Addition in Jacobians of tropical hyperelliptic curves
aa r X i v : . [ n li n . S I] N ov Addition in Jacobians of tropical hyperelliptic curves
Atsushi
Nobe ∗ Department of Mathematics, Faculty of Education, Chiba University,1-33 Yayoi-cho Inage-ku, Chiba 263-8522, Japan.
Abstract
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 theRiemann-Roch theorem. We then show that the restriction of the surjection is reducedto 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 structureinduced from the Jacobian. We finally realize the addition in Jacobian of a tropicalhyperelliptic curve of genus g via the intersection with a tropical curve of degree 3 g/ g − / Let P be a point on a tropical elliptic curve. Suppose P to be the point such that P = P + T , where T is also a point on the tropical elliptic curve and “+” is the additionin the curve [12, 9]. By applying the addition + T repeatedly, we obtain the sequenceof points { P , P , P , . . . } = { P , P + T, P + 2 T, . . . } on the tropical elliptic curve. Thesequence of points thus obtained can be regarded as a dynamical system on the curve and isoften referred as the ultradiscrete QRT system [9]. Since each member of the ultradiscreteQRT system has the tropical elliptic curve as its invariant curve and its general solutioncan be given by using the ultradiscrete theta function [11, 8, 9], it is considered to be atwo-dimensional integrable dynamical system. It should be noted that the evolution ofthe ultradiscrete QRT system is given by a piecewise linear map. On the other hand, ifwe suppose P to be the point such that P = P + P = 2 P and apply the duplicationrepeatedly then we obtain the sequence of points { P , P , P , . . . } = { P , P , P , . . . } on the tropical elliptic curve as well. The sequence of points thus obtained can alsobe regarded as a dynamical system on the curve and is often referred as the solvablechaotic system [7, 6]. The solvable chaotic system also has the tropical elliptic curve asits invariant curve and its general solution can also be given by using the ultradiscretetheta function; nevertheless it can not be considered to be an integrable system becausethe inverse evolution is not uniquely determined. Thus the additive group structure of thetropical elliptic curve leads to two kinds of dynamical systems, one is integrable and theother is not. ∗ e-mail: [email protected]
1n analogy to the theory of plane curves over C , there exists a family of tropical planecurves parametrized with an invariant called the genus of the curve. A paradigmaticexample of such a family of tropical plane curves consists of the tropical hyperellipticcurves. Tropical elliptic curves are of course the members of the family labeled by thelowest genus. Therefore, it is natural to consider that the additive group structure ofthe Jacobian of a tropical hyperelliptic curve also leads to several kinds of dynamicalsystems containing both an integrable one and a solvable chaotic one. In this paper, inorder to investigate a dynamical system arising from the additive group structure of thetropical hyperelliptic curve, we first review several important notions in tropical geometryto 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 theisospectral curves of ( g +1)-periodic ultradiscrete Toda lattice. We then show the existenceof a surjection between a tropical hyperelliptic curve and its Jacobian by using the tropicalversion of the Riemann-Roch theorem. This surjection induces the group structure of anappropriate set of effective divisor of degree g on the hyperelliptic curve from that of theJacobian. We further show that we can realize the addition in the Jacobian of the tropicalhyperelliptic curve of genus g as the addition of the g -tuples of points on the curve interms of the intersection with a curve of genus 0. We briefly review the notions of tropical curves as well as rational functions and divisorson 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] in2006.
Definition 2.1 (Tropical curve) A metric graph is a pair (Γ , l ) consisting of a graph Γ together with a length function l : E (Γ) → R > , 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 metricgraph (Γ , l ) with a length function l : E (Γ) → R > ∪ {∞} ,i.e., a metric graph with possiblyunbounded edges. Definition 2.2 (Divisor) A divisor D on a tropical curve Γ is a formal Z -linear com-bination of finite points on Γ D = X P ∈ Γ a P P, where a P ∈ Z and a P = 0 for all but finitely many P ∈ Γ . The addition of two divisors D = P P ∈ Γ a P P and D ′ = P P ∈ Γ a ′ P P on a tropical curve Γare defined to be D + D ′ = P P ∈ Γ ( a P + a ′ P ) P . All divisors on Γ then naturally compose anabelian group. We call it the divisor group of Γ and denote it by Div(Γ). The degree deg D of a divisor D = P P ∈ Γ a P P is defined to be the integer P P ∈ Γ a P . The support supp D 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 P ∈ Γ a P P are non-negative then the divisor is2alled effective and is written D >
0. We define the canonical divisor K Γ of Γ to be K Γ := X P ∈ Γ (val( P ) − P, where val( P ) is the valence of the point P ∈ Γ [13]. If P is an inner point on an edge ofΓ then val( P ) = 2, therefor such points never appear in K Γ . The canonical divisor K Γ ofa tropical curve Γ of genus g is an effective divisor of degree 2 g − Definition 2.3 (Rational function) A rational function on a tropical curve Γ is acontinuous function f : Γ → R ∪ {±∞} such that the restriction of f to any edge of Γ isa piecewise linear integral function. The values ±∞ can only be taken at the unboundededges of Γ . The order ord P f of a rational function f at P ∈ Γ is defined to be the sum of the outgoingslopes of all segments of Γ emanating from P . If ord P f > P ∈ Γ is calledthe zero of f of order ord P f . If ord P f < P is called the pole of f of order | ord P f | .We define the principal divisor of a rational function f (or divisor associated to f ) onΓ to be ( f ) := X P ∈ Γ (ord P f ) P. Remark 2.4
