[Paper Review] The group law on a tropical elliptic curve

Magnus Dehli Vigeland|ArXiv.org|Nov 22, 2004

Polynomial and algebraic computation2 references21 citations

TL;DR

This paper establishes a group law on tropical elliptic curves by defining the Jacobian as the complement of the curve's tentacles, denoted $̅{C}$. It proves that the group operation satisfies $d_C(\mathcal{O}, P+Q) = d_C(\mathcal{O},P) + d_C(\mathcal{O},Q)$, and shows that $(\u0305{C}, \mathcal{O})$ is isomorphic to $S^1$, generalizing the classical elliptic curve group structure to the tropical setting using a distance function and divisor equivalence.

ABSTRACT

In analogy with the classical group law on a plane cubic curve, we define a group law on a smooth plane tropical cubic curve. We show that the resulting group is isomorphic to $S^1$.

Motivation & Objective

• To define a natural group structure on tropical elliptic curves, analogous to the classical group law on elliptic curves.
• To characterize the Jacobian of a tropical elliptic curve as the complement of its tentacles, rather than the entire curve.
• To establish a distance function $d_C(P,Q)$ that governs the group operation and satisfies a linearity property with respect to the base point.
• To prove that the group $(\u0305{C}, \mathcal{O})$ is isomorphic to the circle group $S^1$ via a distance-based isomorphism.

Proposed method

• Define the tropical semiring $(\mathbb{R}, \oplus, \odot)$ with $a \oplus b = \max\{a,b\}$ and $a \odot b = a + b$, and extend it to $\mathbb{R}^n$ and $\mathbb{P}^n_{tr}$.
• Define a tropical curve $C = V(f)$ as the set of points where a tropical polynomial $f$ is not linear, with $f$ homogeneous of degree $d$.
• Introduce the distance function $d_C(P,Q)$ on $\u0305{C}$, the complement of the tentacles, which measures the displacement along the curve's metric structure.
• Use divisor equivalence and the theory of tropical linear equivalence to define the group law via the map $P \mapsto P - \mathcal{O}$, where $\mathcal{O} \in \u0305{C}$.
• Establish the key relation $d_C(\mathcal{O}, P+Q) = d_C(\mathcal{O},P) + d_C(\mathcal{O},Q)$ using properties of divisor equivalence and the distance function.
• Construct a group isomorphism $\lambda: (\u0305{C}, \mathcal{O}) \to \mathbb{R}/\mathbb{Z} \approx S^1$ via $\lambda(P) = d_C(\mathcal{O},P)/L$, where $L$ is the total length of $\u0305{C}$.

Experimental results

Research questions

• RQ1 How can a group law be defined on a tropical elliptic curve, given that the Jacobian is not the entire curve?
• RQ2 What role does the distance function $d_C(P,Q)$ play in encoding the group operation?
• RQ3 How does the group structure on $\u0305{C}$, the non-tentacle part of the curve, relate to the classical $S^1$ group structure?
• RQ4 Can the tropical group law be geometrically realized using tropical lines and intersection points, similar to the classical chord-tangent construction?
• RQ5 Is the group $(\u0305{C}, \mathcal{O})$ isomorphic to $S^1$, and if so, under what metric or functional correspondence?

Key findings

• The Jacobian of a tropical elliptic curve $C$ is isomorphic to the set $\u0305{C}$, the complement of the tentacles, rather than the entire curve.
• There exists a canonical bijection $\tau_{\mathcal{O}}: \u0305{C} \to \mathrm{Jac}(C)$ given by $P \mapsto P - \mathcal{O}$, which endows $\u0305{C}$ with a group structure.
• The group law on $\u0305{C}$ satisfies the key identity $d_C(\mathcal{O}, P+Q) = d_C(\mathcal{O},P) + d_C(\mathcal{O},Q)$, linking the distance function to the group operation.
• The group $(\u0305{C}, \mathcal{O})$ is isomorphic to the circle group $S^1$ via the map $\lambda(P) = d_C(\mathcal{O},P)/L$, where $L$ is the total length of $\u0305{C}$.
• The group law can be geometrically realized using tropical lines: $P+Q$ is the third intersection point of the line through $P$ and $Q$ with $\u0305{C}$, followed by a second line through $R$ and $\mathcal{O}$, with adjustments for non-good pairs via parallel displacement.
• The isomorphism $\lambda$ is a group isomorphism, as shown by the additive property $\lambda(P+Q) = \lambda(P) + \lambda(Q)$, which follows directly from the distance identity.