Billiards, heights and modular symbols

Curtis T McMullenHarvard University

Weil, Manin, Birch, Leutbecher, Veech, Masur, Forni, Möller, Leininger, Hubert, Lanneau, Davis, Lelievre, ….

A dense set of slopes are periodic.

Billiards in a regular pentagon

How do the periodic trajectories behave?

Slopes and lengths

sL(s) = 5

4sL(s) = 469

20sL(s) = 2338

6765sL(s) = 1025

Slopes, lengths and heights

s

Theorem 1The periodic slopes coincide with Q(√5)s,

and log L(xs) = O(h(x)2).

exponent 2 is sharp

Example

L(10ns) = O(10Cn2

)

Method: descent on a Hilbert modular surface

Every trajectory isperiodic or uniformly distributed.

Billiards in a regular pentagon

How are the periodic trajectories distributed?

Every trajectory isperiodic or uniformly distributed.

Billiards in a regular pentagon

How are the periodic trajectories distributed?

Davis-Lelievre: Not always uniformly!

Theorem IIFor each periodic slope s, the limit measures Ms form a countable set, homeomorphic to ωω + 1.

describe scarring

ComplementWe have uniform distribution iff the lengths of the golden continued fractions of the slopes tend to infinity.

Limit Measures

Method: modular symbols for Teichmüller curves

Limit Measures M0uniform measure

Modular symbols

V = H/Γ hyperbolic surface

modular symbol of degree d: formal product

σ = γ1 * γ2 * …. * γd

a0, a1, …, ad = cusps of V

γi geodesic from ai-1 to ai

a0 a1 a2 ad

γ1 γ2 γi

Modular symbols

S(V) = ∪ Sd(V)

degree d

= morphisms in a graded category whose objects are the cusps of V

geometric topology

Sd(V) = ∪ Se(V)e ≥ d

S(V) ≃ ωω

a

b

c

γn

Modular symbols: topology

a c b

δ1 δ2

γn ⟶ δ1 * δ2

Modular symbols for V = H/SL2(Z)

S1(V) = { [a1, …, an] } = { 1/a1 + 1/a2 + … + 1/an }

∞ p/q in [0,1] γ ∞

2

3

S(V) = { [a1, …, an] : some ai = ∞ }

[a1, …, an] * [b1, …, bm] = [a1, …, an, ∞, b1, …, bm]

{ [a1, …, an] : n ≤ N } is compact

Aside: Classical Modular symbols

Q ∪ ∞ = cusps of Γ(N) in SL2(Z)X(N) = completion of H / Γ(N)

Theorem (Manin-Drinfeld)

The difference of any 2 cusps of X(N) istorsion in Jac(X(N)).

{p,q} : Q×Q ⟶ Ω(X)* ≃ H1(X(N), R)abelian

Teichmüller curves

(X,ω) = holomorphic 1-form of genus g

cusp of SL(X,ω) ⇔ periodic slope s for (X,|ω|)

V = H / SL(X,ω) ⟶ Mg

lattice

parabolic

⇔ cylinder system A = (A1, …, An) +

fundamental twist τA, DτA ∈ SL(X,ω)

Thurston’s multi curve systems

Every Teichmüller curve V can be specifiedby a pair of topological multicurves (Ai), (Bj).

Modular symbols for V organize allthe curves systems encoding V.

Usually of infinite index!

Gives ⟨DτA, DτB⟩ = Γ ⊂ SL(X,ω)

Thurston’s multi curve systems

Every Teichmüller curve V can be specifiedby a pair of topological multicurves (Ai), (Bj).

Usually of infinite index!

Gives ⟨DτA, DτB⟩ = Γ ⊂ SL(X,ω)

Theorem: There is a natural inclusionS1(V) ⟶ MLg × MLg / Modg

whose image is the set of all (A,B) specifying V.

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Q

hA

hB

!= µ

hA

hB

!

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TopologyGeometry

SL2(R

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⇣Xai ·Ai,

Xbj ·Bj

⌘

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Q =

0 mAJ

mBJ t 0

!

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J = i(Ai,Bj)

modular symbol multicurves

heights

Discovering the golden table

A1

A2

B2 B1

The A4 Coxeter diagram

SL(X,ω) is a lattice, ≃ Δ(2,5,∞)i(Ai,Bj) = 0 1

1 1( )non-arithmetic group

Twists and limit measures

τAn(B1) τAn(B2)

Measures predicted by i(A,B)

B1

A

B2

A

Modular symbols to measures

We have a continuous functor I : S(V) ⟶ L(V)

category of matricesup to scale

given by I(γ) = [mod(Ai) i(Ai,Bj)].

