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arXiv:0803.4021v2 [math.CO] 31 Mar 2008

THE TROPICAL j-INVARIANT

ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

Abstract. If (Q,A) is a marked polygon with one interior point, then a

general polynomial f ∈ K[x,y] with support A defines an elliptic curve Cfon

the toric surface XA. If K has a non-archimedean valuation into R we can

tropicalize Cf to get a tropical curve Trop(Cf). If the Newton subdivision

induced by f is a triangulation, then Trop(Cf) will be a graph of genus one

and we show that the lattice length of the cycle of that graph is the negative

of the valuation of the j-invariant of Cf.

1. Introduction

Previous work by Grisha Mikhalkin [13], by Michael Kerber and Hannah Markwig

[11] and by Magnus Vigeland [18] shows that the length of the cycle of a tropical

curve of genus one has properties which one classically attributes to the j-invariant

of an elliptic curve without giving a direct link between these two numbers. In [9]

we established such a direct link for plane cubics by showing that the tropicalization

of the j-invariant is in general the negative of the cycle length. In the present paper

we generalize this result to elliptic curves on other toric surfaces using the same

methods.

More precisely, if (Q,A) is a marked polygon with one interior point, then a general

polynomial f ∈ K[x,y] with support A defines an elliptic curve Cf on the toric

surface XA. If K has a non-archimedean valuation we can tropicalize Cf to get a

tropical curve Trop(Cf). If the Newton subdivision induced by f is a triangulation,

then Trop(Cf) will be a graph of genus one and we show in our main result in

Theorem 6.4 that the lattice length of the cycle of the graph is the negative of the

valuation of the j-invariant.

In the case where the triangulation is unimodular, i.e. all the triangles have area1

this result was independently derived by David Speyer [17, Proposition 9.2] using

Tate uniformization of elliptic curves. David Speyer’s result is more general though

in the sense that it applies to curves in arbitrary toric varieties.

2,

This paper is organized as follows. In Section 2 we consider toric surfaces defined

by a marked lattice polygon with one interior point, we recall the classification of

these polygons and we consider the impact on the j-invariant for the corresponding

elliptic curves. Section 3 recalls the notion of tropicalization and of plane tropical

curves. We then introduce in Section 4 the notion of tropical j-invariant and give

a formula to compute it. Section 5 shows that the tropical j-invariant is preserved

2000 Mathematics Subject Classification: 14H52, 51M20.

The third author would like to thank the Institute for Mathematics and its Applications in

Minneapolis for hospitality.

1

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2 ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

by integral unimodular affine transformations. With this preparation we are able

to state our main result in Section 6. Section 7 is then devoted to the reduction

of the proof to considering only three marked polygons and Section 8 shows how

these three cases can be dealt with using procedures from the Singular library

jinvariant.lib (see [10]) which is available via the URL

http://www.mathematik.uni-kl.de/˜ keilen/en/jinvariant.html.

The actual computations are done using polymake [4], TOPCOM [16] and Singu-

lar [6]. The tropical curves in this paper and their Newton subdivisions were

produced using the procedure drawtropicalcurve from the Singular library

tropical.lib (see [8]) which can be obtained via the URL

http://www.mathematik.uni-kl.de/˜ keilen/en/tropical.html.

The authors would like to thank Vladimir Berkovich, Jordan Ellenberg, Bjorn Poo-

nen, David Speyer, Charles Staats, Bernd Sturmfels and John Voigt for valuable

discussions.

2. Toric surfaces

Throughout this paper we consider mainly marked polygons (Q,A) such that Q

contains a single interior lattice point, where by a marked polygon we mean a

convex lattice polygon Q in R2together with a subset A ⊆ Q ∩ Z2of the lattice

points of Q containing at least the vertices of Q (cf. [5, Section 2.A]). Fixing a base

field K such a polygon defines a polarized toric surface

XA⊂ P|A|−1

K

.

In the torus (K∗)2⊂ XAthe hyperplane section, say Cf, defined by the linear form

?

f =

?

(cf. [5, Chapter 5]). Since the arithmetical genus of the hyperplane sections is the

number of interior lattice points of Q (cf. [3, p. 91]), the general hyperplane section

will be a smooth elliptic curve. The j-invariant of such a curve is an element of

the base field which characterizes the curve up to isomorphism.