For any rational function f on a tropical curve Γ we have deg( f ) = X P ∈ Γ (ord P f ) = 0 . Definition 2.5 (Linear system)
Let D be a divisor of degree n on a tropical curve Γ .We denote by R ( D ) the set of all rational functions f on Γ such that the divisor ( f ) + D is effective: R ( D ) := { f | ( f ) + D > } . For any f ∈ R ( D ) the divisor ( f )+ D is a sum of exactly deg (( f ) + D ) = deg( f )+deg D = n points by the above remark. When formulating a statement about the dimensions of the linear systems R ( D ) wehave to be careful since R ( D ) is in general not a vector space but a polyhedral complexand hence its dimension is ill-defined. The following definition serve as a replacement. Definition 2.6 (Rank)
Let D be a divisor of degree n on a tropical curve Γ . We definethe rank r ( D ) of the divisor D to be the maximal integer k such that for all choices of (notnecessarily distinct) points P , P , . . . , P k ∈ Γ we have R ( D − P − P − · · · − P k ) = ∅ . If R ( D ) = ∅ then we define r ( D ) = − . If we want to specify the curve Γ in the notation of R ( D ) and r ( D ) we also write them as R Γ ( D ) and r Γ ( D ), respectively. 3 efinition 2.7 (Equivalence of divisors) Two divisors D and D ′ on a tropical curve Γ are called equivalent and written D ∼ D ′ if there exists a rational function f on Γ suchthat D ′ = D + ( f ) . If D ∼ D ′ then D ′ = D + ( f ) for some rational function f on Γ. Then the map R ( D ) → R ( D ′ ), g g + f is a bijection because we have ( g ) + D ′ = ( g ) + D + ( f ) = ( g + f ) + D .Thus we have r ( D ) = r ( D ′ ). Lemma 2.8 (See [3])
Let ¯Γ be a tropical curve and let Γ be the metric graph obtainedfrom ¯Γ by removing all unbounded edges. Then every divisor D ∈ Div(¯Γ) is equivalent on ¯Γ to a divisor D ′ with supp D ′ ⊂ Γ . If D is effective then D ′ can be chosen to be effectiveas well. Moreover, R Γ ( D ′ ) = ∅ if and only if R ¯Γ ( D ′ ) = ∅ . Let ¯Γ, Γ, and D ′ as in lemma 2.8. By lemma 2.8 any effective divisor on ¯Γ is equivalentto a divisor of the same degree with the support on Γ. Therefore we conclude that r ¯Γ ( D ′ ) = r Γ ( D ′ ).We then have the Riemann-Roch theorem. Theorem 2.9 (Riemann-Roch theorem for tropical curves [3, 8, 1])
For any di-visor D on a tropical curve ¯Γ of genus g we have r ( D ) − r ( K ¯Γ − D ) = deg D + 1 − g. It immediately follows a corollary of theorem 2.9.
Corollary 2.10 If deg D > g − then r ( D ) = deg D − g . (Proof) Since deg K ¯Γ = 2 g −
2, we have deg( K ¯Γ − D ) <
0. This implies R ( K ¯Γ − D ) = ∅ and hence r ( K ¯Γ − D ) = −
1. By the Riemann-Roch theorem we have r ( D ) = deg D + 1 − g − D − g as desired. By applying the Riemann-Roch theorem to a tropical curve of genus g , one can obtainseveral propositions concerning a surjection from the set of effective divisors of degree g ofthe tropical curve to its Jacobian. Let ¯Γ be a tropical curve of genus g . We define D (¯Γ)to be the subgroup of the divisor group Div(¯Γ) of ¯Γ generated by the divisors of degree 0.We also define D l (¯Γ) to be the subgroup of D (¯Γ) generated by the principal divisors ofrational functions on ¯Γ. Definition 3.1 (Picard group)
We define the Picard group
Pic (¯Γ) of a tropical curve ¯Γ to be the residue class group Pic (¯Γ) := D (¯Γ) / D l (¯Γ) . In particular, by lemma 2.8 wehave Pic (¯Γ) = Pic (Γ) := D (Γ) / D l (Γ) . Note that any principal divisor is of degree 0. ∗ be the vector space of R -valued linear functions on Ω(Γ). Then the integral cycles H (Γ , Z ) form a lattice inΩ(Γ) ∗ by integrating over them. Definition 3.2 (Jacobian)
We define the Jacobian J (Γ) of a tropical curve ¯Γ to be J (Γ) := Ω(Γ) ∗ /H (Γ , Z ) . Remark 3.3
Since there exists an isomorphism between
Pic (Γ) and J (Γ) [8], we canidentify them: J (Γ) ≃ Pic (Γ) = D (Γ) / D l (Γ) . We define the subset D + g (Γ) of Div(¯Γ) to be D + g (Γ) := { D ∈ Div(Γ) | deg D = g, D > } . Fix an element D ∗ of D + g (Γ). Then we can define two maps φ : D + g (Γ) → D (Γ) and¯ φ : D + g (Γ) → J (Γ) to be φ ( A ) = A − D ∗ and ¯ φ ( A ) ≡ A − D ∗ (mod D l (Γ)) , for A ∈ D + g (Γ), respectively. We then have the following theorem. Theorem 3.4
The map ¯ φ : D + g (Γ) → J (Γ) is surjective. (Proof) It is sufficient to show that there exists an element A of D + g (Γ) such that A − D ∗ ∼ D for any D ∈ D (Γ). By the Riemann-Roch theorem we have r ( D + D ∗ ) = deg( D + D ∗ ) + 1 − g + r ( K ¯Γ − D − D ∗ )= g + 1 − g + r ( K ¯Γ − D − D ∗ ) ≥ r ( K ¯Γ − D − D ∗ ) ≥ − R ( D + D ∗ ) = ∅ and hence thereexists a rational function h satisfying ( h ) + D + D ∗ >
0. Let A = ( h ) + D + D ∗ . Thendeg A = 0 + 0 + g = g and A > A ∈ D + g (Γ). Moreover we have¯ φ ( A ) ≡ ( h ) + D ≡ D (mod D l (Γ)) . This completes the proof.As in the non-tropical case, for a tropical curve of genus one the surjection ¯ φ reducesto the bijection. This fact was first found by Vigeland in 2004 [12]. Theorem 3.5
The map ¯ φ : D + g (Γ) → J (Γ) is bijective if and only if g = 1 . (Proof) If we assume ¯ φ ( P ) ≡ ¯ φ ( Q ) for some P, Q ∈ D + g (Γ) then we have P − D ∗ ∼ Q − D ∗ and hence P − Q ∼
