Z dx dx Z A Little History Q ( x ) 1 /d Q ( x ) Q ( x ) a polynomial Z dx Billiards and curves in moduli 1 − x 2 = sin − 1 x √ space Z dx 1 − x 3 = √ 3 Curtis T McMullen q q ⇣ ⌘ Harvard University x − 1 x − 1 2 3 ; 1 3 , 1 3 ; 5 x − 1 x − 1 3 3 √− 1 + 1 3 1 − ( − 1) 2 / 3 + 1( x − 1) F 1 3 ; − 1 − ( − 1) 2 / 3 , − √− 1 1+ 3 1+ 3 √ 1 − x 3 2 3 F 1 = Appel hypergeometric function Avila, Hubert, Kenyon, Kontsevich, Lanneau, Masur, Smillie, Yoccoz, Zorich, ... Z dx Moduli space A Little History Q ( x ) 1 /d M g = moduli space of Riemann surfaces X of genus g { } Z dx 1 − x 2 = sin − 1 x √ -- a complex variety, dimension 3g-3 (X, ω ) = (The curve y d = P(x), the form dx/y) Teichmüller metric: every holomorphic map periods of (X, ω ) = { ∫ ω } C f : H 2 → M g is distance-decreasing. � Riemann surfaces, homology, Hodge theory, TOTALLY unsymmetric automorphic forms, ...
How to describe X in M g ? How to describe X in M g ? g=1: X = C / Λ g=1: X = C / Λ g>1: X = ? g>1: X = ? Uniformization Theorem Uniformization Theorem Every (X, ω ) in Ω M g can be built from a polygon Every X in M g can be built from a polygon in C in C M g P P X = P / gluing (X, ω ) = (P ,dz) / gluing by translations by translations Moduli space Ω M g Teichmüller curves Dynamical: SL 2 ( R ) acts on Ω M g SL 2 ( R ) orbit of (X, ω ) in Ω M g projects to a complex geodesic in M g : Polygon for A ⋅ (X, ω ) = A ⋅ (Polygon for (X, ω )) M g H Complex geodesics f : H ⟶ M g f stabilizer of τ 2 V = H / SL(X, ω ) (X, ω ) τ 1 V → M g is an algebraic, SL(X, ω ) lattice ⇔ f : isometrically immersed Teichmüller curve . f( τ 1 ) f( τ 2 )
Complex geodesics in genus two Rigidity Conjecture Theorem The closure of any complex geodesic f( H ) ⊂ M g Let f : H → M 2 be a complex geodesic. is an algebraic subvariety. Then f( H ) is either: dim • A Teichmüller curve, 1 • A Hilbert modular surface H D , or 2 Celebrated theorem of Ratner (1995) ⇒ • The whole space M 2. 3 true for H ⟶ locally symmetric spaces X = K\G/ Γ Recent progress towards general g Eskin -- Mirzakhani Billiards in polygons Classification Problem V → M 2 ? What are the Teichmüller curves Neither periodic nor evenly distributed
Optimal Billiards Billiard theorists (Veech) Theorem. In a regular n-gon, every billiard path is either periodic or uniformly distributed. Billiards and Riemann surfaces P is a Lattice Polygon ⇔ SL(X, ω ) is a lattice ⇔ (X, ω ) generates a Teichmüller curve Theorem (Veech, Masur): If P is a lattice polygon, then billiards in P is optimal. P (X, ω ) = P/~ (renormalization) X has genus 2 ω has just one zero!