Decouples as γ ⟶∞. ~ [h(Ai) c(Bj)]

⇒ closure of image is ωω union a finite set

⇒ limit measures form a copy of ωω + 1.

QED Theorem IIhidden multiplicative structure

TheoremIf the ray generated by (X,ω) spends at least time T in a compact set K in Mg, then the unit norm positive currents

[P1(ω)] ⊂ H1(X,R)

carried by F(ω) have Hodge diameter O(exp(-C(K) T)).

⇒ decoupling

P

H1(X,R)

Square-tiled case

Sometimes Ms is ωω + 1, sometimes it is one point!

Figure 4. Periodic geodesics near slopes s = 0 and s = 1.

I. The square L. Consider a symmetric L–shaped polygon P made up ofthree squares. By identifying parallel edges, we obtain a square–tiled surface(X, !) 2 ⌦M2(2) with

SL(X, !) =

* 1 2

0 1

!,

0 1

�1 0

!+⇢ SL2(Z).

The corresponding Teichmuller curve V = H/ SL(X, !) is the (2, 1, 1)orbifold; in the terminology of [Mc1], it is the Weierstrass curve WD ⇢ M2

for discriminant D = 9. This is the simplest square–tiled surface of genusg > 1.

For this example, Theorem 9.1 implies:

Mp/q is a single point when p and q are both odd; otherwise,

Ms⇠= !! + 1.

To see this, note that V has two cusps, a and b, corresponding to the slopes1 and 0 respectively. The first cusp has rank one – indeed, C(1) is a singlecylinder; and the corresponding slopes are the ratios of odd integers p/q.The second has rank two; for example, the horizontal and vertical cylindersystems A and B of (X, !) satisfy

i(Ai, Bj) =

1 1

1 0

!.

By Theorem 9.1, all closed geodesics with slopes sn ! 1 are uniformlydistributed, but some with slopes sn ! 0 are not; see Figure 4.

II. The quaternion surface. Our second example is a surface (X, !) ofgenus 3 tiled by 8 squares, studied in [HS], [FMZ, Figure 6], [Mo2] and [Mc3,

38

Slopes, lengths and heights

s

Theorem 1The periodic slopes coincide with Q(√5)s,

and log L(xs) = O(h(x)2).

Method: descent, using a new height on P1(K)

h(p/q + r/s √5) ≃ log max (|p|,|q|,|r|,|s|) ≥ 0

Curves on a Hilbert modular surface

K = real quadratic field

XK = (H⇥H)/ SL(O�O_)

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V = H/� # XK

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geodesic curve

Theorem I

Either V is a Shimura curve, or the cusps of Vcoincide with and satisfy quadraticheight bounds.

P1(K)

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cf. M, Möller-Viehweg

Relation to the pentagon

Holomorphic pentagon-to-star map

FH H

(t,F(t))

H2 / SL2(Z[γ])

XK

V

Symmetries of Teichmüller curves

Theorem (M,Möller)SL(X,ω) a lattice with trace field K ⇒

A=Jac(X) admits real multiplication by K

(a factor of) K ⊂ End(A) ⊗ Q

Idea: g + g�1 =

a b

c d

!+

d �b

�c a

!= Tr(g) · I

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(�+ ��1)⇤! = (Tr�)!

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H1(A,Q) ⇠= K2

K ⇢ End(A)⌦Q

The projective lineHeight HA(x) on P1A(K)

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Height HA(x) on P1A(K)

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= space of K-lines in

H1(A,Q)

Classical height on Pn(K)

For example, if v is an real place of K, and ⇢ : K ! Kv = R is the associatedcompletion, then

|x|v = |⇢(x)|1/g.

Heights on projective space. The absolute multiplicative height onPn(K) is given by

H(x) = H(x0 : x1 : · · · : xn) =Y

v

maxi

|xi|v.

It is well–defined by the product formula, which also implies that H(x) � 1.Our normalizations were chosen so that H(x) remains constant under finiteextensions.

A closely related height can be defined by

eH(x) = infa

Y

v|1

maxi

|ai|v, (2.1)

where the infimum is taken over vectors of integers a 2 On+1 such that

[a0 : · · · : an] = [x]. This height is comparable to the standard one; indeed,using finiteness of the class number, one can show that

H(x) eH(x) C(K, n)H(x)

for all x, and equality holds when O is a UFD.

Abelian varieties. Let A be a polarized Abelian variety of dimension g.We can naturally identify A with the quotient space

A = ⌦(A)⇤/H1(A,Z),

where ⌦(A) ⇠= Cg is the space of holomorphic 1–forms on A, and its paring

with H1(A,Z) ⇠= Z2g is given by hC, !i =

RC !.