0. This means that there exists a rational function h satisfying( h ) = P − Q . Since P is effective, ( h )+ Q is effective as well. Therefore the rational function5 is an element of R ( Q ). If we assume g = 1 then we have deg Q = 1 > g −
2. Bycorollary 2.10 we obtain r ( Q ) = deg Q − g = 1 − . This implies that h ∈ R ( Q ) is a constant function, and hence P − Q = ( h ) = 0. Thus themap ¯ φ is injective.On the other hand, if we assume g ≥ Q = g ≤ g − K ¯Γ − Q ) ≥
0. Let 1 be a constant function then (1) + K ¯Γ − Q = K ¯Γ − Q >
0. Thisimplies 1 ∈ R ( K ¯Γ − Q ) = ∅ and hence r ( K ¯Γ − Q ) ≥
0. Thus we have r ( Q ) = deg Q + 1 − g + r ( K ¯Γ − Q ) ≥ . Therefore there exists a rational function h satisfying P − Q = ( h ) = 0.Any element of D + g (Γ) is given by P + P + · · · + P g , where P , P , · · · , P g are thepoints on the metric graph Γ not necessarily distinct. Let us denote the image of P + P + · · · + P g ∈ D + g (Γ) with respect to the map ¯ φ by( P P · · · P g ) := ¯ φ ( P + P + · · · + P g ) . Since the map ¯ φ : D + g (Γ) → J (Γ) is surjective, we have J (Γ) = { ( P P · · · P g ) | P , P , · · · , P g ∈ Γ } . Thus the addition in J (Γ) can be pulled back in Γ by ¯ φ . Actually, we define the additionof g -tuples P = ( P , P , . . . , P g ) and Q = ( Q , Q , . . . , Q g ) of the points on Γ to be P + Q := ( P P · · · P g ) + ( Q Q · · · Q g ) , where the addition in the right-hand-side is considered to be the one in J (Γ). In this and the subsequent sections we concentrate on a ( g +2)-parameter family of smoothtropical curves of genus g called the tropical hyperelliptic curves.Let us consider the (non-tropical) hyperelliptic curve of genus g defined by the poly-nomial of degree 2 g + 2: f ( x, y ) := y − P ( x ) + 4 c − ,P ( x ) := g +1 X i =0 c i x i , c g +1 = 1 , c i ∈ C for i = − , , . . . , g, where we assume that f ( x,
0) = − P ( x ) + 4 c − has no multiple root. By applying to f ( x, y ) the birational transformation ( x, y ) ( X, Y ) X = x, Y = 12 y − P ( x ) ,
6e obtain the hyperelliptic curve given by the polynomial F ( X, Y ) := Y + Y P ( X ) + c − . Now we tropicalize F ( X, Y ). Replace the addition + and the multiplication × in F ( X, Y ) with the tropical addition ⊕ and the tropical multiplication ⊗ , respectively. Ifwe define these tropical operations to be A ⊕ B := max( A, B ) , A ⊗ B := A + B for A, B ∈ T := R ∪ {−∞} then we obtain the following tropical polynomial˜ F ( X, Y ) := Y ⊗ ⊕ Y ⊗ ˜ P ( X ) ⊕ c − = max (cid:16) Y, Y + ˜ P ( X ) , c − (cid:17) , ˜ P ( X ) := g +1 M i =0 c i ⊗ X ⊗ i = g +1 max i =0 ( c i + iX ) , where we assume c g +1 = 0, c i ∈ T for i = − , , . . . , g . Moreover assume c g +1 = c g = 0 ,c − < c ,c i + c i +2 < c i +1 for i = 0 , , . . . , g − ,c g − < c g = 0 . Then the tropical polynomial ˜ F ( X, Y ) defines a smooth tropical curve ¯Γ of genus g to bethe set of its all non-differentiable points with respect to X or Y (see figure 1).The curve ¯Γ thus obtained is called the tropical hyperelliptic curve and is knownas the isospectral curve of the ultradiscrete periodic Toda lattice, or the periodic box-ballsystem [5].Let v e be the primitive tangent vector along the edge e ∈ E (¯Γ), where E (¯Γ) is theedge set of the tropical hyperelliptic curve ¯Γ. Also let | e | and | v e | be the Euclidean lengthof e and v e , respectively. The length function l : E (¯Γ) → R > ∪ {∞} of ¯Γ is defined to be l ( e ) := | e || v e | . The vertex of ¯Γ whose Y -coordinate is greater than c − / V i and itsconjugate (defined below) by V ′ i ( i = 0 , , . . . , g ): V = (0 , , V ′ = (0 , c − ) ,V i = ( c g − i − c g − i +1 , ( g − i + 1) c g − i − ( g − i ) c g − i +1 ) for i = 1 , , . . . , g,V ′ i = ( c g − i − c g − i +1 , c − − ( g − i + 1) c g − i + ( g − i ) c g − i +1 ) for i = 1 , , . . . , g. The outgoing slope of the edge −−−−→ V i V i +1 , which emanates from V i and connects it with V i +1 ,is g − i for i = 0 , , . . . , g − A tropical curve is called hyperelliptic if there is a linear system with degree 2 and rank 1 [4]. V V V g − V g − V g V ′ V ′ V ′ V ′ g − V ′ g − V ′ g V ′∞ V ∞ V −∞ V ′−∞ A A A A g − A g − A g Y = c − XY O
Figure 1: The tropical hyperelliptic curve ¯Γ.We define the bifurcation point of the tropical hyperelliptic curve ¯Γ to be the inter-section point of ¯Γ and the line Y = c − /
2. Note that ¯Γ is symmetric under the refectionwith respect to Y = c − /
2. There exist exactly g + 1 bifurcation points on ¯Γ. We denotethe g + 1 bifurcation points by A , A , . . . , A g (see figure 1). We also define the conjugate of a point P on ¯Γ to be the point which coincide with P under the reflection with respectto Y = c − /
2. The conjugate of a point P on ¯Γ is denoted by P ′ . Thus the bifurcationpoint is characterized as the point P on ¯Γ such that P = P ′ .The canonical divisors of ¯Γ and Γ are K ¯Γ = g X i =0 (cid:0) V i + V ′ i (cid:1) − V −∞ − V ′−∞ − V ∞ − V ′∞ ,K Γ = g − X i =1 (cid:0) V i + V ′ i (cid:1) , where V −∞ , V ′−∞ , V ∞ , and V ′∞ are the end points of the unbound edges (see figure 1). Ofcourse, deg K ¯Γ = deg K Γ = 2 g − D + g (Γ) on which the surjection¯ φ : D + g (Γ) → J (Γ) reduces to the bijection. Let α i be the basis of the fundamental group π (Γ) of Γ for i = 1 , , . . . , g (see figure 2). Also let α ij = α i ∩ α j \ { end points of α i ∩ α j } α i V i V ′ i V i − V ′ i − Figure 2: The basis of the fundamental group π (Γ).for i = j ∈ { , , . . . , g } . We define the subset ˜ D + g (Γ) of D + g (Γ) to be˜ D + g (Γ) = (cid:26) P + · · · + P g (cid:12)(cid:12) P i ∈ α i for all i = 1 , , . . . , g andthere exists at most one point on α ij (cid:27) . We then have the following theorem.
