Akash 22 o c - Lecture 14- -

-

May I 5, 2020

101 Last time X Riemann surface,X = U U

,- - -

a

Tad ( Ua, I, Ga ) charts , V - E G , OIA : Ua → Va

162 transition functions .

top Ea' '

: Eta (uan up) - Ep Cleanup)are bihol@morphia

Examples Q, affine curves

,

I,tori Xh = %

.

- - -

I Basic Results-- -

IAI Identity theorem f. g : x→ y. f=g on a set S E x which

has a limit point => f- Ig.

IBI Epe Toping f : x - t not constant ⇒ f open .

ICI MII f : x - G , F p with lfcp> S Z' fcxsl t x EX

→ f constant.

Corollary I- - -

All functions on compact Riemann surfaces are constant .

Coz" YI '

f : X - Y,

X,Y connected

,

'

X compact , f- not constant

=> f surjective [ why ? f- (x) open aclosed

.since X

,fcx) are

both compact .] .

PIof-of.LI.

Sr = { a EX : f- =g in a neighborhood of a } .

We show . 5h open ( clear)f

X connected

• r closed => e =x ⇒ f- Eg. r FOI

f w'

→

^ -

# µ~

,

✓ f =4f&"

.

V

-

r. s€as closed- -

Take year and showy E S2 .

We can find Xn Er, xn → y .

f- (xn) =gC×n) => fcy) =gcy) z -

Take charts (U, ,

v) and (u', y ,

v'

)

near yand 2- .

Whoa, flu) EU ' , glue) EU! U connected

Take I = y f Io' ', g- = 4g E' ! over It cus -_ v Ea

.

Let A = TE (rn u) .

Since f- =g on rn u ⇒ I =g~ .

identityon A.EVEG -_> I =5 an v

. ⇒ f=g on way : ⇒ y c- r

Tnm in VEG

or # ol.

-The set s has an accumulation point * ex .

Then A ER .

identityIndeed , pick a chart U near A , f- = g in

Un S. => f- = g in Uturn in U

⇒ a- c-'

r.

II. We have seen there are no holomorphic functions on compact Riemann surfaces .

Question Study the meromorphic functions instead ?- -

txample Over X = I,all meromorphic functions are rational (220A, B) .

f Cz) = PCI =IT ( 2- - ai)C-

QCZ)Ty ( z -bi)

Question : Are there analogues for Xn ?-

Meromorphic functions on Xa h = Z Wn t Zaz , who,€112

.- - - --

D

Re-made f - meromorphic on X,

= %.

Let it : a → xn .

Let- F -- fo 't

⇐ F meromorphic on Q & doubly periodic .

F- ( 2-) = f- (2- + con) = FEZ toy)

Such functions are called elliptic .

Remark WLOG A = Z t IT,Im E 70

.- - -

Indeed if d'= Z t IT

,I = WI

cuzi

%, - % isomorphism .

Z - Z ah .

Xn = %+ ze

,Im 270 from now on .

Question A Describe all meromorphic functions on Xa.

-

§etInB Describe Space of all meromorphic functions on Xa

with given Zeros 1 poles & multiplicities .

Play general facts about zeros / poles

approaches to Question A & answers

l Question B & towards Riemann- Roch.

in General facts-

- - - -

Renard ( Zeros / Poles)

All zeros d poles will be restricted to the parallelogram .

I = / t, w , + tacy , o Etz El , o Etz Er}

.

Write Ea = a + I = translate of E.

theorem f elliptic Function . then

II # Zeros = # poles counted w/multiplicity in E .

④ IF ai ,b.. are the zeros / poles with multiplicity in E .

then

E ai - Ebi C- A-i r

'

(z wi t cuz

Ioof II.

Ar ament principlew-

s

# Ze-s - # poles -- ÷.. ftp.iz . = .

↳#ta .

JE

1. =L.

- 1. ' 1. +1,

°"

By periodicity # Cz) = # Cztwn) = # (2+02) .Thus

{,

¥ dz = - f,,¥ dz & {

,

¥ dz = - fLa

,

¥ dz ⇒ QED.

If there are Zeros / poles on 2E , translate & work with E,instead .

④ The extended argument principle ( Conway I?. 6)

7

2¥. §,

2- dz = § ai - ? bi E A- .

We claim

÷. ({! # de t !! # da) c- Iwai's

÷.( f,? tf da t f ↳

Z ' #dz) c- Zoo . -Cas .

I " = 2¥ , ( f ,.FI#d2--fg,fz-iwz1f/Izdz) by periodicity= (2¥, I ,

,

t.dz ) l - wa) . Let z be a parametrization of La

= n ( foV , 9) . C - wa).E Z wz

.

as needed.TT '

twinding number of loop fo -8 .

If Question A Describe all meromorphic functions on Xa.

- -

Mae Jacobi

171 Weierstrap-

Tae Jacobi theta function Im z > o,Xa = %

+ ez.

.Let

- - --

G- ( z) = [ e

Fine -125in z

converges but not periodic .

NEZ

G- ( Z ti) = D- (Z)

0-(2+2) = f Cz) . exp ( - 2 Tiz - Tiz).

Fact All meromorphic functions on Xa are of the form

IT D- ( Z - ai) f Zeros lpolesat

c - E - -- for some ai >bi Itt ai , II. +b . '

II a ( Z - bi)~, well - defined

Remark this resembles the case of se' with A ← 2-.

- -

Known for

Jacobi's elliptic functionsJacobianJacobi symbolJacobi identityPopularizing the character ∂

Carl Gustav Jacob Jacobi (1804-1851)

f ( z) = [ e'T'RE +2 'T" "Z

theta function.

n c- 21

⇒ WeierstrassII Jb - function ( even) .

Ty G - function (log"= - ys

TL 3 - function 3"

= - yd .

tarots II Meromorphic functions on Xa .

t.CN Cz) - plait)-even :←--

Ty Cas - jscbi))- odd functions : we need Js

'

.

Do Allmeromorphic functions on X n

are :

ITi f ( z - ai)

r =

IT s Cz - bi)i

Wghap- Js #notion

plz ) - Ia t [ (¥. - ¥)X E N

x # o -

Questions ( next times- - - -

ITI Does this convergetoa meromorphic function ?

-'

TeyIs js elliptic ?L

ud Differential equation

N'2=4 Js-- a JS - b . for some a

,b

.

Thus ( Js , js'

) lies on the cubic curve

y2= 4×3 - ax -b .

I Show all meromorphic functions on Xa are expressed

in terms of T.

Me.

Let Vd = { functions on Xa with poles of order dato} i

Show : dim Xd =D (Question B1 Riemann - Rock)

weie-shapeiticfun-h.no

plz) = # t E ( a

- ¥)X EN

X F o

Jb is N - periodic & even

ft'

CZK = 4 js ( 213 - Go Ga, . Jb (z) - 140 Gs .

p'"

= izp y'

Kd V equationI

{igmafynation

p = - Gogo)"

T Cz ) = 2- IT Ez (÷)X H o .