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

Lecture 7 Modular forms for subgroups Γ Ă SL2pZq & dimension formulas

April 28, 2020

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Modular forms

Γ Ă SL2pZq 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) pf |khqpzq “ f pzq for every h P Γ (iii) f is bounded (resp. vanishing) at cusps, that is for every g P SL2pZq |pf |kgqpzq| “ Op1q (resp. op1q) as Impzq Ñ `8 Recall: right action of g “ ˆa b c d ˙ in weight k pf |kgqpzq “ 1 pcz ` dqk f ˆaz ` b cz ` d ˙

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Why finite index?

MkpΓq :“ t mod. forms of weight k for Γu Y SkpΓq :“ t cusp forms of weight k for Γu T “ ˆ1 1 1 ˙ , Tpzq “ z ` 1 rSL2pZq : Γs ă 8 ñ Dh ě 1 such that T h P Γ f pz ` hq “ f pzq, f pzq “ ř

n an exp

` 2πinz

h

˘ “ ř

n anq

n h

piiiq ñ an “ 0, n ă 0 Other cusps: α “ gp8q, g P SL2pZq ù q-expansion for f |kg

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Weight 0

Quotient Riemann surfaces: Y pΓq “ ΓzH XpΓq “ ΓzpH Y P1pQqq “ Y Y tcuspsu compact f P M0pΓq is a Γ-invariant function on H (by (ii)) ñ f P MpY q, holomorphic (by (i)) piiiq ñ f P MpXq, holomorphic ñ f is constant M0pΓq “ C S0pΓq “ t0u

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Weight 2

Let f pzq be a modular form of weight 2 for Γ. Then f pzqdz is Γ-invariant: f pzqdz ˇ ˇ ˇ

z“ aw`b

cw`d

“ f ˆaw ` b cw ` d ˙ ˆaw ` b cw ` d ˙1 dw “ f ˆaw ` b cw ` d ˙ dw pcw ` dq2 “ f pwqdw ñ f pzqdz descends to a holomorphic differential form ωf P ΩpY q

• n Y pΓq “ ΓzH.

Near cusps: q “ expp2πiz

h q is a local coordinate near r8s P XpΓq

dq “ 2πi h expp2πiz h qdz ñ dz “ h 2πi dq q ωf “ h 2πi f pqqdq q “ h 2πi pa0 ` a1q ` a2q2 ` . . .qdq q

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Weight 2

ωf “ f pzqdz P ΩpY q, holomorphic Near cusps: q “ expp2πiz

h q local coordinate near r8s P XpΓq

dq “ 2πi h expp2πiz h qdz ñ dz “ h 2πi dq q ωf “ h 2πi f pqqdq q “ h 2πi pa0 ` a1q ` a2q2 ` . . .qdq q has at most a simple pole at q “ 0 (no pole when a0 “ 0) ñ ωf P ΩpXq, poles only at cusps Note: f P S2pΓq ô ωf has no poles (holomorphic form)

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Weight 2

Summary: there is a natural injective map M2pΓq Ñ ΩpXq f pzq ÞÑ ωf p“ f pzqdzq What is the image of M2pΓq? Quick answer for cusp forms: S2pΓq – t holomorphic forms on Xu Riemann–Roch ñ dimC S2pΓq “ g pgenus ofXq

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dimC M2pΓq

M2pΓq Ñ ΩpXq f pzq ÞÑ ωf p“ f pzqdzq What is the image of M2pΓq? Pick any 0 ‰ ω P ΩpXq, write ωf “ hω, h P MpXq. Then ImagepM2pΓqq “ thω : divphωq ` ÿ

P:cusp

P ě 0u – th : divphq ` D ě 0u, D “ divpωq ` ÿ

P:cusp

P P DivpXq M2pΓq – LpDq, Riemann–Roch ñ dim M2pΓq “ dim LpDq “ degpDq ` 1 ´ g “ 2g ´ 2 ` ε8 ` 1 ´ g “ g ´ 1 ` ε8

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f P MkpΓq ù ??? on XpΓq (k ‰ 0, 2)

pUi, ziqiPI atlas of coordinate charts X “ Ť

iPI Ui

zi : Ui – zipUiq Ă C wij “ zi ˝ z´1

j

: zjpUi X Ujq Ñ zjpUi X Ujq transition maps Specifically:

• n X “ XpΓq transition

maps are given by linear fractional transformations wij “ azj`b

czj`d ,

ˆa b c d ˙ P Γ

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Transition maps on the quotient surfaces

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Differential k-forms, k P Z

pUi, ziqiPI atlas of coordinate charts on X wij “ zi ˝ z´1

j

transition maps A differential k-form ω P ΩbkpXq is a collection of meromorphic functions ω “ tgipziquiPI satisfying gjpzjq “ gipwijpzjqqpw1

ijpzjqqk,

@i, j. On the quotients X “ XpΓq we have w1

ij “

´ azj`b

czj`d

¯1 “

1 pcz`dq2

f P M2kpΓq ù ωf P ΩbkpXq Similarly to the case of weight 2: if we pick any 0 ‰ ω P ΩbkpXq, then (see Lecture 8 for details) ImagepM2kpΓqq “ thω : h P MpXq, divphq ` D ě 0u – LpDq D “ divpωq ` ÿ

P : ell.pt

• f order2

tk 2u P ` ÿ

P : ell.pt

• f order3

t2k 3 u P ` ÿ

P:cusp

k P.

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M2k`1pΓq? M2k`1pΓ, χq? a generalization of k-forms

pUi, ziqiPI atlas of coordinate charts on X wij “ zi ˝ z´1

j

: zjpUi X Ujq Ñ zipUi X Ujq transition maps A line bundle is given by a collection of non-vanishing holomorphic functions φij : zjpUi X Ujq Ñ zipUi X Ujq φijpzjq ‰ 0 zj P zjpUi X Ujq satisfying certain compatibility conditions on triple intersections Ui X Uj X Uk. Its sections are collections of meromorphic functions g “ tgipziquiPI satisfying gjpzjq “ gipwijpzjqqφijpzjq. Case φij “ wk

ij corresponds to k-forms. Modular forms of odd

weight and modular forms with characters define sections of more general line bundles on XpΓq.

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