# The tropical $j$-invariant

**ABSTRACT** If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation into the real numbers we can tropicalize C to get a tropical curve Trop(C). If the Newton subdivision induced by f is a triangulation, then Trop(C) 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 C. Comment: 15 pages

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**ABSTRACT:**We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space \Gamma_X. We give a geometric interpretation of the cohomology of \Gamma_X in terms of the action of a monodromy operator on the cohomology of X. This gives bounds on the Betti numbers of $\Gamma_X$ in terms of the Betti numbers of $X$. When $X$ is a sufficiently general complete intersection, this allows us to show that the cohomology of Trop(X) vanishes in degree less than dim(X). In addition, we give a description for the top power of the monodromy operator acting on middle cohomology in terms of the volume pairing on $\Gamma_X$.04/2008;

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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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2ERIC 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: