COMPLEX UNIFORMIZATION OF FERMAT CURVES Pilar Bayer University of - PowerPoint PPT Presentation

COMPLEX UNIFORMIZATION OF FERMAT CURVES Pilar Bayer University of Barcelona BMS Student Conference Berlin Mathematical School, 2018-02-22

based on joint work with Jordi Gu` ardia References 1. Bayer, P.; Gu` ardia, J.: Hyperbolic uniformization of Fermat curves. Ramanujan J. 12 (2006), no. 2, 207–223. 2. Bayer, P.: Uniformization of certain Shimura curves. Differential Galois theory (Bedlewo, 2001), 13–26, Banach Center Publ., 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002. 3. Gu` ardia, J.: A fundamental domain for the Fermat curves and their quotients. Contributions to the algorithmic study of problems of arithmetic moduli. Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. 94 (2000), no. 3, 391–396.

Outline 1 Curves and Riemann surfaces 2 Fermat curves 3 The Fermat sinus and cosinus functions: ( sf , cf ) 4 Fermat tables

Outline 1 Curves and Riemann surfaces 2 Fermat curves 3 The Fermat sinus and cosinus functions: ( sf , cf ) 4 Fermat tables

Compact surfaces Theorem Any connected compact surface is homeomorphic to: 1. The sphere ( abb − 1 a − 1 ). 2. The connected sum of g tori ( aba − 1 b − 1 ), for g � 1. 3. The connected sum of k real projective planes ( abab ), for k � 1. P # P = K = abab − 1

Compact connected orientable surfaces g = 0 1 � g � 3 Image source: Henry Segerman Compact connected orientable surfaces are classified by their genus g .

g = 1. Weierstrass’s elliptic functions Λ = { m ω 1 + n ω 2 : m , n ∈ Z } ⊆ C , τ = ω 2 /ω 1 , ℑ ( τ ) > 0 � � ℘ ( z ; ω 1 , ω 2 ) = 1 1 1 z 2 + � ( z + m ω 1 + n ω 2 ) 2 − n 2 + m 2 � =0 ( m ω 1 + n ω 2 ) 2 E Λ : ℘ ′ 2 ( z ) = 4 ℘ 3 ( z ) − g 2 ℘ ( z ) − g 3 , g 2 ( ω 1 , ω 2 ) , g 3 ( ω 1 , ω 2 ) ∈ C z �→ ( ℘ ( z ) , ℘ ′ ( z )) C / Λ ≃ E Λ ( C ) , Y 2 = 4 X 3 − g 2 X − g 3 , ∆ = g 2 2 − 27 g 2 3 � = 0

Hyperbolic geometry P 1 ( C ) = C ∪ {∞} , complex projective line, Riemann sphere � a � ∈ SL (2 , C ) , α ( z ) = az + b b α = cz + d , M¨ obius transformations c d PSL (2 , C ) = SL (2 , C ) / {± I 2 } , conformal transformations of P 1 ( C ) A model for the hyperbolic plane: H = { z = x + iy ∈ C : y > 0 } µ = dx 2 + dy 2 1 + | z 1 − z 2 | 2 � �� � � � , d ( z 1 , z 2 ) = � arc cosh � � y 2 2 z 1 z 2 � PSL (2 , R ) = SL (2 , R ) / {± I 2 } , hyperbolic motions of H

The Poincar´ e disk model for the hyperbolic plane D r = { z ∈ C : zz < r 2 } , r ∈ R , r > 0 �� a � � br / R ∗ . Aut ( D r ) ≃ : a , b ∈ C , | b | < | a | b / r a x 2 + (1 + y ) 2 , x 2 + y 2 − 1 � 2 x � H ≃ D r , z = x + iy �→ z = r x 2 + (1 + y ) 2 geodesics in H geodesics in D r

Types of conformal isometries of H � a � b α = ∈ SL (2 , R ) , α � = ± I 2 . Fixed points: c d ( a + d ) 2 − 4 � α ( z ) = z ⇔ z = a − d ± 2 c P 1 ( C ) α = { z 1 , z 2 } , z 1 , z 2 ∈ P 1 ( R ) | tr ( α ) | > 2 , hyperbolic: P 1 ( C ) α = { z , z } , elliptic: | tr ( α ) | < 2 , z ∈ H , elliptic P 1 ( C ) α = { z } , z ∈ P 1 ( R ) , cusp parabolic: tr ( α ) = ± 2 , Conjugacy classes in SL (2 , R ): � cos θ � λ � � � ± 1 � sin θ h , λ � = 1; , θ ∈ R \ 2 π Z ; λ − 1 − sin θ cos θ 0 ± 1