Optimal Billiards Explicit package: Pentagon example Example: if X = C / Λ , ω =dz, then SL(X, ω ) ≃ SL 2 ( Z ) Theorem (Veech, 1989): For (X, ω ) = (y 2 = x n -1, dx/y), � � (X, ω ) = (y 2 =x 5 -1,dx/y) SL(X, ω ) is a lattice. M g Corollary Any regular polygon is a lattice polygon. V = H /SL(X, ω ) ⊂ SL 2 ( √ 5) g · (X, ω ) = ⟨ a, b ⟩ mb ⇒ Direct proof that SL(X, ω ) is a lattice 20th century lattice billiards Genus 2 SL_2(Z) � Regular 5- 8- and 10-gon Square Problem Tiled by squares ~ SL_2(Z) Are there infinitely many primitiveTeichmüller V in the moduli space M 2 ? curves Regular polygons ~ (2,n, ) triangle group 8 Various triangles triangle groups
Jacobians with real multiplication The Weierstrass curves Theorem W D = {X in M 2 : O D acts on Jac(X) and its (X, ω ) generates a Teichmüller curve V ⇒ eigenform ω has a double zero.} Jac(X) admits real multiplication by O D ⊂ Q ( √ D). Theorem. W D is a finite union of Teichmüller curves. 1 Corollary Corollaries V lies on a Hilbert modular surface P d • P d has optimal billiards γ for all integers d>0. V ⊂ H D ⊂ M 2 = 1 • There are infinitely many H x H /SL 2 ( O D ) primitive V in genus 2. γ = (1 + √ d)/2 Torsion divisors in genus two The regular decagon 4 2 P -2 -1 1 2 Q -2 -4 Theorem (Möller) (X, ω ) generates a Teichmüller curve Theorem. The only other primitive Teichmüller curve in ⇒ [P-Q] is torsion in Jac(X) genus two is generated by the regular decagon.
Teichmüller curves in genus 2 Mysteries Theorem • Is W D irreducible ? • What is its Euler characteristic ? The Weierstrass curves W D account for all the • What is its genus ? primitive Teichmüller curves in genus 2 -- • Algebraic points (X, ω ) in W D ? -- except for the curve coming from the • What is Γ = SL(X, ω ) ? regular decagon. W D = H / Γ , Γ ⊂ SL 2 ( O D ) Euler characteristic of W D Classification Theorem M, 2004 W D is connected except when D=1 mod 8, D>9. Theorem (Bainbridge, 2006) (spin) χ ( W D ) = − 9 2 χ (SL 2 ( O D )) 0 W 17 = coefficients of a modular form Compare: χ ( M g,1 ) = ζ (1-2g) (Harer-Zagier) 1 W 17 Proof: Uses cusp form on Hilbert modular surface with ( α ) = W D - P D, where P D is a Shimura curve
Genus of W D Elliptic points on W D D g ( W D ) e 2 ( W D ) C ( W D ) χ ( W D ) D g ( W D ) e 2 ( W D ) C ( W D ) χ ( W D ) Theorem (Mukamel, 2011) − 3 5 0 1 1 52 1 0 15 − 15 10 − 3 − 21 8 0 0 2 53 2 3 7 4 2 Corollary 9 0 1 2 − 1 56 3 2 10 − 15 2 The number of orbifold points on W D is given − 3 � − 21 2 , − 21 ⇥ 12 0 1 3 57 { 1 , 1 } { 1 , 1 } { 10 , 10 } 2 2 − 3 13 0 1 3 60 3 4 12 − 18 2 by a sum of class numbers for Q ( √ -D). W D has genus − 3 − 33 16 0 1 3 61 2 3 13 2 2 � − 3 2 , − 3 ⇥ 17 { 0 , 0 } { 1 , 1 } { 3 , 3 } 64 1 2 17 − 18 2 0 only for 20 0 0 5 − 3 65 { 1 , 1 } { 2 , 2 } { 11 , 11 } { − 12 , − 12 } 21 0 2 4 − 3 68 3 0 14 − 18 D<50 − 9 24 0 1 6 69 4 4 10 − 18 2 − 3 , − 3 − 45 25 { 