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.
(X,!) + �<latexit sha1_base64="eYRa9ux4GTXjohsVn0fxcOznHYA=">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</latexit>
(Y, ⌘) + [0,1]<latexit sha1_base64="CfFDxStaZUXE1me8p5ySmTw9cPI=">AAACNHicbZDLSgMxGIUz3h1vVVfiJiiCYikzutBl0Y3LCrYqnaFk0owGcxmSf8RhKL6Er+BWd76L4E7c+gymrYK3A4HDOf9Pki/JBLcQBM/eyOjY+MTk1LQ/Mzs3v1BZXGpZnRvKmlQLbc4SYpngijWBg2BnmWFEJoKdJleH/f70mhnLtTqBImOxJBeKp5wScFGnsrJ5Xo0YkC0cVbejKm4H1YirFIq4U1kPasFA+K8JP816fS3avnuuF43OojcadTXNJVNABbG2HQYZxCUxwKlgPT/KLcsIvSIXrO2sIpLZuBz8oYc3XNLFqTbuKMCD9PtGSaS1hUzcpCRwaX93/fC/rp1Duh+XXGU5MEWHF6W5wKBxHwjucsMoiMIZQg13b8X0khhCwWHzo8FiKV2lbY1qKbXqfaU3hbM3hc76LEsiRM/3HbbwN6S/prVTC3drO8eO3wEaagqtojW0iUK0h+roCDVQE1F0i+7RA3r0nrwX79V7G46OeJ87y+iHvPcPS/6r1g==</latexit>
Q
hA
hB
!= µ
hA
hB
!
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TopologyGeometry
SL2(R
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⇣Xai ·Ai,
Xbj ·Bj
⌘
<latexit sha1_base64="fwDCEff/TOgQPpd+s6M0yyywk+A=">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</latexit>
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_)
<latexit sha1_base64="ufr3YCrO7lGZC8elBJnL/LDwLqY=">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</latexit>
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)
<latexit sha1_base64="5icAXsP/ey5Rm5LYsskmVF9t7es=">AAACI3icbVDLSgMxFM20Pur4anXpZrAIdVNmpKLLohvBTQX7wE4tmTRtY/MYkox0GOYv3OrOr3Enblz4L6YPQVsPBA7n3MO9OUFIidKu+2llsiura+u5DXtza3tnN1/YaygRSYTrSFAhWwFUmBKO65poiluhxJAFFDeD0eXEbz5iqYjgtzoOcYfBASd9gqA20p0fSvFw75Wuj7v5olt2p3CWiTcnRTBHrVuwsn5PoIhhrhGFSrU9N9SdBEpNEMWp7UcKhxCN4AC3DeWQYdVJpienzpFRek5fSPO4dqbq70QCmVIxC8wkg3qoFr2J+J/XjnT/vJMQHkYaczRb1I+oo4Uz+b/TIxIjTWNDIJLE3OqgIZQQadOS7U+DCTOWUGUkGBM8/VHHsaHjWIST6hJIaWrbpjZvsaRl0jgpe5Xy6U2lWL2YF5gDB+AQlIAHzkAVXIEaqAMEOHgCz+DFerXerHfrYzaaseaZffAH1tc3ZlejyQ==</latexit>
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
<latexit sha1_base64="F7eoiDocQEvANtySk2UjPAeoqJk=">AAACHHicbVDLSgMxFM20Pur4anXpJlgEV8OMVHRZdeOyoq2FtpRMmmlD8xiSjHQY6h+41Z1f407cCv6N6UPQ1gOBwzn3cG9OGDOqje9/Obn8yuraemHD3dza3tktlvYaWiYKkzqWTKpmiDRhVJC6oYaRZqwI4iEj9+HwauLfPxClqRR3Jo1Jh6O+oBHFyFjp9rF70S2Wfc+fAi6TYE7KYI5at+Tk2z2JE06EwQxp3Qr82HQypAzFjIzddqJJjPAQ9UnLUoE40Z1seusYHlmlByOp7BMGTtXfiQxxrVMe2kmOzEAvehPxP6+VmOi8k1ERJ4YIPFsUJQwaCScfhz2qCDYstQRhRe2tEA+QQtjYetz2NJhxa0ntYcm5FOMfdZRaOkplPOksQ4yNXdfWFiyWtEwaJ15Q8U5vKuXq5bzAAjgAh+AYBOAMVME1qIE6wKAPnsAzeHFenTfn3fmYjeaceWYf/IHz+Q2ow6Ff</latexit>
|C|v =
