Some Complex Function Theory
Holomophicity Meromorphicity f 71 him flat f zo f Q Q holomorphic 2 to Z exists countably many poles Epi where Complex logarithm ftp.ul x
f Z log z rett t floor iotas
logarithmic derivative 7 log f fl f 1 0 f z Z Zo lez g t f Z ta z zo n r non vanishing around to ft I n r f 2 Zo t r Argument Principle
xp Ifk 2T res P tEn
Hf IT'LL zeroes of f poles of f Modular forms Definition First Examples Notation Let Te S 27 e Tbd ad be I a bcode27
Let IH ze Im Cz o Cad Definition f k Itm G modularofweight f VI flat feetd 0 i k 1 ft f f to ft C 1 fto flz C Litmus Test for Modularity S T SLID S L T f l
n t E.is 9 15 i nmueeuiii 1 fi I I
Example Eisenstein series
let k 4 k even
tm tCmH a Cm k go
Ey Eg generate all modular forms of positive weight The Valence formula i.e the base case
Thmi Let F and D be as above and let f be a nonzero modular form of weight k Then
2 t fttzVLffttJVeaiilslHtIEy.y kH
D modular Mk Sh forms of weight k 1st that goal prove Mk Shad is finite dimensional
k 4 f My 1 12711 1 B Idina Can only vanish at eat
K G I 2771 1 dimeMolSbl 1 Can only at I
f k lo I 6 The Valence formula i.e the base case
Thmi Let F and D be as above and let f be a nonzero modular form of weight k Then
f t Vz f t Vezails If t klf
First Dimensionality Results O K dim Mk Sh D odd k 0241b 8 10 The Valence formula i.e the base case
Thmi Let F and D be as above and let f be a nonzero modular form of weight k Then
2 fIttzVzlftttJVezailslHtIEy.y kH 11 I
First Dimensionality Results dimacilkeshlah 89 74 o
The Dimension formula
4 4 k 2 lawd121 dime Mk1541271 4412 I k 2 lawd 12 2 Eyes LA 1728
M 54271 Sk Sbl forms f f cusp w The Valence formula i.e the base case
Thmi Let F and D be as above and let f be a nonzero modular form of weight k Then
f t Vz f t Vez.is Ht V f y
First Dimensionality Results dimacilkeshlah Ii D't w
The Dimension formula
4 4 k 2 lawd121 dime Mk1541271 4412 I k 2 lawd 12
Basis of MkCS D b Mk Sh D Cya E 4atbb ki aibe23,03
all Ey Eg modular forms of positive or zoo of weight
G Cx y Proof of Valence Complex Analysis on Choice of Contour 1 E a al ft.rs damn taxi I 1 I 1 i i i i 1 to o b f l pyob o b f l Yz O k l l Yz 0 112 I
1 zeroes c t case e i e'T Case 2 A Zero d on the Vertical Lives f modular from 1 Vplf C pent D 4 Brief Proof of Valence Riemann Surfaces
Definition of Riemann Surface
ie
1 Collection of charts which map biholomophically to an open set in 2 You holomorphic 4 Brief Proof of Valence Riemann Surfaces
Definition of Riemann Surface
homeomorphism I mom
Stereographic Projection
In
F 4 Brief Proof of Valence Riemann Surfaces
Definition of Riemann Surface
homeomorphism I mom
Stereographic Projection
a
OF FUE becomes a Riemann surface called Charts X11
i g
I N l l 1 4 Brief Proof of Valence Riemann Surfaces R m surface Symmetric k forms on X I X fflzl IC.dz dzmfcz Ldzlk Cz f is www.rphieonx I divlw ordxlfl.fr
deg Cd'wlwH od H wtf Volg Riemann Roch Theorem t g
deg Cd'wlwH 2g 2 ok
Fixed forms of SCH X under SLID For f c Mk Sh D Az z is fixed under transformation
by SLzC2 and so is a k form on X
Then there exists a form Wf on XLI such that f z Wfo f Hk where IH X l T SLIDE 062 1 x OH Cf 4 Brief Proof of Valence Riemann Surfaces
Symmetric k forms on X X
NCH X fczlfdz.lk f meromorphic on X
din w ord If X w f G lolz
degldivlw E ordxHI LEX
Riemann Rock Theorem
deg div w kC2g 2
Form on IH Form on X l for f E MkCS D Az z is fixed under transformation
by SLzC2 and so is a k form on X
Then there exists a form wf on Stk x l such that Wfo f f z Hk where IH X l T SLIDE The Valence formula
06 1 x ord u.CH Ida KISEI in Igodetails f 143 a e2isits od Cf k e A degldivlw 2K
2K adsit K JordezailsHI Kk
H Kya E ordx If XE X K 4cxHEx e g
Lord If t