Theorem 4.1 (See [5])
A reduced map ¯ φ | ˜ D + g (Γ) is bijective ¯ φ | ˜ D + g (Γ) : ˜ D + g (Γ) ∼ → J (Γ) . We define the set Γ + g to be the set of g -tuples P = ( P , P , . . . , P g ) satisfying P + P + · · · + P g ∈ ˜ D + g (Γ):Γ + g := n P = ( P , P , . . . , P g ) | P + P + · · · P g ∈ ˜ D + g (Γ) o . In terms of the map ¯ φ , the additive group structure of Γ + g is induced from J (Γ) . g Now we assume that the genus g of the tropical hyperelliptic curve ¯Γ is even. Let us fix D ∗ ∈ D + g (Γ) as follows D ∗ = g (cid:0) V + V ′ (cid:1) . We have the following proposition. 9 roposition 5.1
For any even g we have (cid:16) P P · · · P g/ P ′ P ′ · · · P ′ g/ (cid:17) = 0 , where P , P , · · · , P g/ are the points on Γ and P ′ , P ′ , · · · , P ′ g/ are their conjugates, re-spectively. (Proof) Assume ( P P · · · P g ) = 0. Then we have P + P + · · · + P g − D ∗ ∼ , and vice versa. Therefore ( P P · · · P g ) = 0 is equivalent to the existence of a rationalfunction h ∈ R ( D ∗ ) on Γ such that ( h ) = P + P + · · · + P g − D ∗ .We show that there exists a rational function h ∈ R ( D ∗ ) on Γ such that ( h ) = P + · · · + P g/ + P ′ + · · · + P ′ g/ − D ∗ . Without loss of generality we can assume the Y -coordinate of P i is greater than or equal to c − / i = 1 , , · · · , g/
2. Let us denote thepath in Γ connecting A i with V i − (resp. V ′ i − ) through V i (resp. V ′ i ) by π i (resp. π ′ i ) for i = 1 , , . . . , g (see figure 3). Also denote the path in Γ connecting A with V (resp. V ′ ) ✯❥ A i V i V ′ i V i − V ′ i − π i π ′ i Figure 3: The paths π i and π ′ i .by π (resp. π ′ ).Owing to the above assumption, there exist g/ P , P , . . . , P g/ on ∪ gi =0 π i . Weconstruct the rational function ¯ h on ∪ gi =0 π i as follows. The value of ¯ h at A g can arbitrarilybe chosen. We start walking the path π g form A g to V g − . If we find the point P g on thepath π g then we set ¯ h ( P g ) = ¯ h ( A g ) and the outgoing slope of ¯ h from P g to be 1. If wefind the next point P g on the path π g then we set the outgoing slope of ¯ h from P g tobe 2. Thus P g and P g are the zeros of ¯ h of order 1. If P g = P g then the part of ¯ h ofslope 1 disappears and P g = P g is the zero of ¯ h of order 2. If there exist exactly k points P g , P g , . . . , P g k on the path π g then the slope of ¯ h at V g − is k . The value of ¯ h at V g − isuniquely determined. By applying the same procedure inductively to the remaining paths π g − , π g − , . . . , π we obtain the rational function ¯ h on ∪ gi =0 π i satisfying(¯ h ) = P + P + · · · + P g/ − g V . Note that the slope of ¯ h at V is g/ g/ P , P , . . . , P g/ on ∪ gi =0 π i .Also note that the value of ¯ h at the bifurcation points A g − , A g − , . . . , A are uniquelydetermined by h ( V g − ) , h ( V g − ) , . . . , h ( V ), respectively.10n the same manner we obtain the rational function ¯ h ′ on ∪ gi =0 π ′ i satisfying(¯ h ′ ) = P ′ + P ′ + · · · + P ′ g/ − g V ′ . If we set ¯ h ( A g ) = ¯ h ′ ( A g ) then h = ¯ h ∪ ¯ h ′ = ( ¯ h on ∪ gi =0 π i , ¯ h ′ on ∪ gi =0 π ′ i is the rational function on Γ = ( ∪ gi =0 π i ) ∪ ( ∪ gi =0 π ′ i ) satisfying ( h ) = P + · · · + P g/ + P ′ + · · · + P ′ g/ − D ∗ as desired.Combining theorem 4.1 and proposition 5.1, we obtain the following proposition con-cerning the unit of addition O of the group (Γ + g , O ). Proposition 5.2
For any even g , the g -tuple ( V , V , · · · , V g − , V ′ , V ′ , · · · , V ′ g − ) is theunit of addition O of the group (Γ + g , O ) : O = ( V , V , · · · , V g − , V ′ , V ′ , · · · , V ′ g − ) . (Proof) Since V i ∈ α i and V ′ i ∈ α i +1 and V i , V ′ i α i,i +1 for i = 1 , , . . . , g −
1, wehave ( V , V , · · · , V g − , V ′ , V ′ , · · · , V ′ g − ) ∈ Γ + g . By proposition 5.1 we have( V V · · · V g − V ′ V ′ · · · V ′ g − ) = 0 . On the other hand, ( V , V , · · · , V g − , V ′ , V ′ , · · · , V ′ g − ) is the only choice which satisfiesboth ( V , V , · · · , V g − , V ′ , V ′ , · · · , V ′ g − ) ∈ Γ + g and ( V V · · · V g − V ′ V ′ · · · V ′ g − ) = 0 be-cause the map ¯ φ | ˜ D + g (Γ) is bijective. For any element P = ( P , P , . . . , P g ) of Γ + g the inverse − P with respect to the additionis simply given as follows. Proposition 5.3
For any even g we have P + P ′ = O , where P = ( P , P , . . . , P g ) ∈ Γ + g and P ′ := ( P ′ , P ′ , . . . , P ′ g ) . Therefore we write − P = P ′ . (Proof) Note first that P ′ ∈ Γ + g since P ∈ Γ + g . By proposition 5.1 we have P + P ′ = ( P P · · · P g ) + ( P ′ P ′ · · · P ′ g ) ≡ g X i =1 P i − D ∗ + g X i =1 P ′ i − D ∗ (mod J (Γ))= g/ X i =1 ( P i + P ′ i ) − D ∗ + g X i = g/ ( P i + P ′ i ) − D ∗ ≡ J (Γ))= O . This completes the proof. 11 .1.3 Tropical curves passing through given points
