Lecture 7 Modular forms for subgroups SL 2 p Z q & dimension - PowerPoint PPT Presentation

Lecture 7 Modular forms for subgroups Γ Ă SL 2 p Z q & dimension formulas April 28, 2020 1 / 12

Modular forms Γ Ă SL 2 p Z q of finite index A modular ( resp. cusp) form of weight k for Γ is a function f : H Ñ C satisfying: (i) f is holomorphic (ii) p f | k h qp z q “ f p z q for every h P Γ (iii) f is bounded (resp. vanishing) at cusps, that is for every g P SL 2 p Z q |p f | k g qp z q| “ O p 1 q (resp. o p 1 q ) as Im p z q Ñ `8 ˆ a b ˙ Recall: right action of g “ in weight k c d ˆ az ` b ˙ 1 p f | k g qp z q “ p cz ` d q k f cz ` d 2 / 12

Why finite index? M k p Γ q : “ t mod. forms of weight k for Γ u Y S k p Γ q : “ t cusp forms of weight k for Γ u ˆ 1 ˙ 1 T “ , T p z q “ z ` 1 0 1 D h ě 1 such that T h P Γ r SL 2 p Z q : Γ s ă 8 ñ ` 2 π inz n ˘ f p z ` h q “ f p z q , f p z q “ ř n a n exp “ ř n a n q h h p iii q ñ a n “ 0 , n ă 0 Other cusps: α “ g p8q , g P SL 2 p Z q ù q -expansion for f | k g 3 / 12

Weight 0 Quotient Riemann surfaces: Y p Γ q “ Γ z H X p Γ q “ Γ zp H Y P 1 p Q qq “ Y Y t cusps u compact f P M 0 p Γ q is a Γ-invariant function on H (by (ii)) ñ f P M p Y q , holomorphic (by (i)) p iii q ñ f P M p X q , holomorphic ñ f is constant M 0 p Γ q “ C S 0 p Γ q “ t 0 u 4 / 12

Weight 2 Let f p z q be a modular form of weight 2 for Γ. Then f p z q dz is Γ-invariant: ˙ 1 ˆ aw ` b ˙ ˆ aw ` b ˇ f p z q dz “ f dw ˇ cw ` d cw ` d ˇ z “ aw ` b cw ` d ˆ aw ` b ˙ dw “ f p cw ` d q 2 “ f p w q dw cw ` d ñ f p z q dz descends to a holomorphic differential form ω f P Ω p Y q on Y p Γ q “ Γ z H . Near cusps: q “ exp p 2 π iz h q is a local coordinate near r8s P X p Γ q dq “ 2 π i h exp p 2 π iz h dq h q dz ñ dz “ 2 π i q 2 π i f p q q dq h 2 π i p a 0 ` a 1 q ` a 2 q 2 ` . . . q dq h ω f “ q “ q 5 / 12

Weight 2 ω f “ f p z q dz P Ω p Y q , holomorphic Near cusps: q “ exp p 2 π iz h q local coordinate near r8s P X p Γ q dq “ 2 π i h exp p 2 π iz h dq h q dz ñ dz “ 2 π i q 2 π i f p q q dq h 2 π i p a 0 ` a 1 q ` a 2 q 2 ` . . . q dq h ω f “ q “ q has at most a simple pole at q “ 0 (no pole when a 0 “ 0) ñ ω f P Ω p X q , poles only at cusps Note: f P S 2 p Γ q ô ω f has no poles (holomorphic form) 6 / 12

Weight 2 Summary: there is a natural injective map M 2 p Γ q Ñ Ω p X q f p z q ÞÑ ω f p“ f p z q dz q What is the image of M 2 p Γ q ? Quick answer for cusp forms: S 2 p Γ q – t holomorphic forms on X u Riemann–Roch ñ dim C S 2 p Γ q “ g p genus of X q 7 / 12

dim C M 2 p Γ q M 2 p Γ q Ñ Ω p X q f p z q ÞÑ ω f p“ f p z q dz q What is the image of M 2 p Γ q ? Pick any 0 ‰ ω P Ω p X q , write ω f “ h ω , h P M p X q . Then ÿ Image p M 2 p Γ qq “ t h ω : div p h ω q ` P ě 0 u P : cusp ÿ – t h : div p h q ` D ě 0 u , D “ div p ω q ` P P Div p X q P : cusp M 2 p Γ q – L p D q , Riemann–Roch ñ dim M 2 p Γ q “ dim L p D q “ deg p D q ` 1 ´ g “ 2 g ´ 2 ` ε 8 ` 1 ´ g “ g ´ 1 ` ε 8 8 / 12

f P M k p Γ q ù ??? on X p Γ q ( k ‰ 0 , 2) p U i , z i q i P I atlas of coordinate charts X “ Ť i P I U i z i : U i – z i p U i q Ă C w ij “ z i ˝ z ´ 1 : z j p U i X U j q Ñ j z j p U i X U j q transition maps Specifically: on X “ X p Γ q transition maps are given by linear fractional transformations ˆ a ˙ b w ij “ az j ` b cz j ` d , P Γ c d 9 / 12

Transition maps on the quotient surfaces 10 / 12

Differential k -forms, k P Z p U i , z i q i P I atlas of coordinate charts on X w ij “ z i ˝ z ´ 1 transition maps j A differential k-form ω P Ω b k p X q is a collection of meromorphic functions ω “ t g i p z i qu i P I satisfying g j p z j q “ g i p w ij p z j qqp w 1 ij p z j qq k , @ i , j . ´ az j ` b ¯ 1 1 On the quotients X “ X p Γ q we have w 1 ij “ “ cz j ` d p cz ` d q 2 ω f P Ω b k p X q f P M 2 k p Γ q ù Similarly to the case of weight 2: if we pick any 0 ‰ ω P Ω b k p X q , then (see Lecture 8 for details) Image p M 2 k p Γ qq “ t h ω : h P M p X q , div p h q ` D ě 0 u – L p D q t k t 2 k ÿ ÿ ÿ D “ div p ω q ` 2 u P ` 3 u P ` k P . P : cusp P : ell . pt P : ell . pt of order 3 of order 2 11 / 12

M 2 k ` 1 p Γ q ? M 2 k ` 1 p Γ , χ q ? a generalization of k -forms p U i , z i q i P I atlas of coordinate charts on X w ij “ z i ˝ z ´ 1 : z j p U i X U j q Ñ z i p U i X U j q transition maps j A line bundle is given by a collection of non-vanishing holomorphic functions φ ij : z j p U i X U j q Ñ z i p U i X U j q φ ij p z j q ‰ 0 z j P z j p U i X U j q satisfying certain compatibility conditions on triple intersections U i X U j X U k . Its sections are collections of meromorphic functions g “ t g i p z i qu i P I satisfying g j p z j q “ g i p w ij p z j qq φ ij p z j q . Case φ ij “ w k ij corresponds to k -forms. Modular forms of odd weight and modular forms with characters define sections of more general line bundles on X p Γ q . 12 / 12