The polarization of A is recorded by a positive–definite Hermitian innerproduct on ⌦(A)⇤, with the property that the symplectic form

[C, D] = � ImhC, Di (2.2)

takes integral values on H1(A,Z). We denote the associated Hodge norm onH1(A,R) by kCkA = hC, Ci

1/2. The polarization also determines a normand inner product on ⌦(A), via duality.

11

For example, if v is an real place of K, and ⇢ : K ! Kv = R is the associatedcompletion, then

|x|v = |⇢(x)|1/g.

Heights on projective space. The absolute multiplicative height onPn(K) is given by

H(x) = H(x0 : x1 : · · · : xn) =Y

v

maxi

|xi|v.

It is well–defined by the product formula, which also implies that H(x) � 1.Our normalizations were chosen so that H(x) remains constant under finiteextensions.

A closely related height can be defined by

eH(x) = infa

Y

v|1

maxi

|ai|v, (2.1)

where the infimum is taken over vectors of integers a 2 On+1 such that

[a0 : · · · : an] = [x]. This height is comparable to the standard one; indeed,using finiteness of the class number, one can show that

H(x) eH(x) C(K, n)H(x)

for all x, and equality holds when O is a UFD.

Abelian varieties. Let A be a polarized Abelian variety of dimension g.We can naturally identify A with the quotient space

A = ⌦(A)⇤/H1(A,Z),

where ⌦(A) ⇠= Cg is the space of holomorphic 1–forms on A, and its paring

with H1(A,Z) ⇠= Z2g is given by hC, !i =

RC !.

The polarization of A is recorded by a positive–definite Hermitian innerproduct on ⌦(A)⇤, with the property that the symplectic form

[C, D] = � ImhC, Di (2.2)

takes integral values on H1(A,Z). We denote the associated Hodge norm onH1(A,R) by kCkA = hC, Ci

1/2. The polarization also determines a normand inner product on ⌦(A), via duality.

11

For example, if v is an real place of K, and ⇢ : K ! Kv = R is the associatedcompletion, then

|x|v = |⇢(x)|1/g.

Heights on projective space. The absolute multiplicative height onPn(K) is given by

H(x) = H(x0 : x1 : · · · : xn) =Y

v

maxi

|xi|v.

It is well–defined by the product formula, which also implies that H(x) � 1.Our normalizations were chosen so that H(x) remains constant under finiteextensions.

A closely related height can be defined by

eH(x) = infa

Y

v|1

maxi

|ai|v, (2.1)

where the infimum is taken over vectors of integers a 2 On+1 such that

[a0 : · · · : an] = [x]. This height is comparable to the standard one; indeed,using finiteness of the class number, one can show that

H(x) eH(x) C(K, n)H(x)

for all x, and equality holds when O is a UFD.

Abelian varieties. Let A be a polarized Abelian variety of dimension g.We can naturally identify A with the quotient space

A = ⌦(A)⇤/H1(A,Z),

where ⌦(A) ⇠= Cg is the space of holomorphic 1–forms on A, and its paring

with H1(A,Z) ⇠= Z2g is given by hC, !i =

RC !.

The polarization of A is recorded by a positive–definite Hermitian innerproduct on ⌦(A)⇤, with the property that the symplectic form

[C, D] = � ImhC, Di (2.2)

takes integral values on H1(A,Z). We denote the associated Hodge norm onH1(A,R) by kCkA = hC, Ci

1/2. The polarization also determines a normand inner product on ⌦(A), via duality.

11

(ai are integers)

comparable to

only requires knowledge of integers andinfinite places

Height HA(x) on P1A(K)

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H(x) = infC

Y

v|1

|C|v

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A

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|C|v =

����Z

C!v

����1/g

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Hodge norm at vx ∈ PA1(K)

C ∈ H1(X, Z)

x = [K・C]

Why a height?

eH(x) = infa

Y

v|1

maxi

|ai|v

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Theorem. Given a linear isomorphism

◆ : P1A(K) ! P1(K)

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H(◆(x)) ⇣ HA(x)

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we have .

H(x) = infC

Y

v|1

|C|v

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A

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Case of a torus

K = Q

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A = C/Z� Z⌧

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H1(A,Z) ⇠= Z2

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Hτ(x) = length of geodesic with slope x = a/b

kCk2A =

����Z

C!

