u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Faculty of Science A Distribution Result Related to Automorphic Forms Flemming von Essen Department of Mathematical Sciences April 2013 Slide 1/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Fuchsian groups � a � b For γ = let c d γ z = az + b cz + d . Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 2/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Fuchsian groups � a � b For γ = let c d γ z = az + b cz + d . We will consider Fuchsian groups, i.e. discrete subgroups of SL 2 ( R ). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 2/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Fuchsian groups � a � b For γ = let c d γ z = az + b cz + d . We will consider Fuchsian groups, i.e. discrete subgroups of SL 2 ( R ). Let Γ be such a group. We say that γ ∈ Γ \{± I } is • elliptic if | Tr γ | < 2 (or if γ fixes a point in the upper half plane H ). • parabolic if | Tr γ | = 2 (or if γ fixes one point in R ∪ {∞} ). • hyperbolic if | Tr γ | > 2 (or if γ fixes two points in R ∪ {∞} ). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 2/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Hyperbolic matrices and geodesics Let γ be a hyperbolic (i.e. | Tr γ | > 2).Then there exists | λ | > 1 and A ∈ SL 2 ( R ) s.t. � λ � 0 A − 1 , γ = A λ − 1 0 and we define N ( γ ) := λ 2 and l ( γ ) = log N ( γ ). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 3/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Hyperbolic matrices and geodesics Let γ be a hyperbolic (i.e. | Tr γ | > 2).Then there exists | λ | > 1 and A ∈ SL 2 ( R ) s.t. � λ � 0 A − 1 , γ = A λ − 1 0 and we define N ( γ ) := λ 2 and l ( γ ) = log N ( γ ). Let Γ be a discrete subgroup of SL 2 ( R ).Then Γ \ H is a Riemann surface, and there is a bijection between conjugacy classes { γ } = { τγτ − 1 | τ ∈ Γ } of hyperbolic elements in Γ and closed geodesics on Γ \ H . Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 3/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Hyperbolic matrices and geodesics Let γ be a hyperbolic (i.e. | Tr γ | > 2).Then there exists | λ | > 1 and A ∈ SL 2 ( R ) s.t. � λ � 0 A − 1 , γ = A λ − 1 0 and we define N ( γ ) := λ 2 and l ( γ ) = log N ( γ ). Let Γ be a discrete subgroup of SL 2 ( R ).Then Γ \ H is a Riemann surface, and there is a bijection between conjugacy classes { γ } = { τγτ − 1 | τ ∈ Γ } of hyperbolic elements in Γ and closed geodesics on Γ \ H . The geodesic associated with γ has length l ( γ ). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 3/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Automorphic forms Let Γ be a discrete subgroup of SL 2 ( R ), and f : H = { z ∈ C | ℑ z > 0 } → C be holomorphic with � az + b � ( cz + d ) k f ( z ) , f ( γ z ) = f = cz + d � a � b for γ = ∈ Γ and some k ∈ R . We say that f is c d an (classical) automorphic form wrt. Γ of weight k . Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 4/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Automorphic forms Let Γ be a discrete subgroup of SL 2 ( R ), and f : H = { z ∈ C | ℑ z > 0 } → C be holomorphic with � az + b � = ν ( γ )( cz + d ) k f ( z ) , f ( γ z ) = f cz + d � a � b for γ = ∈ Γ and some k ∈ R . We say that f is c d an (classical) automorphic form wrt. Γ of weight k with multiplier system ν . Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 4/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Automorphic forms Let Γ be a discrete subgroup of SL 2 ( R ), and f : H = { z ∈ C | ℑ z > 0 } → C be holomorphic with � az + b � = ν ( γ )( cz + d ) k f ( z ) , f ( γ z ) = f cz + d � a � b for γ = ∈ Γ and some k ∈ R . We say that f is c d an (classical) automorphic form wrt. Γ of weight k with multiplier system ν . If ν is a multiplier system on Γ the following should hold 1) | ν ( γ ) | = 1, for all γ ∈ Γ, 2) ν ( − I ) = exp ( − π ik ) if − I ∈ Γ, 3) ν ( γ 1 γ 2 ) = σ k ( γ 1 , γ 2 ) ν ( γ 1 ) ν ( γ 2 ). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 4/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Zero free automorphic forms Let f : H → C be a zero free automorphic form, so � az + b � = ν ( γ )( cz + d ) k f ( z ) , f cz + d � a � b for γ = ∈ Γ. c d Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 5/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Zero free automorphic forms Let f : H → C be a zero free automorphic form, so � az + b � = ν ( γ )( cz + d ) k f ( z ) , f cz + d � a � b for γ = ∈ Γ. We can take a holomorphic c d logarithm � az + b � log f = 2 π ik Φ( γ ) + k log( cz + d ) + log f ( z ) , (1) cz + d where exp(2 π ik Φ( γ )) = ν ( γ ). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 5/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Zero free automorphic forms Let f : H → C be a zero free automorphic form, so � az + b � = ν ( γ )( cz + d ) k f ( z ) , f cz + d � a � b for γ = ∈ Γ. We can take a holomorphic c d logarithm � az + b � log f = 2 π ik Φ( γ ) + k log( cz + d ) + log f ( z ) , (1) cz + d where exp(2 π ik Φ( γ )) = ν ( γ ). Multiplying with m ∈ R in (1) and taking the exponential gives us a m ’th power of f , which is an automorphic form of weight km , and multiplier system exp(2 π ikm Φ( γ )). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 5/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The Dedekind η -function Let η : H → C be defined by � ∞ � π iz � η ( z ) = exp (1 − exp(2 π inz )) , 12 n =1 then η is a zero free weight 1 / 2 automorphic form on SL 2 ( Z ). Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 6/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s The Dedekind η -function Let η : H → C be defined by � ∞ � π iz � η ( z ) = exp (1 − exp(2 π inz )) , 12 n =1 then η is a zero free weight 1 / 2 automorphic form on SL 2 ( Z ).Taking logarithms as on the previous slide we get � az + b � (log η ) = π i Φ( γ ) + log( cz + d ) / 2 + (log η )( z ) , cz + d where 12Φ is the so called Rademacher function. Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 6/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Ghys’ theorem Γ \ H is homeomorphic to { ( x , y ) ∈ C 2 | | x | 2 + | y | 2 = 1 }\ τ , where τ is a trefoil knot. Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 7/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Ghys’ theorem Γ \ H is homeomorphic to { ( x , y ) ∈ C 2 | | x | 2 + | y | 2 = 1 }\ τ , where τ is a trefoil knot. Theorem (´ E Ghys) For γ ∈ SL 2 ( Z ) hyperbolic, there is a (oriented) curve γ ′ in S 3 \ τ associated with γ , and 12Φ( γ ) is the linking number between γ ′ and τ . � � az + b (log η ) = π i Φ( γ ) + log( cz + d ) / 2 + (log η )( z ) , cz + d Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 7/15
u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Ghys’ theorem Γ \ H is homeomorphic to { ( x , y ) ∈ C 2 | | x | 2 + | y | 2 = 1 }\ τ , where τ is a trefoil knot. Theorem (´ E Ghys) For γ ∈ SL 2 ( Z ) hyperbolic, there is a (oriented) curve γ ′ in S 3 \ τ associated with γ , and 12Φ( γ ) is the linking number between γ ′ and τ . � � az + b (log η ) = π i Φ( γ ) + log( cz + d ) / 2 + (log η )( z ) , cz + d The number of times the curves wind around each other. Flemming von Essen — A Distribution Result Related to Automorphic Forms — April 2013 Slide 7/15
Recommend
More recommend