Now we realize the addition P + Q = M (1)of the group (Γ + g , O ), where P = ( P , P , . . . , P g ), Q = ( Q , Q , . . . , Q g ), and M =( M , M , . . . , M g ) are the elements of Γ + g , by using the intersection of the tropical hyper-elliptic curve ¯Γ and a tropical curve of degree 3 g/ P + Q + M ′ = O . Therefore there exists a rational function h on Γ satisfying( h ) = g X i =1 (cid:0) P i + Q i + M ′ i (cid:1) − D ∗ . (2)This implies h ∈ R (3 D ∗ ).Define the rational functions x and y on the metric graph Γ to be x ( P ) = p, y ( P ) = q, where P = ( p, q ) is a point on Γ. We can easily see( x ) = V g + V ′ g − V − V ′ , ( y ) = ( g + 1) V ′ − ( g + 1) V . Consider the tropical monomial a i ⊗ x ⊗ i in x for a i ∈ T . Then we have a i ⊗ x ⊗ i = a i + ix ∈ R (3 D ∗ ) for i = 0 , , . . . , g , where a ⊗ x ⊗ = a is the constant function. Because the principal divisor is computedas follows ( a i + ix ) = ( ix ) = i (cid:0) V g + V ′ g (cid:1) − i (cid:0) V + V ′ (cid:1) , and hence we have( a i + ix ) + 3 D ∗ = ( ix ) = i (cid:0) V g + V ′ g (cid:1) + (cid:18) g − i (cid:19) (cid:0) V + V ′ (cid:1) > g if and only if i = 0 , , . . . , g/
2. Similarly, for the tropical monomial b i ⊗ x ⊗ i ⊗ y in x, y for b i ∈ T we have b i ⊗ x ⊗ i ⊗ y = b i + ix + y ∈ R (3 D ∗ ) for i = 0 , , . . . , g − . Because we have( b i + ix + y ) + 3 D ∗ = ( ix ) + ( y ) + 3 D ∗ = i (cid:0) V g + V ′ g (cid:1) + (cid:18) g − i (cid:19) V ′ + (cid:16) g − − i (cid:17) V > g if and only if i = 0 , , . . . , g/ − M g spanned by the 2 g + 1 rational functions x ⊗ i ( i = 1 , , . . . , g/
2) and x ⊗ i ⊗ y ( i = 0 , , . . . , g/ −
1) [8]: M g := g/ M i =0 a g/ − i ⊗ x ⊗ i ⊕ g/ − M i =0 b i ⊗ x ⊗ i ⊗ y | a i , b i ∈ T = (cid:26) max (cid:18) g/ max i =0 (cid:0) a g/ − i + ix (cid:1) , g/ − max i =0 ( b i + ix + y ) (cid:19) | a i , b i ∈ T (cid:27) . Then we have the following proposition.
Proposition 5.4
For any even g we have M g ( R (3 D ∗ ) . (Proof) Denote the set of poles of a rational function h on the metric graph Γ by S h .Then for the set S f ⊕ h of the poles of the linear combination f ⊕ h = max( f, h ) of tworational functions f and h on Γ we have S f ⊕ h ⊂ S f ∪ S h . Because if it does not hold then there exists a point P ∈ S f ⊕ h which is the pole of neither f nor h . Since P is the pole of f ⊕ h , the sum of its outgoing slope at P is negative, whilethose of both f and h are non-negative. Let the edges outgoing from P be e , e , . . . , e k for k = 2 ,
3. (Note that the point P on Γ is at most trivalent.) Also let the slope of f and h on e i be f i and h i , respectively. We then have P ki =1 f i ≥ P ki =1 h i ≥
0. Bydefinition the outgoing slope of f ⊕ h at P is k X i =1 max ( f i , h i ) ≥ max k X i =1 f i , k X i =1 h i ! ≥ . This is a contradiction. Thus if f, g ∈ R (3 D ∗ ) then f ⊕ h ∈ R (3 D ∗ ). Therefore we have M g ⊂ R (3 D ∗ ).We can easily find a rational function h ∈ R (3 D ∗ ) not included in M g , e.g. , the onewhose two zeros are on the edge −−→ V V ′ .Let h ∈ M g . Since h is a tropical polynomial in x and y of degree 3 g/
2, it can beextended as a rational function on the (
X, Y )-plane. If P is the zero of h then the function h is not differentiable with respect to X or Y at P . Thus the tropical curve C of degree3 g/ h passes through the zero P of h . If P is on the tropical hyperellipticcurve ¯Γ as well then P is the intersection points of ¯Γ and C .Assume that the coefficients of h satisfy the generic condition a i + a i +2 < a i +1 for i = 0 , , . . . , g − ,b i + b i +2 < b i +1 for i = 0 , , . . . , g − ,a − a < b − b . (3)13 U U g/ − U g/ U U − g/ U − g/ U − Figure 4: The tropical curve C of degree 3 g/ h ∈ M g .Then C has exactly 2 g − C satisfying the generic condition.The vertex from which an unbound edge emanating upward is denoted by U i for non-positive i = 0 , − , . . . , − g/ U i for positive i = 1 , , . . . , g/
2. The coordinate of the vertices are givenas follows U i = (cid:18) a i − a i − , (cid:18) g − i + 1 (cid:19) a i − (cid:18) g − i (cid:19) a i − − b (cid:19) for i = 1 , , . . . , g/ ,U − i = ( b i − b i +1 , ( i − b i +1 − ib i + a ) for i = 0 , , . . . , g − / − . The outgoing slope of the edge −−−−→ U i U i +1 is 3 g/ − i for i = 0 , , . . . , g/ − −−−−→ U i − U i is 3 g/ i − i = 0 , − , . . . , − ( g − / U − g/ and U g/ are g + 1 and 0, respectively. Remark 5.5 If h ∈ M g satisfies the generic condition (3) then the curve C defined by h is smooth. Therefore the dimension of the tropical module M g is exactly g + 1 [8]. Since deg 3 D ∗ = 3 g > g − for any g ≥ , by corollary 2.10, we have r (3 D ∗ ) = 3 g − g = 2 g = dim M g − . This means that the maximal dimension of the cells of R (3 D ∗ ) is at least dim M g [3]. In order to investigate the intersection points of ¯Γ and C , we consider the homogeneouscoordinate ( x : x : x ) of the tropical projective plane TP [8]. Denote the point( −∞ : 0 : −∞ ) at infinity by P ∞ . Then we can see ¯Γ passes through P ∞ . Actually, the14nbound edge emanating from the vertex V is given as follows( g + 1) x − x − gx = 0 (4)in the homogeneous coordinate. Note that this unbound edge goes to the direction x = −∞ . If we set x = −∞ in (4) then x must be −∞ since g is positive. We can also seethat the g/ − U − i for i = 0 , , . . . , g/ − C pass through P ∞ . Thus the two curves ¯Γ and C intersect at P ∞ . We then have thefollowing lemma concerning the intersection number of ¯Γ and C at P ∞ . Lemma 5.6
