Lecture 6 Analysis on compact Riemann surfaces: meromorphic - PowerPoint PPT Presentation

Lecture 6 Analysis on compact Riemann surfaces: meromorphic functions, differential forms and the Riemann–Roch Theorem April 21, 2020 1 / 9

Meromorphic functions X Riemann surface A meromorphic function is a homomorphic map f : X Ñ P 1 p C q . Equivalent description: p U i , z i q i P I atlas of coordinate charts X “ Ť z i : U i – z i p U i q Ă C i P I U i , w ij “ z i ˝ z ´ 1 : z j p U i X U j q Ñ z j p U i X U j q transition maps j f Øt f i p z i qu i P I f i p z i q : “ f ˝ z ´ 1 : z i p U i q Ñ P 1 p C q i f i p w ij p z j qq “ f j p z j q M p X q “ t f : X Ñ P 1 p C qu field of meromorphic functions 2 / 9

Meromorphic functions & compactness X is compact ñ a non-constant f : X Ñ P 1 p C q has a well-defined degree d : f assumes every value exactly d times counting multiplicities ` 1 ˘ d “ ř P P f ´ 1 p c q ν P p f ´ c q , @ c P C “ ř P P f ´ 1 p8q ν P f Div p X q “ t ř j n j P j u free abelian group deg p ř j n j P j q “ ř j n j , Div p X q Ñ Z div p f q : “ ř P P X ν P p f q P P Div p X q divisor of f div p f ¨ g q “ div p f q ` div p g q div : M p X q ˆ Ñ Div p X q homomorphism (we put div p const q “ 0) Observe: principal divisors have degree 0. deg p div p f qq “ # of zeroes ´ # of poles (counting multiplicities!) “ d ´ d “ 0 3 / 9

Differential forms p U i , z i q i P I atlas of coordinate charts on X X “ Ť z i : U i – z i p U i q Ă C i P I U i , w ij “ z i ˝ z ´ 1 : z j p U i X U j q Ñ z j p U i X U j q transition maps j A differential form ω on X 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 qq w 1 ij p z j q , @ i , j . This transition or glueing rule comes from differential calculus: ˇ z i “ w ij p z j q “ g i p w ij p z j qq w 1 g i p z i q dz i ij p z j q dz j “ g j p z j q dz j . ˇ ˇ Ω p X q “ t diff . forms u is an M p X q -vector space of dimension 1. 4 / 9

Divisors of differential forms ω “ t g i p z i qu i P I g j p z j q “ g i p w ij p z j qq w 1 ij p z j q @ P P X ν P p ω q : “ ν z i p P q p g i q @ i : P P U i w 1 ij p z j q ‰ 0 ñ ν P p ω q is well-defined div p ω q : “ ř P P X ν P p ω q P P Div p X q Ω p X q “ t diff . forms u is an M p X q -vector space of dimension 1 ñ divisors of differential forms belong to one linear equivalence class Div p X q{ div p M p X q ˆ q , called the canonical class . When X is compact: § deg p div p ω qq “ 2 g ´ 2 § dim C t ω : div p ω q ě 0 u “ g (holomorphic diff. forms) These facts are consequences of the Riemann–Roch Theorem. 5 / 9

Riemann–Roch Theorem ÿ D “ n i P i P Div p X q L p D q “ t f P M p X q ˆ : div p f q ` D ě 0 u Y t 0 u ν P i p f q ě ´ n i @ i , ν P p f q ě 0 , P ‰ P i Exercises: § L p D q – L p D ` div p g qq for any g P M p X q ˆ § X compact ñ L p D q “ t 0 u when deg p D q ă 0 § X compact ñ ℓ p D q : “ dim C L p D q ă 8 Theorem. Assume that X is compact. Let K : “ div p ω q for some 0 ‰ ω P Ω p X q . Then for all D P Div p X q ℓ p D q “ deg p D q ` 1 ´ g ` ℓ p K ´ D q . 6 / 9

Residues of differential forms & compactness ω “ t g i p z i qu i P I g j p z j q dz j “ g i p w ij p z j qq w 1 ij p z j q dz j “ g i p z i q dz i ñ residues are well-defined P g j p z j q dz j @ j : P P U j (here i “ ?´ 1) 1 ű @ P P X Res P p ω q : “ 2 π i Lemma. If X is compact, then for any ω P Ω p X q one has ř P Res P p ω q “ 0. As an application, one can get another proof of existence of degree (valence) for meromorphic functions. It is sufficient to show that for a non-constant f P M p X q one has deg p div p f qq “ 0. ` df ˘ Note that Res P “ ν p p f q , thus f ˆ df ˙ ÿ ÿ deg p div p f qq “ ν P p f q “ Res P “ 0 . f P P 7 / 9

Example from 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 8 / 9

Example from modular forms 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 ω f “ q P Ω p X p Γ qq ! 9 / 9