(i,j)∈Aaij· zijis the vanishing locus of the Laurent polynomial

(i,j)∈A

aij· xiyj

An integral unimodular affine transformation of R2is an affine map

φ : R2−→ R2: α ?→ A · α + τ

with τ ∈ Z2and A ∈ Gl2(Z) invertible over the integers. Such an integral uni-

modular affine transformation φ maps each face of Q to a face of the convex lattice

polygon φ(Q) and preserves thereby the number of lattice points on each face.

Moreover, φ induces an isomorphism of the polarized toric surfaces XAand Xφ(A)

(cf. [5, Proposition 5.1.2]). From the point of view of toric surfaces it therefore suf-

fices to consider the marked polygon (Q,A) only up to integral unimodular affine

transformations, and if we suppose A = Q ∩ Z2then there are precisely sixteen of

them which we divide into two groups, Qa, Qband Qcrespectively Qca,...,Qcm

(see Figure 1, cf. [15] or [14]). We fix the interior point at position (1,1).

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THE TROPICAL j-INVARIANT3

(Qa,Aa)(Qb,Ab)(Qc,Ac)

(Qca,Aca)(Qcb,Acb)(Qcc,Acc)(Qcd,Acd)(Qce,Ace)

(Qcf,Acf)(Qcg,Acg)(Qch,Ach)(Qci,Aci)(Qcj,Acj)

(Qck,Ack)(Qcl,Acl)(Qcm,Acm)

Figure 1. The 16 convex lattice polygons with one interior lattice point

The marked polygon (Qc,Ac) corresponds to P2

uple Veronese embedding. The marked polygon (Qb,Ab) corresponds to P1

embedded into P8

Kvia the (2,2)-Segre embedding. The marked polygon (Qa,Aa)

describes the singular weighted projective plane PK(2,1,1) embedded into P8.

Kembedded into P9

Kvia the 3-

K× P1

K

If a polygon marked polygon (Q′,A′) is derived from (Q,A) by cutting off one

lattice point (k,l), like (Qca,Aca) and (Qc,Ac), then the toric surface XA′ is a

blow up of XA in a single point. Moreover, in the torus (K∗)2the hyperplane

sections corresponding to

f =

?

(i,j)∈A′

aij· xiyj=

?

(i,j)∈A

aij· xiyj

with akl = 0 coincide. In particular, if they are both smooth their j-invariant

coincides since two birationally equivalent curves are already isomorphic (cf. [7,

Section I.6]). Since the 13 polygons Qca,...,Qcmin the second group in Figure 1

are all subpolygons of Qcthe corresponding toric surfaces are all obtained from the

projective plane by a couple of blow ups. When we want to compute the j-invariant

of the curve corresponding to some Laurent polynomial with support in one of these

13 polygons, we can instead consider the plane curve with support in Acbut with

the appropriate coefficients being zero.

Once we are able to compute the j-invariant for polynomials with support Aa,

Ab and Ac we are therefore able to compute the j-invariant for every Laurent

polynomial with support on the lattice points of a lattice polygon with only one

interior point.

We still assume that (Q,A) is a marked lattice polygon with only one interior

lattice point as above. Moreover, we use the notation a = (aij| (i,j) ∈ A), and we

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4 ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

suppose that

f =

?

(i,j)∈A

aij· xiyj,

then the j-invariant

j(Cf) = j(f) =AA

BA

of the curve Cf in XAdefined by f can be expressed as a quotient of two homo-

geneous polynomials AA,BA∈ Q[a] of degree 12. In the case of A = Ac AAhas

1607 terms and BAhas 2040. In the case of A = Ab AAhas 990 terms and BA

has 1010. And finally in the case A = Aa AA has 267 terms and BA has 312.

Every other case can be reduced to these three via some integral unimodular affine

transformation and by setting some coefficients equal to zero. The reader interested

in seeing or using the polynomials can consult the procedure invariantsDB in the

Singular library jinvariant.lib (see [10]). The proof of our result relies heavily

on the investigation of the combinatorics of these polynomials.

3. Tropicalization

In this section we want to pass from the algebraic to the tropical side. For this we

specify a field K with a non-archimedean valuation val : K∗→ R as base field and

we extend the valuation to K by val(0) = ∞. We call val(k) also the tropicalization

of k. In the examples that we consider K will always be the field of Puiseux series

∞

?

N=1

Quot

?

