Mod Forms Number

7/17/2019 Mod Forms Number

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Modular Forms in NumberTheory

Karl Mahlburg (HMC ’01)C.L.E. Moore Instructor (MIT)

December ! "00#



$u%line

& 'ums o suares

& Di*isor 'ums& Ferma%’s +as% Theorem& ,erec% -o.er Fibonacci numbers

& ,ar%i%ion congruences& Modular Forms





+agrange’s Theorem

& /e obser*ed %ha% se*eral in%egers can be

.ri%%en as %he sum o 4 suares9

Theorem (+agrange 10): Every -osi%i*e

in%eger is %he sum o a% mos% 4 suares9

,roo idea: Norm ormulas or ua%ernions;



<acobi’s =numera%ion

& 'o! any in%eger can be .ri%%en as %he sum o 4suares> bu% in ho. many dieren% .ays

& +e%’s ?ee- %rac? o bo%h orders and signs

Deini%ion: For a -osi%i*e in%eger n,

( ){ }nnnnnnnnnnr =+++∈= 2

4

2

3

2

2

2

1

4

43214 |,,,#:)( Z



=@am-le:

1 (A1)" 3 0" 3 0" 3 0"

0" 3 (A1)" 3 0" 3 0"

0" 3 0" 3 (A1)" 3 0"

0"

3 0"

3 0"

3 (A1)"

Thus! r 4(1) 89

'imilarly! r 4(") "4! r 4(5) 5"! r 4(4) "49



Theorem (<acobi 18"7):

∑∑ −=

nd nd

d d nr |4|

4 .328)(

,roo s?e%ch: Deine %he genera%ing unc%ion

∑∈

+++==Zn

n qqqq .221:)( 42

θ

Then∑≥

=0

4

4 .)()(n

n qqnr θ



& <acobi’s -roo uses Belli-%ic unc%ions %oind %he +amber% series e@-ansions

& =lli-%ic unc%ions symme%ries in Fourier%ransorms θ (more on %his la%er)9

& No%e: This .as %he -recursor %o modernmodular orms;;;



E B"2line -roo

• θ (q)4 is a modular form o B.eigh% " and Ble*el49

& The seriesn

n nd nd

qd d ∑ ∑∑≥

−

0 |4|

328

is also a modular orm o %his %y-e9



& BNice modular orms lie in ini%e2dimensional *ec%or s-aces (ac%ually!

graded rings)

& /eigh% " and le*el 4 orms are only a "2dimensional *ec%or s-ace

" ma%ching coeicien%s gi*es euali%y;;;



Di*isor 'ums

Deini%ion: ( ) .:|∑=

nd

k

k d nσ

=@am-le: σ 5(4) 5! σ (") 1"7

& ecall %ha% σ k (n) is mul%i-lica%i*e9 2 Gu% %ha%’s no% all>



Fac%:( ) ( ) ( ) ( )∑

−

=

−+=1

1

3337 120:n

j

jn jnn σ σ σ σ

/hy

( )

( ) ...6192048014801

...216024012401

2

1

7

2

1

3

+++=+

+++=+

∑

∑

≥

≥

qqqn

qqqn

n

n

n

n

σ

σ

are modular orms o .eigh% 4 and 89

Dimension one "nd is 1s% suared;



=lli-%ic Cur*es

& +e% E deno%esolu%ions %o %he

eua%ion:

y" + y = x5 - x"



,oin%s o*er ini%e ields

& /e’re ac%ually in%eres%ed in Blocalbeha*ior:– F p ini%e ield .i%h p elemen%s

– E (F p) solu%ions %o y" + y = x5 - x" in F p

=@am-le: E (F5) { (0!0)! (0!")! (1!0)! (1!") }



En unrela%ed() series

& Deine a q-series

( ) ( ) .11:)(2112

1

n

n

n qqqq f −−= ∏≥

& This is an eta-product, .hich are modular orms9

Three main %y-es:19 The%a unc%ions (uadra%ic orms)"9 =isens%ein series (di*isor sums)59 =%a2-roduc%s (inini%e -roduc%s)