����Z
C!v
����1/g
<latexit sha1_base64="F/mPzVeNBZcH5IfhFIwTWBHbjFk=">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</latexit>
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
<latexit sha1_base64="C58TjfnIuHdxwH3GjqN8l98tis8=">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</latexit>
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
<latexit sha1_base64="F7eoiDocQEvANtySk2UjPAeoqJk=">AAACHHicbVDLSgMxFM20Pur4anXpJlgEV8OMVHRZdeOyoq2FtpRMmmlD8xiSjHQY6h+41Z1f407cCv6N6UPQ1gOBwzn3cG9OGDOqje9/Obn8yuraemHD3dza3tktlvYaWiYKkzqWTKpmiDRhVJC6oYaRZqwI4iEj9+HwauLfPxClqRR3Jo1Jh6O+oBHFyFjp9rF70S2Wfc+fAi6TYE7KYI5at+Tk2z2JE06EwQxp3Qr82HQypAzFjIzddqJJjPAQ9UnLUoE40Z1seusYHlmlByOp7BMGTtXfiQxxrVMe2kmOzEAvehPxP6+VmOi8k1ERJ4YIPFsUJQwaCScfhz2qCDYstQRhRe2tEA+QQtjYetz2NJhxa0ntYcm5FOMfdZRaOkplPOksQ4yNXdfWFiyWtEwaJ15Q8U5vKuXq5bzAAjgAh+AYBOAMVME1qIE6wKAPnsAzeHFenTfn3fmYjeaceWYf/IHz+Q2ow6Ff</latexit>
Case of a torus
K = Q
<latexit sha1_base64="oMhE2aOnHXf5DUK+vAOKyTwbu/c=">AAACI3icbVDLSgMxFM20Pur4anXpJlgEV2VGFN0IRTeCmwr2gZ1SMmnahuYxJBnpMPQv3OrOr3Enblz4L6btCNp6IHA45x7uzQkjRrXxvE8nl19ZXVsvbLibW9s7u8XSXkPLWGFSx5JJ1QqRJowKUjfUMNKKFEE8ZKQZjq6nfvORKE2luDdJRDocDQTtU4yMlR5u4SUMFDJMd4tlr+LNAJeJn5EyyFDrlpx80JM45kQYzJDWbd+LTCdFylDMyMQNYk0ihEdoQNqWCsSJ7qSzkyfwyCo92JfKPmHgTP2dSBHXOuGhneTIDPWiNxX/89qx6V90Uiqi2BCB54v6MYNGwun/YY8qgg1LLEFYUXsrxEOkEDa2JTeYBVNuLakrWHIuxeRHHSeWjhMZTatLEWMT17W1+YslLZPGScU/rZzdnZarV1mBBXAADsEx8ME5qIIbUAN1gIEAT+AZvDivzpvz7nzMR3NOltkHf+B8fQNu9KPR</latexit>
A = C/Z� Z⌧
<latexit sha1_base64="6tptk376kWH2P3SOCq6Mnq8cVn4=">AAACOXicbVBJSwMxGM20LnXcWj0KEiyCpzojFb0IVS8eK9gFOqVk0lRDswxJRjoOc/PXeNWbv8SjN/HqHzBdBLcHgZf3vkfyvTBiVBvPe3Fy+bn5hcXCkru8srq2XixtNLWMFSYNLJlU7RBpwqggDUMNI+1IEcRDRlrh8Hzst26J0lSKK5NEpMvRtaADipGxUq+4fQpPYIBH+8Ed6cOgT5WOOZxeDIp7xbJX8SaAf4k/I2UwQ71XcvJBX+KYE2EwQ1p3fC8y3RQpQzEjmRvEmkQID9E16VgqECe6m04WyeCuVfpwIJU9wsCJ+j2RIq51wkM7yZG50b+9sfif14nN4LibUhHFhgg8fWgQM2gkHLcC7d4EG5ZYgrCi9q8Q3yCFsLHducEkmHJrSV3BknMpsi91lFg6SmQ0LjRFjGWua2vzf5f0lzQPKn61cnhZLdfOZgUWwBbYAXvAB0egBi5AHTQABvfgATyCJ+fZeXXenPfpaM6ZZTbBDzgfn2miq9g=</latexit>
H1(A,Z) ⇠= Z2
<latexit sha1_base64="0Gy3RKO5S5Vxwu8ernsHMvrINws=">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</latexit>
Hτ(x) = length of geodesic with slope x = a/b
kCk2A =
����Z
C!
����2
=|a+ b⌧ |2
Im ⌧
<latexit sha1_base64="qWYQYAgtzhP49ZTWZhkpnaz0hOU=">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</latexit>
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