C??t

1

N???

=

?

∞

?

ν=m

cν· t

ν

N

??? cν∈ C,N ∈ Z>0,m ∈ Z

?

and the valuation of a Puiseux series is its order.

If f =?aij· xiyj∈ K[x,y,x−1,y−1] is any Laurent polynomial, we call the set

supp(f) = {(i,j) ∈ Z2| aij?= 0}

the support of f and the convex hull N(f) of supp(f) in R2is called the Newton

polygon of f. If supp(f) ⊆ A ⊆ N(f) ∩ Z2then f defines a curve Cf in the toric

surface XAas described in Section 2 and we define the tropicalization of Cf as

Trop?Cf

?= val?Cf∩ (K∗)2) ⊆ R2,

i.e. the closure of val?Cf∩ (K∗)2?with respect to the Euclidean topology in R2.

val : (K∗)2−→ Q2: (k1,k2) ?→?val(k1),val(k2)?

denotes the Cartesian product of the above valuation map.

A better way to compute the tropicalization of Cf is as the tropical curve defined

by the tropicalization of the polynomial f, i.e. the piecewise linear map

Here by abuse of notation

trop(f) : R2−→ R : (x,y) ?→ min{val(aij) + i · x + j · y | (i,j) ∈ supp(f)}.

Given any plane tropical Laurent polynomial

F : R2−→ R : (x,y) ?→ min{uij+ i · x + j · y | (i,j) ∈ A′}

with support supp(F) = A′⊂ Z2finite and uij ∈ R, we call the locus CF of

non-differentiability of F, i.e. the set of points (x,y) ∈ R2where the minimum

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THE TROPICAL j-INVARIANT5

is attained at least twice, the plane tropical curve defined by F. The convex hull

N(F) of supp(F) is again called the Newton polygon of F.

By Kapranov’s Theorem (see [2, Theorem 2.1.1]), Trop(Cf) coincides with the

plane tropical curve defined by the plane tropical polynomial trop(f). In particular,

Trop(Cf) is a piece-wise linear graph.

The plane tropical Laurent polynomial F induces a marked subdivision (cf. [5,

Definition 7.2.1]) of the marked polygon?N(F),A?with supp(F) ⊆ A ⊆ N(F)∩Z2

{(i,j,uij| (i,j) ∈ supp(F)}

in the following way: project the lower faces of the convex hull of

into the xy-plane to subdivide of N(F) into smaller polygons and mark those lattice

points for which (i,j,uij) is contained in a lower face.

This subdivision is dual to the tropical curve CF in the following sense (see [12,

Prop. 3.11]): Each marked polygon of the subdivision is dual to a vertex of CF, and

each facet of a marked polygon is dual to an edge of CF. Moreover, if the facet, say

e, has end points (x1,y1) and (x2,y2) then the direction vector v(E) of the dual

edge E in CF is defined (up to sign) as

v(E) = (y2− y1,x1− x2)t

and points in the direction of E. In particular, the edge E is orthogonal to its dual

facet e. Finally, the edge E is unbounded if and only if its dual facet e is contained

in a facet of N(F).

Example 3.1

Consider the polynomial

f = xy + t · (y + x + x2+ x2y2) + t3

The following diagram shows the support of f and its marked Newton polygon.

supp(f)

?N(f),supp(f)?

The tropicalization of f is

trop(f) : R2→ R : (x,y) ?→ min{x + y,1 + y,1 + x,1 + 2x,1 + 2x + 2y,3}.

The support and Newton polygon of f respectively of trop(f) coincide. In order to

compute the marked subdivision of the Newton polygon note that the points

(0,1,1),(1,0,1),(2,0,1),(2,2,1)

lie in a plane while watching from below the point (1,1,0) sticks out from this plane

and the point (0,0,3) lies way above it. We therefore get the following subdivision

of the Newton polygon:

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6ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

e

The polygon spanned by (0,0), (1,0) and (0,1) is dual to the vertex of the tropical

curve where the terms 3, 1+x and 1+y take their common minimum, which is at

the point (x,y) = (2,2). Similarly the polygon spanned by (1,0), (1,1) and (0,1)

corresponds to the point (x,y) = (1,1), and the common face e of the two polygons

then is dual to the edge connecting these two points. Note that the direction

vector of this edge E is v(E) = (1,1) is orthogonal to the face e connecting the

points (1,0) and (0,1) and points from the starting point (1,1) of E to its end

point (2,2). Computing the remaining vertices and edges of Trop(Cf) we get the

following graph.