0 , 0 } { 0 , 1 } { 5 , 3 } � ⇥ 72 4 1 16 2 2 � − 33 2 , − 33 ⇥ 28 0 2 7 − 6 73 { 1 , 1 } { 1 , 1 } { 16 , 16 } 2 Proof: (X, ω ) correponds to an orbifold point ⇒ − 9 − 57 29 0 3 5 76 4 3 21 2 2 32 0 2 7 − 6 77 5 4 8 − 18 � − 9 2 , − 9 ⇥ 33 { 0 , 0 } { 1 , 1 } { 6 , 6 } 80 4 4 16 − 24 X covers a CM elliptic curve E ⇒ 2 � − 18 , − 27 ⇥ 36 0 0 8 − 6 81 { 2 , 0 } { 0 , 3 } { 16 , 14 } 2 37 0 1 9 − 15 84 7 0 18 − 30 2 (X, ω ), p: X → E and Jac(X) can be described explicitly. − 21 40 0 1 12 85 6 2 16 − 27 2 − 69 41 { 0 , 0 } { 2 , 2 } { 7 , 7 } { − 6 , − 6 } 88 7 1 22 2 44 1 3 9 − 21 89 { 3 , 3 } { 3 , 3 } { 14 , 14 } � − 39 2 , − 39 ⇥ 2 2 45 1 2 8 − 9 92 8 6 13 − 30 48 1 2 11 − 12 93 8 2 12 − 27 (table by Mukamel) 49 { 0 , 0 } { 2 , 0 } { 10 , 8 } { − 9 , − 6 } 96 8 4 20 − 36 Algebraic points on W D Computing W D X ∈ M 2 D=5 y 2 = x 5 -1 D=8 y 2 = x 8 -1 i Mukamel D=13 y 2 = (x 2 -1)(x 4 -ax 2 +1) 0 √ 11 -11+ √ 11 - √ 11 a = 2594 + 720 √ 13 Mukamel .... D=44 D=108 96001 + 48003 a + 3 a 2 + a 3 = 0
Higher genus? Theorem (Möller, Bainbridge-Möller) Finiteness holds... Conjecture: for hyperelliptic statum (g-1,g-1) Teichmüller curves in M g There are only finitely many for g=3, stratum (3,1) with deg(trace field SL(X, ω )) = g = 3 or more. Methods: Variation of Hodge structure; rigidity (avoids echos of lower genera) theorems of Deligne and Schmid; Neron models; arithmetic geometry [Rules out quadratic fields] • Jac(X) admits real multiplication by K, • P-Q is torsion in Jac(X) for any two zeros of ω . However... Exceptional triangular billiards 1/4 E E 6 Theorem 5/12 1/3 2/9 There exist infinitely many primitive E E 7 Teichmüller curves in M g for genus g = 2, 3 and 4. 4/9 1/3 1/5 E E 8 7/15 1/3
Prym systems in genus 2, 3 and 4 Higher genus? 1 1 3 2 4 2 3 4 Question. Are there only finitely many primitive Teichmüller curves 1 6 6 1 5 3 4 2 in M g for each g ≥ 5? 3 5 2 4 1 1 7 2 5 2 3 8 5 3 8 7 6 4 6 4 W D for g=3,4: Lanneau--Nguyen but still quadratic fields What about the Hilbert modular surfaces Pentagon-to-star map H D ⊂ M 2 ? ~ H D H × H foliated by complex geodesics ~ ↓ W D F H D ⊂ M 2 each leaf is the graph of a holomorphic function ~ F : H → H W 5 = graph of F
Slice of H D Action on slices of H D ~ Slice { τ 1 } × H H D points of W D ρ = ∫ ab ω = relative period q = (d ρ ) 2 quadratic differential SL( H ,q) = SL 2 ( O D ) acts on slice points of P D { τ 1 } × H golden table gives picture of action of SL 2 (R) on Ω M 2 Exotic leaves Slice of Hilbert modular surface q = Q | { τ 1 } × H D=5 12 10 8 6 4 2 � 10 � 5 0 5 10
Möller-Zagier formula × Billiards in a pentagon, � � � � � reprise d ϑ m d τ − 1 d τ 2 Q = ( τ , 0) ϑ m ( τ , 0) 2 . 1 dz 2 m even m odd products taken over spin strs m (6 odd, 10 even) (Q) = W D - P D on the Hilbert modular surface X D = H × H /SL 2 ( O D )
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