Fuchsian groups and Riemann surfaces Γ ⊆ SL (2 , R ) discrete subgroup , Γ ⊆ PSL (2 , R ) H ∗ = H ∪ P Γ , π : H ∗ → Γ \H ∗ P Γ set of cusps , Γ \H ∗ ≃ X (Γ)( C ) compact Riemann surface  ∞ if z is a cusp   ♯ Γ z = e π ( z ) > 1 if z is elliptic  1 otherwise  Γ ′ ⊆ Γ , [Γ : Γ ′ ] = n , ϕ : X (Γ ′ ) → X (Γ) , ′ e w ,ϕ = [Γ ϕ ( w ) : Γ w ] Hurwitz formula: 2 g ′ − 2 = n (2 g − 2) + � w ∈ X (Γ ′ ) ( e w ,ϕ − 1) Laszlo Fuchs obtained his PhD in 1858 under Ernst Kummer in Berlin.

Fundamental domains F ⊆ H connected domain Γ Fuchsian group, (i) H = � γ ∈ Γ γ ( F ), (ii) F = U , U = int ( F ), U open set, (iii) γ ( U ) ∩ U = ∅ , for any γ ∈ Γ, γ � = ± 1. v1 b b a a v3 v2 v4 � � � � 0 1 1 1 SL (2 , Z ) = � S = , T = � v0 − 1 0 0 1 fundamental domain for the modular group

Hyperbolic tesselations by SL (2 , Z )

Γ-Automorphic forms � a � b ∈ GL + (2 , R ) , α = k ∈ Z , j ( α, z ) := cz + d c d f : H → P 1 , ( f | k α )( z ) = det ( α ) k / 2 j ( α, z ) − k f ( α z ) , z ∈ H Definition A meromorphic function f ( z ) on H is called a Γ- automorphic form of weight k if it is meromorphic at all cusps and satisfies f | k γ = f , for all γ ∈ Γ. f ( z )( dz ) m = ω f ◦ π A 2 m (Γ) ≃ Ω m ( X (Γ)) , f �→ ω f , A 0 (Γ) = C ( X (Γ)) field of Γ-automorphic functions

g = 0. Klein’s j invariant Γ = SL (2 , Z ) modular group, C ( X ( SL (2 , Z ))) = C ( j ) ( m , n ) � =(0 , 0) ( m + nz ) − 4 g 2 = 60 � modular form of weight 4 ( m , n ) � =(0 , 0) ( m + nz ) − 6 g 3 = 140 � modular form of weight 6 j ( z ) = 1728 g 3 ∆ = g 3 2 − 27 g 2 2 3 ∈ S 12 (Γ), ∆ ∈ A 0 (Γ), z ∈ H j ( q ) = 1 q + 744 + 196 884 q + O ( q 2 ) q = e 2 π iz local parameter, 2 π i 3 ) = 0 , j ( e j ( i ) = 1728

Schwarzian derivatives Theorem (a) The derivative f ′ of an automorphic function f is an automorphic form of weight 2. (b) If f is an automorphic form of weight k , then kff ′′ − ( k + 1)( f ′ ) 2 is an automorphic form of weight 2 k + 4. Definition The Schwarzian derivative with respect to z , { w , z } , of a non-constant smooth function w ( z ) is defined by { w , z } = 2 w ′ w ′′′ − 3( w ′′ ) 2 w ′ = dw , dz . 4( w ′ ) 2 Hermann Schwarz obtained his PhD in 1864 under Kummer and Weierstrass in Berlin.

Automorphic derivatives Definition The Γ-automorphic derivative { w , z } Γ of a non-constant smooth function w ( z ) with respect to z is defined by { w , z } Γ := { w , z } w ′ = dw , dz . w ′ 2 Proposition If w ( z ) is a Γ-automorphic function on H , so is { w , z } Γ = 2 w ′ w ′′′ − 3( w ′′ ) 2 = −{ z , w } . 4( w ′ ) 4 That is: { w , γ ( z ) } Γ = { w , z } Γ , for all γ ∈ Γ.