The tropical hyperelliptic curve ¯Γ intersects the tropical curve C defined by h ∈ M g with multiplicity g / at the point P ∞ at infinity. (Proof) Let the unbound edge of ¯Γ emanating from the vertex V be e . Also let theedge of C outgoing upward from the vertex U i be ε i for i = 0 , − , . . . , − g/ C outgoing from U − g/ be ε − g/ . The primitive tangentvector of these unbound edges are given as follows e : ( g, g + 1) ,ε i : (1 ,
1) for i = 0 , − , . . . , − g/ ,ε − g/ : ( g, g + 1) , where we choose the basis ( − ,
0) and (1 , P ∞ of two curves, whose primitive tangent vectors of theunbound edges passing through P ∞ are ( a, b ) and ( c, d ) respectively, are defined to be w w min ( ad, bc ) , where w and w are the weights of the two edges, respectively [2]. Note that the weightsof the edges e and ε i ( i = 0 , − , . . . , − g/ C at P ∞ can be computed as followsmin ( g ( g + 1) , ( g + 1) g ) + g/ − X i =0 min ( g × , ( g + 1) ×
1) = g ( g + 1) + (cid:16) g − (cid:17) g = 3 g . This completes the proof.The degree of the curves ¯Γ and C are g + 2 and 3 g/
2, respectively. By the B´ezout theo-rem for tropical curves [10, 2] ¯Γ intersects C at ( g + 2)3 g/ C do not intersect at the point at infinity other than P ∞ . Therefore, inthe affine part, ¯Γ intersects C at 3 g g + 2) − g g points, counting multiplicities. 15 .1.5 Realization of addition Let P = ( P , P , . . . , P g ) and Q = ( Q , Q , . . . , Q g ) be in Γ + g . Assume the 2 g points in P and Q to be in generic position. Then there exists a unique tropical curve C passingthrough these 2 g points and defined by a rational function h ∈ M g . Note that the rationalfunction h ∈ M g is uniquely determined by the 2 g points P and Q up to a constant because h has 2 g + 1 parameters. Consider the intersection of the tropical hyperelliptic curve ¯Γ andthe curve C . Since ¯Γ intersects C at 3 g points, there further exist g intersection points,counting multiplicities. Let these intersection points be M ′ , M ′ , . . . , M ′ g . Figure 5 showsan example of the intersection ¯Γ and C for g = 2. P P ∞ Q M ′ P M ′ Q V V ′ M M Figure 5: An intersection of the tropical hyperelliptic curve ¯Γ (broken one) of genus 2 andthe curve C (solid one) of degree 3. The addition ( P , P ) + ( Q , Q ) = ( M , M ) of thecouples of points on ¯Γ is realized by the intersection of ¯Γ and C with the unit O = ( V , V ′ ).Then the principal divisor of the rational function h defining C has the form (2). Itfollows that the g -tuples P , Q , and M = ( M , M , . . . , M g ) satisfy the addition formula P + Q = M of the group (Γ + g , O ) (see figure 5). g Next we assume the genus g of the tropical hyperelliptic curve ¯Γ to be odd. Let us fix D ∗ ∈ D + g (Γ) as follows D ∗ = g − (cid:0) V + V ′ (cid:1) + V . .2.1 Unit of addition We have the following proposition in analogy to the case of even g . Proposition 5.7
For any odd g we have (cid:16) V P P · · · P ( g − / P ′ P ′ · · · P ′ ( g − / (cid:17) = 0 , where P , P , · · · , P ( g − / are the points on Γ and P ′ , P ′ , · · · , P ′ ( g − / are their conjugates,respectively. (Proof) It is sufficient to show the existence of the rational function h satisfying( h ) = V + P + P + · · · P ( g − / + P ′ + P ′ + · · · + P ′ ( g − / − D ∗ = V + P + · · · P ( g − / + P ′ + · · · + P ′ ( g − / − (cid:18) g − (cid:0) V + V ′ (cid:1) + V (cid:19) = P + P + · · · + P ( g − / + P ′ + P ′ + · · · + P ′ ( g − / − g − (cid:0) V + V ′ (cid:1) . It is obvious from proposition 5.1.Combining theorem 4.1 and proposition 5.7, it immediately follows the propositionconcerning the unit O of addition the group (Γ + g , O ). Proposition 5.8
The g -tuple ( V , V , V , · · · , V g − , V ′ , V ′ , · · · , V ′ g − ) is the unit of addi-tion O of the group (Γ + g , O ) : O = ( V , V , V , · · · , V g − , V ′ , V ′ , · · · , V ′ g − ) . Since the case of g = 1 is distinctive, we first consider the case. Let g = 1 then D ∗ = V and O = V . Note that the map ¯ φ : D +1 (Γ) = Γ → J (Γ) is bijective, and hence Γ +1 = D +1 (Γ) = Γ ≃ J (Γ). For the rational functions y and x + y we have( y ) + 3 D ∗ = 2 V ′ + V > , ( x + y ) + 3 D ∗ = V + V ′ + V ′ > . Define the tropical module M to be M = { max( a, b + y, c + x + y ) | a, b, c ∈ T } . It is easy to see that M = R (3 D ∗ ).Let h ∈ M . Suppose h to pass through the points P, Q ∈ Γ +1 = Γ ≃ J (Γ), which arein generic position. Then h is uniquely determined up to a constant. The tropical curve C defined by h is of degree 2 and passes through the points P ∞ and P ′∞ := (0 : −∞ : −∞ )at infinity. The tropical elliptic curve ¯Γ defined by the tropical polynomial˜ F ( X, Y ) = max (2