����2

=|a+ b⌧ |2

Im ⌧

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Hodge norm

Level sets of height

decreases like exp(-s) alonggeodesic rays descending to a/b

a/b

Ht(a/b)

t ∈ H

g=1

a/b

Descent on aHilbert modular surface

t = γ(s) ∈ H

a/b ∈ Q(√D)Holomorphic pentagon-to-star map

F

γ(s)

Hτ(a/b) τ = (t,F(t))

Aτ = C2 / O ⊕ τOv

g=2

To show a/b is a cusp

When t lies over Vthick :

Hτ(a/b) ≥ 1|F´(t)| < δ < 1

So γ spends only a finite amount of time over Vthick

Hτ(a/b) ~ (t term) x (F(t) term)

≤ exp(-s) exp(|F´| s)

⇒a/b is a cusp

Cor: Triangle groups

0 γ11/γ

The cusps of the (2,5,∞) triangle group Γ coincide with K.

� = hz 7! �1/z and z 7! z + �i

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Cor: Continued fractions1. The cusps of �(2, 5,1) coincide with Q(�) [ {1} (see Figure 1).

2. Every s 2 Q(�) can be expanded as a finite golden continued fraction,

s = [a1, a2, a3, . . . , aN ] = a1� +1

a2� +1

a3� + · · ·1

aN�

with ai 2 Z.

3. Every s 2 Q(�) can be expressed as a golden fraction s = a/c, charac-terized by the property that

�a bc d

�2 � for some b, d. This expression

is unique up to a sign change, s = (�a)/(�c).

Let us elaborate the last point. Since K has class number one, we cancertainly write s = A/B as a ratio of relatively integers A, B 2 Z[�]. Infact, since � is a unit, there are many such expressions: we also have s =(�kA)/(�kB) for any k 2 Z.

The golden fraction expression s = a/c uses the thin group � to pick outa particular value of k. The complexity of this expression is controlled bythe height bounds in Theorem 1.1; in this case, they yield:

Corollary 1.3 The height of any nonzero golden fraction s = a/c satisfies

h(a) + h(c) = O(h(s)2). (1.1)

Here h(x) is the absolute logarithmic height on K = Q�Q�; it satisfies

h((p/q) + (r/s)�) ⇣ log max{1, |p|, |q|, |r|, |s|}.

One can readily verify that for k � 0 the denominator of the goldenfraction for s = �2k is c = �k2�k+1, so the exponent 2 in equation (1.1) issharp. We will also see that the length of the golden continued fraction fors satisfies N = O(h(s)).

Matrix coe�cients. Let M ⇢ Z[�] denote the set of all matrix entriesthat occur in �. The discussion of golden fractions above shows that

Z[�] =[

k

�kM.

As noted by Leutbecher in the 1970s [Le], there is no known characteri-zation of the elements of M . The next result gives a qualitative descriptionof M and also reveals its hidden multiplicative structure.

3

Height bounds: length N and ai are O(1+h(x)) .

Theorem

The cusps of the (p,q,∞) triangle group coincidewith P1(K) whenever deg(K/Q) = 1 or 2.

non arithmetic

Open problem

Converse? For triangle groups and for Veech groups?

(2,q,∞) known

Cor: Billiards in a pentagon

The periodic slopes coincide with Q(√5)s, and log L(xs) = O(h(x)2).

s = tan(2π/5)

1

1�

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�

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Figure 3. Long periodic billiard paths, each with over 200 segments, withinitial slopes 5 and 8

p3 respectively.

Computer experimental quickly reveal that even small, rational slopeslead to very long trajectories in P ; for example, L(5) ⇡ 479, while L(6765)is on the order of 1025. This suggest that the exponent 2 in (1.2) is sharp,and indeed this is the case.

The 1–form associated to this polygon satisfies SL(X, !) = �(2, 5,1).Using this connection, we will give a simple dynamical proof that

a + b� 2 M =) ab � 0

and hence���2

m0/m 1 (1.3)

for all matrix entries m 6= 0 in �(2, 5,1). Equality arises is when m = 1and m = �.

Example 2. The golden arrow. A second lattice polygon, also based onthe golden ratio, is shown at the right in Figure 3; its internal angles are⇡(1, 1, 2, 8)/6, and its periodic slopes are given by S(P ) =

p3 ·Q(�)[ {1}.

Both examples belong to infinite families, discussed in [Mc1, §9] and[EMMO, §8] respectively, and their side lengths can be varied to produceinfinitely many di↵erent quadratic trace fields.

6

Applies to all families of optimal billiards

…since these are quadratic: Eskin - Filip - Wright

Open problem

In a regular heptagon, (i) characterize the periodic slopes, and (ii) bound L(x s), x in K.

Shown: L(s)=7, L(2 s) =

s = tan(2π/7)

K = Q(cos(2π/7))(cubic)

2190

References

Teichmüller dynamics and unique ergodicity via currents and Hodge theory

Modular symbols for Teichmüller curves

Billiards, heights, and the arithmetic of non-arithmetic groups in preparation

math.harvard.edu/~ctm/papers

Preprints, 2019/2020