En amaing coincidence

f (q) = q + 2q2 – q3 + 2q4 + q5 + 2q6 - 2q7 - 2q9 - 2q10 +

q11 - 2q12 + 4q13 + 4q14 – q15 - 4q16 - 2q17 + …

p E (F p) b(p=1s%2"nd

5 4 21

6 4 1

7 2"11 10 1

15 7 4

1 17 2"



En amaing coincidence

f (q) = q + 2q2 – q3 + 2q4 + q5 + 2q6 - 2q7 - 2q9 - 2q10 +

q11 - 2q12 + 4q13 + 4q14 – q15 - 4q16 - 2q17 + …

p E (F p) b(p1s%2"nd

5 4 21

6 4 1

7 2"11 10 1

15 7 4

1 17 2"



Modulari%y o =lli-%ic Cur*es

& The -a%%ern con%inues 22 i f (q) = Σ !(n) qn!%hen

!( p) b( p) or (almos% all) -rimes9

ela%ion %o %he coeicien%s o modular ormJ E is modular.



Theorem (Taniyama'himura2/iles 1777):

=*ery elli-%ic cur*e is modular9

& In ac%! %he modular orms al.ays ha*eB.eigh% "9

& The %echnical s%a%emen% in*ol*es Bmodular "2unc%ions9



Ferma%’s +as% Theorem

Theorem (/iles2Taylor 1774): I n # 5! %hen%here are no in%eger solu%ions %o

xn

+ yn

= $ n

9

,roo Idea: E solu%ion (!, b, %)

E non2modular elli-%ic cur*e Con%radic%ing Taniyama2'himura;



& This is no. ?no.n as %he modularity

approach9

&The Frey curves are E& y" = x( x - !n)( x - bn)

& Im-or%an%: No re-ea%ed roo%s



!n + bn = %n

BDiscriminan% o E is im-ossible

& Im-ossibili%y comes rom com-aringBLalois re-resen%a%ions o E and BmodularLalois re-resen%a%ions9

E--roach has o%her a--lica%ions>



,erec% ,o.er Fibonacci ’s

Deini%ion: The Fibonacci numbers are gi*enby

' 0 0! ' 1 1!

' n3" ' n31 3 ' n 9

The Lucas numbers s%ar% .i%h

"0 "! "1 19



The seuences begin:

{ ' n } 0! 1! 1! "! 5! 6! 8! 15! "1! 54! 66! 87!

144! "55! 5! #10! 78! 167! >

{ "n } "! 1! 5! 4! ! 11! 18! "7! 4! #! 1"5!

177! 5""! 6"1! 845! 15#4! >



The seuences begin:

{ ' n} 0! 1! 1! "! 5! 6! 8! 15! "1! 54! 66! 87!

144! "55! 5! #10! 78! 167! >

{ "n} "! 1! 5! 4! ! 11! 18! "7! 4! #! 1"5!

177! 5""! 6"1! 845! 15#4! >

& There are a e. suares



The seuences begin:

{ ' n} 0! 1! 1! "! 5! 6! 8! 15! "1! 54! 66! 87!

144! "55! 5! #10! 78! 167! >

{ "n} "! 1! 5! 4! ! 11! 18! "7! 4! #! 1"5!

177! 5""! 6"1! 845! 15#4! >


& There are cubes



The seuences begin:

{ ' n} 0! 1! 1! "! 5! 6! 8! 15! "1! 54! 66! 87!

144! "55! 5! #10! 78! 167! >

{ "n} "! 1! 5! 4! ! 11! 18! "7! 4! #! 1"5!

177! 5""! 6"1! 845! 15#4! >


& There are cubes& E--ears %o be no more>



The seuences begin:

{ ' n} 0! 1! 1! "! 5! 6! 8! 15! "1! 54! 66! 87!

144! "55! 5! #10! 78! 167! >

{ "n} "! 1! 5! 4! ! 11! 18! "7! 4! #! 1"5!