Y, Y + c , Y + X, Y + 2
X, c − )17 ¯ M P ∞ P ′∞ O = V QM Figure 6: An intersection of the tropical elliptic curve ¯Γ (broken one) and the curve C (solid one) and ˜ C (dotted one) of degree 2. The addition P + Q = M of the points on ¯Γis realized by the intersection of ¯Γ and C and ˜ C with the unit O = V .is of degree 3 and passes through both P ∞ and P ′∞ as well.The primitive tangent vector of the unbound edge of C passing through P ∞ is (1 , − ,
0) and (1 , P ∞ is(1 , C and ¯Γ at P ∞ is min(1 × , ×
1) = 1. Theintersection number of C and ¯Γ at P ′∞ can also be computed as min(2 × , ×
3) = 2.Therefore ¯Γ intersects C at 2 × − − M ∈ Γ +1 satisfying P + Q + ¯ M = O is the third intersection point of ¯Γ and C (see figure 6).Let us consider the rational function f ∈ M passing through both ¯ M and O = V .Then the third intersection point M (see figure 6) of ¯Γ and the curve ˜ C defined by f satisfies the addition ¯ M + M = O . This implies that we have P + Q = M. Thus the addition in (Γ +1 , O ) can be realized as the intersection of ¯Γ and tropical curves C and ˜ C of degree 2 (see figure 6). 18 .2.3 Tropical curves passing through given points Next we consider the case of g ≥
3. For the rational function x we have( ix ) + 3 D ∗ = i (cid:0) V g + V ′ g (cid:1) + (cid:18) g + 1)2 − i (cid:19) V + (cid:18) g − − i (cid:19) V ′ > g ≥ i = 0 , , . . . , g − /
2. Moreover we have( ix + y ) + 3 D ∗ = i (cid:0) V g + V ′ g (cid:1) + (cid:18) g + 12 − i (cid:19) V + (cid:18) g − − i (cid:19) V ′ > g ≥ i = 0 , , . . . , ( g + 1) / M g to be the linear combinations of the 2 g + 1 rationalfunctions x ⊗ i ( i = 0 , , . . . , g − /
2) and x ⊗ i ⊗ y ( i = 0 , , . . . , g + 1 / M g := g − / M i =0 a g − / − i ⊗ x ⊗ i ⊕ g +1 / M i =0 b i ⊗ x ⊗ i ⊗ y | a i , b i ∈ T = (cid:26) max (cid:18) g − / max i =0 (cid:0) a g − / − i + ix (cid:1) , ( g +1) / max i =0 ( b i + ix + y ) (cid:19) | a i , b i ∈ T (cid:27) . Then we have the following proposition in analogy to the case of even g . Proposition 5.9
For any odd g we have M g ( R (3 D ∗ ) . Assume that the rational function h ∈ M g satisfies the generic condition a i + a i +2 < a i +1 for i = 0 , , . . . , g − − ,b i + b i +2 < b i +1 for i = 0 , , . . . , g + 12 − ,a − a < b − b . Then the curve C of degree 3( g − / h has exactly 2 g − g , the vertex from which an unbound edge emanating upwardis denoted by U i for non-positive i = 0 , − , . . . , − ( g + 1) / U i for positive i = 1 , , . . . , g − /
2. Thecoordinates of the vertices are given as follows U i = (cid:18) a i − a i − , (cid:18) g − − i (cid:19) ( a i − a i − ) + a i − b (cid:19) for i = 1 , , . . . , g − / ,U − i = ( b i − b i +1 , ( i − b i +1 − ib i + a ) for i = 0 , , . . . , ( g + 1) / − . The slope of the edge −−−−→ U i U i +1 is 3( g − / − i for i = 0 , , . . . , g − / − −−−−→ U i − U i is 3( g − / i − i = 0 , − , . . . , − ( g + 1) / U − ( g +1) / and U g − / are g − U U g − / − U g − / U U − ( g +1) / U − ( g +1) / U − Figure 7: The tropical curve C of degree 3( g − / h ∈ M g . We have the following lemma concerning the intersection number of ¯Γ and C at the pointat infinity. Lemma 5.10
For any odd g ≥ the tropical hyperelliptic curve ¯Γ of genus g intersectsthe tropical curve C defined by h ∈ M g with multiplicity g + 1)( g − / at the point P ∞ at infinity. (Proof) Let the unbound edge of ¯Γ emanating from the vertex V be e . Also let theedge of C outgoing upward from the vertex U i be ε i for i = 0 , − , . . . , − ( g + 1) / C outgoing from U − ( g +1) / be ε − ( g +1) / . The primitivetangent vector of these unbound edges are given as follows e : ( g, g + 1) ,ε i : (1 ,
1) for i = 0 , − , . . . , − ( g + 1) / ,ε − ( g +1) / : ( g − , g − , where we choose the basis ( − ,
0) and (1 , C at P ∞ can be computed as followsmin ( g ( g − , ( g + 1)( g − g +1 / − X i =0 min ( g, g + 1) = ( g + 1)( g −
3) + g + 12 g = 3( g + 1)( g − . This completes the proof.For any odd g ≥ C are g + 2 and 3( g − /