Connection with second order linear differential equations Theorem (Poincar´ e) Let Γ be a Fuchsian group of the first kind, w ( z ) ∈ A 0 (Γ) a non-constant automorphic function and ζ ( w ) be its inverse function. Then ζ ( w ) = η 1 ( w ) η 2 ( w ) , where { η 1 , η 2 } is a fundamental system of solutions of the ordinary differential equation d 2 η dw 2 = { w , z } Γ η. Moreover, { w , z } Γ is an algebraic function of w .

The genus zero case: Hauptmoduln If X (Γ) is of genus g = 0, then C ( X (Γ)) = C ( w ) where w ( z ) is a Hauptmodul for X (Γ). Thus there is a rational function R ( w ) ∈ C ( w ) such that w ( z ) is a solution of the third order differential equation { w , z } Γ = R ( w ) . We can obtain ζ ( w ) by integrating the linear differential equation d 2 η dw 2 = R ( w ) η. Remark A key point is always the computation of R ( w ).

How to obtain the Hauptmodul j ? v1 b b a a v3 v2 v4 SL (2 , Z ) \H ∗ j ( z ) v0 → P 1 ( C ) − = 1 q + 744 + 196 884 q + 21 493 7602 2 + O ( q 3 ) j ( z ) = 1728 g 3 2 q = e 2 π iz , z ∈ H ∆ ,

Dedekind’s valence function (1877) d 2 � dv − 4 dz = 4 { z , v } SL (2 , Z ) [ v , z ] = � dv dv 2 dz 2 π i 3 ) = 0 , v ( i ) = 1 , v ( e v ( ∞ ) = ∞ 1 dv 1 dv 1 dv dz , dz , (1 − v ) 1 / 2 v 2 / 3 v dz Fuchs’ theory 3 8 36(1 − v ) + 23 b 23 R ( v ) = 4(1 − v ) 2 + 9 v 2 + 36 v 36 v 2 − 41 v + 32 = 36 v 2 (1 − v ) 2 d 2 η dv 2 = − 1 z ( v ) = η 1 ( v ) [ v , z ] = R ( v ) , 4 R ( v ) η, η 2 ( v )

Dedekind’s valence function versus Klein’s j invariant The function � 1 � dv 2 z ( v ) := const. v − 1 3 (1 − v ) − 1 4 dz satisfies the hypergeometric differential equation � dz v (1 − v ) d 2 z � 2 3 − 7 v z dv 2 + dv − 144 = 0 6 whose solutions are c 1 η 1 ( v ) + c 2 η 2 ( v ), where η 1 ( v ) = F (1 / 12 , 1 / 12 , 2 / 3; v ) , η 2 ( v ) = F (1 / 12 , 1 / 12 , 1 / 2; 1 − v ) 1728 v ( z ) = j ( z )

Hypergeometric differential equation z (1 − z ) d 2 w dz 2 + [ c − ( a + b + 1) z ] dw dz − ab w = 0 It has regular singular points at 0, 1, and infinity. Its solutions are obtained in terms of the hypergeometric series ∞ z n ( a ) n ( b ) n � F ( a , b , c ; z ) = n ! , | z | < 1 , (Wallis,1655) ( c ) n n =0 where � 1 , n = 0 ( q ) n = q ( q + 1) · · · ( q + n − 1) , n > 0 denotes de Pochhammer symbol. Leo Pochhammer obtained his PhD in 1863 under Kummer in Berlin.

Outline 1 Curves and Riemann surfaces 2 Fermat curves 3 The Fermat sinus and cosinus functions: ( sf , cf ) 4 Fermat tables

The Fermat curves F N N � 4 a positive integer X N + Y N = Z N F N : deg ( F N ) = N , g ( N ) = ( N − 1)( N − 2) / 2 D r = { z ∈ C : zz < r 2 } ∆ a Fuchsian triangle group of signature ( N , N , N ) acting on D r : ∆ = � α, β, γ : α N = β N = γ N = Id , αβγ = Id � Theorem A hyperbolic model for the Fermat curve F N is given through an isomorphism Γ \D ∗ r ≃ F N ( C ) , where Γ = [∆ , ∆] denotes the commutator subgroup of ∆.

� � � � � � � � First idea of the proof C N = F N ( x , y ) C A x y C B x N � y N � C C g ( C ) = g ( C A ) = g ( C B ) = 0