E

4. The tropical j-invariant of an elliptic plane tropical curve

For the purpose of this paper we want to define an elliptic plane tropical curve in

the following way.

Definition 4.1

An elliptic plane tropical curve is a tropical curve CF defined by a plane tropical

Laurent polynomial F whose Newton polygon has precisely one interior lattice

point.

The plane tropical curve CF in Example 3.1 is elliptic in this sense. Moreover, the

graph CF has genus one, where the genus of a graph is the number of independent

cycles of the graph. Obviously a cycle in the graph corresponds to an interior

lattice point of the subdivision being a vertex of at least three polygons in the

subdivision of the Newton polygon. We want to make this more precise in the

following definition.

Definition 4.2

Let C be a plane tropical curve with marked Newton polygon (Q,A) and with dual

marked subdivision {(Qi,Ai) | i = 1,...,l}. Suppose that ˜ ω ∈ Int(Q)∩Z2and that

the (Qi,Ai) are ordered such that ˜ ω is a vertex of Qifor i = 1,...,k and it is not

contained in Qifor i = k+1,...,l (see Figure 2). We then say that ˜ ω determines a

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THE TROPICAL j-INVARIANT7

ω2

ω1= ωk+1

ωk

ω0

ωk−1

ω3

ω4

˜ ω

Qk

Q1

Q2

Q3

Qk+1

Qk+2

Figure 2. Marked subdivision determining a cycle

cycle of C, namely the union of the edges of C dual to the facets emanating from ˜ ω,

and we say that these edges form the cycle determined by ˜ ω. We define the lattice

length of the cycle to be the sum of the lattice lengths of the edges which form the

cycle, where for an edge E with direction vector v(E) (see p. 5) the lattice length

of E is

l(E) =

||E||

||v(E)||

the Euclidean length of E divided by that of v(E).

Example 4.3

Coming back to our Example 3.1 the curve has one cycle dual to the interior lattice

point (1,1) and it consists of four edges E1,...,E4.

E4

E3

E2

E1

The edge E1is dual to the edge e1from (0,1) to (1,1) in the Newton subdivision

in Example 3.1, so that its direction vector is v(E1) = (0,−1) of Euclidean length 1

and that the lattice length of E1is l(E1) = ||E1|| = 3. Doing similar computations

for the other edges the cycle length is

l(E1) + l(E2) + l(E3) + l(E4) = 3 + 2 + 1 + 1 = 7.

Definition 4.4

If C is an elliptic plane tropical curve then C has at most one cycle, and we define

its tropical j-invariant jtrop(C) to be the lattice length of this cycle if it has one. If

C has no cycle we define its tropical j-invariant to be zero.

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8ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

In Example 4.3 the elliptic plane tropical curve has tropical j-invariant 7.

If a we fix the part of a Newton subdivision which determines the cycle then there

is a nice formula to compute the cycle length, and thus the tropical j-invariant. For

the proof we refer to [9, Lemma 3.7].

Lemma 4.5

Let (Q,A) be a marked polygon in R2with a marked subdivision {(Qi,Ai) | i =

1,...,l} and suppose that ˜ ω ∈ Int(Q) ∩ Z2is a vertex of Qifor i = 1,...,k and it

is not contained in Qifor i = k + 1,...,l.

If u ∈ RAis such that the plane tropical curve

F = min{uij+ i · x + j · y | (i,j) ∈ A}

induces this subdivision (as described in Section 3), then ˜ ω determines a cycle in

the plane tropical curve CF and, using the notation in Figure 2, its length is

k

?

j=1

(u˜ ω− uωj) ·Dj−1,j+ Dj,j+1+ Dj+1,j−1

Dj−1,j· Dj,j+1

where Di,j= det(wi,wj) with wi= ωi− ˜ ω and wj= ωj− ˜ ω.

This formula implies in particular the following corollary.

Corollary 4.6

If (Q,A) is a marked lattice polygon in R2with precisely one interior lattice point,

then

jtrop: RA−→ R : u ?→ jtrop(u) := jtrop(CFu)

with

Fu= min{uij+ i · x + j · y | (i,j) ∈ A}

is a piecewise linear function which is linear on cones of the secondary fan (cf. [5,

Chapter 7]) of A.