177! 5""! 6"1! 845! 15#4! >

Theorem (Gugeaud! Migno%%e! 'i?se? 0#):

There are no o%her -erec% -o.ers9



& ,roo uses modulari%y a--roach>

combined .i%h many o%her %echniues:

19 Combina%orics o ' n and "n

"9 Elgebraic Number Theory 2ac%oria%ion in Z (136) O " P

59 Dio-han%ine Bheigh% bounds

49 Com-u%a%ional bounds



,ar%i%ions

Deini%ion: E partition o n is a non2decreasing seuence o -osi%i*e in%egers 1 # 2 # … # k # 1 %ha% sum %o n,

n = 1 + 2 + … + k )

The -ar%i%ion unc%ion p(n) coun%s %henumber o -ar%i%ions o n)



=@am-le: The -ar%i%ions o 6 are

6! 431! 53"! 53131! "3"31!

"313131! 131313131!

so p(6) = 9

emar?: In a -ar%i%ion! %he order o -ar%sdoesn’% ma%%er in con%ras% %o r 4(n) romearlier9



amanuQan Congruences

Theorem (amanuQan 1717): For n # 0!

p(6n+4) ≡ 0 (mod 6)

p(n+6) ≡ 0 (mod ) p(11n+#) ≡ 0 (mod 11)

emar?: These are s%ri?ing mul%i-lica%i*e-ro-er%ies or a -urely addi%i*e unc%ion;;



,ar%i%ion genera%ing unc%ion

& 'im-le combina%orics

∏∑≥≥

+++++=−

=1

432

0

.5321)1(

1)(

nn

n

n

qqqqq

qn p

emar?: This is an inini%e -roduc%>hin%s o a modular orm



Dyson’s Cran?

ConQec%ure (Dyson 1744): There is a crank s%a%is%ic %ha% e@-lains %he congruences9

Theorem (Endre.s2Lar*an 178): The

crank e@is%s;



Deini%ion: 'u--ose a -ar%i%ion has r ones91) r 0 %r!nk larges% -ar%!

2) r R 0 %r!nk * r,

.here * -ar%s R r 9

=@am-le:

%r!nk (53"3131) 1 " 1

%r!nk (4353") 4



Deini%ion: ( ! ! n) -ar%i%ions o n .i%h%r!nk S (mod )

Endre.s2Lar*an2Dyson:

( ! 6! 6n34) p(6n34) O 6

( ! ! n36) p(n36) O

( ! 11! 11n3#) p(11n3#) O 11



Modular orms and cran?

& In ac%!n

n

q-

n pn- , + ∑

≥

−

0

)(),,(

is al.ays a modular orm;

emar?: For %he amanuQan congruences! %hismodular orm is iden%ically 09



$no’s congruences

Theorem ($no "000): For any -rime R 5!%here are .! / so %ha%

p( .n 3 /) ≡ 0 (mod )

,roo idea: p(n) are %he coeicien%s o a

modular orm! so ari%hme%ic comes rom:2 Lalois re-resen%a%ions ('erre)! combina%orics(Hec?e)! -rime dis%ribu%ions (Tchebo%are*)



Theorem (M9 "006): For any -rime R 5!%here are .! / so %ha%

( ! ! .n 3 /) ≡ 0 (mod )

Corollary: $no’s congruences;

emar?: The amanuQan congruences are*ery s-ecial in general %he %r!nk isuneual9



,roo idea: ( ! ! n) and p(n) are rela%ed%hrough %he modular orm

n

n

q

n pn ∑

≥

−

0

)(),,(

Gu% i%’s no% *ery Bnice →

+o%s o .or? beore using %heearlier %ools;



Modular Forms

& The q2series are ac%ually Fourier series:

q = 0"$ or $ in H

& E modular orm f ( $ ) o .eigh% k has %.oBsymme%ries:

1) f ( $ 3 1) f ( $ ) (-eriodici%y)2) f (21 O $ ) B $ k f ( $ ) (Mellin %ransorm)



& Com-osi%ion Lrou- o %ransorma%ions

( ) )(:)( $ f d %$ d %$ b!$ f

d %

b! $ f k += ++=

or " " ma%rices .O de%erminan% 19

& B'ie o ma%ri@ subgrou- le*el9

& El%erna%i*ely *ie. as (nearly) in*arian% unc%ionson "2dimensional la%%ices



Fac%: BNice modular orms lie in ini%e2dimensional *ec%or s-aces9

No%e: BNice %echnical analy%ic condi%ions

& 'eries or p(n) and %r!nk ail badly No longer ini%e2dimensional; