty$ which coincides with the barycenter of $0, 1/2,1$. The upper point is the barycenter of $0,1,\infty$ that is $1+\omega$.
  • Figure 4: Projection of the ideal triangle $1/3, 2/3,\infty$ to $\mathbb{H}/\Gamma(2)$.
  • Figure 5: Ford circles with tangent points and curvatures. Recall that the curvature of a euclidean circle is twice the reciprocal of the square of its radius.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Theorem 1.1: Fermat
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 4.1
  • Lemma 4.2