5. Unimodular transformations preserve lattice length

We want to relate the classical j-invariant to the tropical j-invariant, and we would

again like to reduce the consideration of all possible Newton polygons with one

interior point to the 16 polygons in Figure 1, or even better, to the three basic

ones in the first group there. For that we have to understand the impact of an

integral unimodular affine transformation on a plane tropical Laurent polynomial

respectively the induced plane tropical curve.

Given a linear form l = u + i · x + j · y = u + (i,j) · (x,y)twith i,j ∈ Z and u ∈ R

and given an integral unimodular affine transformation

φ : R2−→ R2: α ?→ A · α + τ

with A ∈ Gl2(Z) and τ ∈ Z2, we let φ act on l via

lφ= u + (x,y) · φ?(i,j)t?

and we let φ act on a plane tropical Laurent polynomial F = min{uij+ i · x + j ·

y | (i,j) ∈ A′} via the linear forms, i.e.

Fφ= min{uij+ (x,y) · φ?(i,j)t?| (i,j) ∈ A′}.

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THE TROPICAL j-INVARIANT9

Note, that the translation by τ does not change the piecewise linear function defined

by F at all and the Newton polygon of F is just translated by τ. So τ has neither

any impact on the Newton subdivision of F nor on the tropical curve defined by F.

Moreover, it is obvious that if {(Qi,Ai) | i = 1,...,k} is the marked subdivision

of?N(F),supp(F)) induced by F, then {φ(Qi),φ(Ai) | i = 1,...,k} is the marked

It is a well-known fact that an integral unimodular affine transformation preserves

lattice length, which implies the following corollary.

subdivision of?N(Fφ),supp(Fφ)?induced by Fφ.

Corollary 5.1

Let F be a plane tropical Laurent polynomial such that CF is elliptic with positive

tropical j-invariant and let φ be an integral unimodular affine transformation of

R2, then CFφ is elliptic with the same tropical j-invariant

jtrop(CF) = jtrop(CFφ).

Example 5.2

Consider the polynomial

f = x2y + xy2+1

t· xy + x + y

inducing the following subdivision of its Newton polygon and the corresponding

tropical curve:

N(f) subdividedTrop(Cf)

The plane tropical curve Trop(Cf) is elliptic with tropical j-invariant 8. If we now

apply the integral unimodular affine transformation

φ : R2−→ R2: α ?→

?21

11

?

· α

we get

fφ= x5y3+ x4y3+1

t· x3y2+ x2y + xy

with the following subdivision of its Newton polygon and the corresponding elliptic

plane tropical curve having again tropical j-invariant 8.

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10ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

N(fφ) subdividedTrop(Cfφ)

The considerations in Section 2 together with this corollary will allow us to reduce

the study of the tropicalization of an elliptic curve in a toric surface with an arbi-

trary Newton polygon with one interior point to the study of those whose Newton

polygons are among the 16 polygons in Figure 1.

6. The main result

Let us suppose now that (Q,A) is a lattice polygon with only one interior point.

Remark 6.1

In Section 2 we have seen that the j-invariant of a curve Cf with supp(f) ⊆ A can

be computed by plugging the coefficients aij of f into a suitable quotient j =AA

of homogeneous polynomials AA,BA ∈ Q[a]. This means in particular, that the

valuation of the j-invariant can be read off AAand BAdirectly, unless some unlucky

cancellation of leading terms occurs.

BA

This leads to the following definition.

Definition 6.2

The generic valuation of a polynomial 0 ?= H =?

valH: RA−→ R : u ?→ valH(u) = min{u · ω | Hω?= 0},

ωHω· aω∈ Q[a] with a =

(aij| (i,j) ∈ A) is

where

u · ω =

?

(i,j)∈Ac

uij· ωij.

The generic valuation of the j-invariant is the function

valj: RA−→ R : u ?→ valj(u) = valAA(u) − valBA(u).

Note that the tropical j-invariant is a tropical rational function in the sense of [13,

Sec. 2.2] and [1, Def. 3.1].

Remark 6.3

As mentioned above, unless some unlucky cancellation of the leading terms occurs

for any f =?

valj(u) = val?j(f)?.

(i,j)∈Aaij· xiyj∈ K[x,y] with uij = val(aij) for all (i,j) ∈ A we

have

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THE TROPICAL j-INVARIANT11

Note also, that if D is a cone of the Gr¨ obner fan of AA and D′is a cone of the

Gr¨ obner fan of BAthen

valj|: D ∩ D′−→ R

is linear by definition, and if both are top-dimensional, then no unlucky cancellation

of leading terms can occur. In particular, the generic valuation of the j-invariant

valjis a piece linear function.

We can now state the main result of our paper whose proof is discussed in the

subsequent sections.

Theorem 6.4

Let (Q,A) be a lattice polygon with only one interior lattice point.

If u ∈ RAis such that CF with

F : R2−→ R : (x,y) ?→ min{uij+ i · x + j · y | (i,j) ∈ A}

has a cycle, then

valj(u) = −jtrop(u).

Moreover, if u is in a top-dimensional cone of the secondary fan of A and f =

?

val?j(f)?= −jtrop(CF) = −jtrop

Example 6.5

Consider the curve Cf defined by

(i,j)∈Aaij· xiyjwith val(aij) = uij, then

?Trop(Cf)?.

f = t

3

2· (y + x2+ xy2) + xy

with the following subdivision of the Newton polygon and the corresponding elliptic

plane tropical curve Trop(Cf):

N(f) = Qcm

Trop(Cf)

The vertices of Trop(Cf) are

?3

2,3

?

,

?3

2,−3

2

?

and

?

−3,−3

2

?

,

so that its tropical j-invariant is jtrop

?Trop(f)?=27

2, while its j-invariant

j(f) =1 + 72 · t

9

2 + 1728 · t9+ 13824· t

27

2

t

27

2 + 27 · t18

has valuation −27

2.

An immediate consequence of the above theorem is the following corollary.

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12ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

Corollary 6.6

If f =?

(i,j)∈Aaij· xiyj∈ K[x,y] defines a smooth elliptic curve in XA whose

j-invariant has non-negative valuation, then Trop(Cf) has no cycle.

7. Reduction to A ∈ {Aa,Ab,Ac}

Using an integral unimodular transformation we may assume that Q is one of the

16 polygons in Figure 1, since the application of such a transformation does not

effect the statement of Theorem 6.4 due to Corollary 5.1 and Section 2.

Next we want to reduce to the cases A ∈ {Aa,Ab,Ac}.

If f =?

(i,j)∈Aaij· xiyj∈ K[x,y] with supp(f) ⊆ A and we replace f by

f′= f + tα·

?

(i,j)∈A\supp(f)

xiyj

where α is much larger than max{val(aij) | (i,j) ∈ supp(f)}, then obviously

val?j(f)?= val?j(f′)?.

Moreover, if we allow to plug in into valj points u where some of the uij are ∞

(as long as the result still is a well defined real number), then we can evaluate valj

at u with uij= val(aij) ∈ R ∪ {∞} , f defines a smooth elliptic curve and we get

obviously

valj(u) = valj(u′)

where u′

ij= uijfor (i,j) ∈ supp(f) and else u′

ij= α with α sufficiently large.

Finally, if in the definition of Fuwe allow some uij to be ∞ then with the above

notation the cycle of CFuand CFu′will not change, so that

jtrop(u) = jtrop(CFu) = jtrop(CFu′) = jtrop(u′).

This shows that whenever we may as well assume that A ∈ {Aa,Ab,Ac}.

8. The cases Aa, Aband Ac

The case Achas been treated in [9], and the two other cases work along the same

lines. We therefore will be rather short in our presentation. Instead of considering

all the cases by hand, as was done in [9] we will refer to computations done using

the computer algebra systems polymake [4], TOPCOM [16] and Singular [6]. The

code that we used for this is contained in the Singular library jinvariant.lib

(see [10]) and it is available via the URL

http://www.mathematik.uni-kl.de/˜ keilen/en/jinvariant.html.

Fix now (Q,A) with A ∈ {Aa,Ab,Ac}.

We first of all observe that by Corollary 4.6 the tropical j-invariant is linear on

the cones of the secondary fan of A and that by Lemma 4.5 we can read off the

assignment rule on each cone from the Newton subdivision of (Q,A). Moreover,

for the statement in Theorem 6.4 we only have to consider such cones for which the

interior lattice point of Q is visible in the subdivision.