2, respectively. By theB´ezout theorem ¯Γ intersects C at 3( g + 2)( g − / C do not intersect at the point at infinity other than P ∞ . Therefore, in theaffine part, ¯Γ intersects C at3( g + 2)( g − − g + 1)( g − g points, counting multiplicities. P Q P ∞ V V V ′ ¯ M P Q ¯ M P Q ¯ M Figure 8: An intersection of the tropical hyperelliptic curve ¯Γ (broken one) of genus3 and the curve C (solid one) of degree 3. The addition ( P , P , P ) + ( Q , Q , Q ) +( ¯ M , ¯ M , ¯ M ) = O of the triples of points on ¯Γ is realized by the intersection of ¯Γ and C with the unit O = ( V , V , V ′ ). Let P = ( P , P , . . . , P g ) and Q = ( Q , Q , . . . , Q g ) be in Γ + g . Assume the 2 g points in P and Q to be in generic position. Then there exists a tropical curve C passing throughthese 2 g points and defined by a rational function h ∈ M g . Consider the intersection ofthe tropical hyperelliptic curve ¯Γ and the curve C . Since ¯Γ intersects C at 3 g points, therefurther exist g intersection points, counting multiplicities. Let these intersection points be¯ M , ¯ M , . . . , ¯ M g . Then the principal divisor of the rational function h defining C has theform ( h ) = g X i =1 (cid:0) P i + Q i + ¯ M i (cid:1) − D ∗ . P + Q + ¯ M = O . Figure 8 shows an example of the intersection of ¯Γ of g = 3 and C of degree 3.Moreover consider the curve ˜ C defined by a rational function f ∈ M g and passingthrough the 2 g points in ¯ M = ( ¯ M , ¯ M , . . . , ¯ M g ) and O . Then ¯Γ intersects ˜ C at 3 g points,counting multiplicities. Let the remaining g intersection points be M , M , . . . , M g . Thenwe have M + ¯ M = O , where M = ( M , M , . . . , M g ). It follows that the g -tuples P , Q , and M satisfy theaddition formula P + Q = M of the group (Γ + g , O ). We show that there exists a surjection between the set of effective divisors of degree g on the tropical hyperelliptic curve of genus g and its Jacobian. We also show that thesurjection is bijective if and only if g = 1. We then show that there exists a subset ofthe set of effective divisors of degree g on which the surjection reduces to the bijection.It follows that the additive group structure of the subset is induced by the bijection fromthe Jacobian, which is isomorphic to the Picard group. The addition in the Jacobian of atropical hyperelliptic curve of genus g thus induced can be interpreted geometrically as theaddition of g -tuples of points on the curve. We realize the addition of g -tuples of pointson the curve in terms of the intersection of the hyperelliptic curve and a curve of degree3 g/ g − /
2) for even (resp. odd) g .If g = 1 then the addition of points on a tropical hyperelliptic curve induces two kindsof dynamical systems realized as the evolutions of points on the curve; one is an integrablesystem referred as the ultradiscrete QRT system and the other is a solvable chaotic system.Since we can realize the addition of g -tuples of points on the tropical hyperelliptic curveof genus g via the intersection with a curve, we can construct several dynamical systemsrealized as the evolutions of points on the tropical hyperelliptic curve. We will report suchdynamical systems in a forthcoming paper. Acknowledgments
This work was partially supported by Grants-in-Aid for Scientific Research, Japan Societyfor the Promotion of Science (JSPS), No. 22740100.
References [1] Baker, M. and Norine, S., Riemann-Roch and Abel-Jacobi theory on a finite graph,
Adv. Math. , (2007), 766–788. 222] Gathmann, A., Tropical algebraic geometry, Preprint , (2006),arXiv:math/ 0601322v1 (24 pp).[3] Gathmann, A. and Kerber, M., A Riemann-Roch theorem in tropical geometry,
Preprint , (2006), arXiv:math/0612129v2 (14 pp).[4] Haase, C., Musiker, G., and Yu, J., Linear systems on tropical curves,
Preprint ,(2009), arXiv:0909.3685v1 (35 pp).[5] Inoue, R. and Takenawa, T., Tropical spectral curves and integrable cellular au-tomata,
Int. Math. Res. Not. , (2008), Art ID. rnn019 (27 pp).[6] Kajiwara, K., Kaneko, M., Nobe, A. and Tsuda, T., Ultradiscretization of a solvabletwo-dimensional chaotic map associated with the Hesse cubic curve, Kyushu J. Math. , (2009), 315-338.[7] Kajiwara, K., Nobe, A. and Tsuda, T., Ultradiscretization of solvable one-dimensionalchaotic maps, J. Phys. A: Math. Theor. , (2008), 395202 (13pp).[8] Mikhalkin, G. and Zharkov, I., Tropical curves, their Jacobians and theta functions, Preprint , (2006), arXiv:math/0612267v1 (27 pp).[9] Nobe, A., Ultradiscrete QRT maps and tropical elliptic curves,
J. Phys. A: Math.Theor. , (2008), 125205 (12 pp).[10] Richter-Gebert, J., Sturmfels, B. and Theobald, T., First steps in tropical geometry,Idempotent Mathematics and Mathematical Physics (Litvinov, G. and Maslov, V.eds.), Proceedings Vienna 2003, American Mathematical Society, Contemp. Math. (2005), 289–317.[11] Takahashi, D., Tokihiro, T., Grammaticos, B., Ohta, Y. and Ramani, A., Constructingsolutions to the ultra-discrete Painlev´e equations, J. Phys. A.: Math. Gen. , (1997),7953–7966.[12] Vigeland, M. D., The group law on a tropical elliptic curve, Preprint , (2004),arXiv:math.AG/ 0411485 (13 pp).[13] Zhang, S., Admissible pairing on a curve,
Invent. Math. ,112