We show that there exists a surjection from the set of effective divisors of degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical version of the Riemann-Roch theorem. We then show that the restriction of the surjection is reduced to the bijection on an appropriate subset of the set of effective divisors of degree $g$ on the curve. Thus the subset of effective divisors has the additive group structure induced from the Jacobian. We finally realize the addition in Jacobian of a tropical hyperelliptic curve of genus $g$ via the intersection with a tropical curve of degree $3g/2$ or $3(g-1)/2$.
Let P 0 be a point on a tropical elliptic curve. Suppose P 1 to be the point such that P 1 = P 0 + T , where T is also a point on the tropical elliptic curve and "+" is the addition in the curve [12,9]. By applying the addition +T repeatedly, we obtain the sequence of points {P 0 , P 1 , P 2 , . . .} = {P 0 , P 0 + T, P 0 + 2T, . . .} on the tropical elliptic curve. The sequence of points thus obtained can be regarded as a dynamical system on the curve and is often referred as the ultradiscrete QRT system [9]. Since each member of the ultradiscrete QRT system has the tropical elliptic curve as its invariant curve and its general solution can be given by using the ultradiscrete theta function [11,8,9], it is considered to be a two-dimensional integrable dynamical system. It should be noted that the evolution of the ultradiscrete QRT system is given by a piecewise linear map. On the other hand, if we suppose P 1 to be the point such that P 1 = P 0 + P 0 = 2P 0 and apply the duplication repeatedly then we obtain the sequence of points {P 0 , P 1 , P 2 , . . .} = {P 0 , 2P 0 , 4P 0 , . . .} on the tropical elliptic curve as well. The sequence of points thus obtained can also be regarded as a dynamical system on the curve and is often referred as the solvable chaotic system [7,6]. The solvable chaotic system also has the tropical elliptic curve as its invariant curve and its general solution can also be given by using the ultradiscrete theta function; nevertheless it can not be considered to be an integrable system because the inverse evolution is not uniquely determined. Thus the additive group structure of the tropical elliptic curve leads to two kinds of dynamical systems, one is integrable and the other is not.
In analogy to the theory of plane curves over C, there exists a family of tropical plane curves parametrized with an invariant called the genus of the curve. A paradigmatic example of such a family of tropical plane curves consists of the tropical hyperelliptic curves. Tropical elliptic curves are of course the members of the family labeled by the lowest genus. Therefore, it is natural to consider that the additive group structure of the Jacobian of a tropical hyperelliptic curve also leads to several kinds of dynamical systems containing both an integrable one and a solvable chaotic one. In this paper, in order to investigate a dynamical system arising from the additive group structure of the tropical hyperelliptic curve, we first review several important notions in tropical geometry to describe the Riemann-Roch theorem for tropical curves. We also introduce a (g + 2)-parameter family of tropical hyperelliptic curves of genus g known as the set of the isospectral curves of (g+1)-periodic ultradiscrete Toda lattice. We then show the existence of a surjection between a tropical hyperelliptic curve and its Jacobian by using the tropical version of the Riemann-Roch theorem. This surjection induces the group structure of an appropriate set of effective divisor of degree g on the hyperelliptic curve from that of the Jacobian. We further show that we can realize the addition in the Jacobian of the tropical hyperelliptic curve of genus g as the addition of the g-tuples of points on the curve in terms of the intersection with a curve of genus 0.
We briefly review the notions of tropical curves as well as rational functions and divisors on them. By using these tools we mention the Riemann-Roch theorem for tropical curves, which was independently found by Gathemann-Kerber [3] and Mikhalkin-Zharkov [8] in 2006.
A metric graph is a pair (Γ, l) consisting of a graph Γ together with a length function l : E(Γ) → R >0 , where E(Γ) is the edge set of the graph Γ. The first Betti number of Γ is called the genus of Γ. A tropical curve is a metric graph (Γ, l) with a length function l : E(Γ) → R >0 ∪ {∞},i.e., a metric graph with possibly unbounded edges. where a P ∈ Z and a P = 0 for all but finitely many P ∈ Γ.
The addition of two divisors D = P ∈Γ a P P and D ′ = P ∈Γ a ′ P P on a tropical curve Γ are defined to be D + D ′ = P ∈Γ (a P + a ′ P )P . All divisors on Γ then naturally compose an abelian group. We call it the divisor group of Γ and denote it by Div(Γ). The degree deg D of a divisor D = P ∈Γ a P P is defined to be the integer P ∈Γ a P . The support suppD of D is defined to be the set of all points of Γ occurring with a non-zero coefficient. If all the coefficients a P of a divisor D = P ∈Γ a P P are non-negative then the divisor is called effective and is written D > 0. We define the canonical divisor K Γ of Γ to be
where val(P ) is the valence of the point P ∈ Γ [13]. If P is an inner point on an edge of Γ then val(P ) = 2, therefor such points never appear in K Γ . The canonical divisor K Γ of a tropical curve Γ of genus g is an effective divisor of degree 2g -2 [8].
A rational function on a tropical curve Γ is a continuous function f : Γ → R ∪ {±∞} such that the restriction of f to any edge of Γ is
This content is AI-processed based on open access ArXiv data.