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THE TROPICAL j-INVARIANT13

If UA ⊆ RAis the union of these cones, then it was in each of the cases A ∈

{Aa,Ab,Ac} computed by the procedure testInteriorInequalities in the li-

brary jinvariant.lib that UAis contained in a single cone of the Gr¨ obner fan of

AAand that

valAA|: UA−→ R : u ?→ 12 · u11.

It suffices therefore to show that valBAis linear on the cones of the secondary fan

of A and to compare the assignment rules for valjand jtropon each of these cones.

The two marked polygons (Qb,Ab) and (Qc,Ac) define smooth toric surfaces and

in these cases BA= ∆Ais the A-discriminant of A (cf. [5, Chapter 9]). Therefore,

by the Prime Factorization Theorem (see [5, Theorem 10.1.2]) the secondary fan of

Abrespectively Acis a refinement of the Gr¨ obner fan of the BAbrespectively BAc.

In view of Remark 6.3 and by the above considerations this shows in particular

that valj is linear on each cone of the secondary fan of A for A ∈ {Ab,Ac} which

is contained in UA.The comparison of the assignment rules of the two linear

functions valj and jtrop on each of the cones contained in UA was done by the

procedure displayFan from the Singular library jinvariant.lib using TOPCOM

and polymake. It produces two postscript files which show all the different cases

together with the assignment rules. The files are available via

http://www.mathematik.uni-kl.de/˜ keilen/download/Tropical/secondary fan of 2x2.ps

for A = Abrespectively via

http://www.mathematik.uni-kl.de/˜

keilen/download/Tropical/secondary fan of cubic.ps

for A = Acrespectively. 849 cases have to be considered for A = Acand 255 for

A = Ab.

In the case A = Aathe toric surface XAis not smooth, but a quadric cone. More-

over, in this case BAis not the A-discriminant ∆A, but instead

BA= u2

02· ∆A.

Thus, the Gr¨ obner fan of BAcoincides with the Gr¨ obner fan of ∆Aand it is still true

by the Prime Factorization Theorem that the secondary fan of A is a refinement of

the Gr¨ obner fan of BA. We can therefore argue as above, and the case distinction

(202 cases) can be viewed via

http://www.mathematik.uni-kl.de/˜ keilen/download/Tropical/secondary fan of 4x2.ps.

This finishes our proof, where for the “moreover” part we take Remark 6.3 into

account.

Remark 8.1

It follows from the proof that jtrop is indeed linear on each cone of the Gr¨ obner

fan of BAin the above cases. This could have been proved directly with the same

argument as in [9, Lemma 5.2].

If we denote by DAthe regular A-determinant (cf. [5, Section 11.1]), then DAb=

∆Aband DAc= ∆Acby [5, Theorem 11.1.3] since XA is smooth in these cases.

Even though in general the regular A-determinant is not a polynomial, it is so for

Page 14

14 ERIC KATZ, HANNAH MARKWIG, THOMAS MARKWIG

A = Aaby [5, Theorem 11.1.6] since Aais quasi-smooth by [5, Theorem 5.4.12] in

the sense of that theorem. More precisely, we have

DAa= u02· ∆Aa

and by [5, Theorem 11.1.3] it is a divisor of the principal A-determinant EA(cf. [5,

Chapter 9]). Therefore, the secondary fan of A (which is the Gr¨ obner fan of EA)

is a refinement of the Gr¨ obner fan of DAand thus of the Gr¨ obner fan of BA.

One therefore could have used the description of the vertices of the Newton polytope

of DAin [5, Theorem 11.3.2] in order to show that on each cone of the Gr¨ obner fan

of BA contained in UA the two functions valj and jtrop coincide by a direct case

study as was done in [9, Lemma 5.5] for the case A = Ac.

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Eric Katz, Department of Mathematics, The University of Texas at Austin, 1 University

Station, C1200, Austin, TX 78712

E-mail address: eekatz@math.utexas.edu

Page 15

THE TROPICAL j-INVARIANT15

Hannah Markwig, University of Michigan, Department of Mathematics, 2074 East Hall,

530 Church Street, Ann Arbor, MI 48109-1043

E-mail address: markwig@umich.edu

Thomas Markwig, Fachbereich Mathematik, Technische Universit¨ at Kaiserslautern,

Postfach 3049, 67653 Kaiserslautern, Germany

E-mail address: keilen@mathematik.uni-kl.de

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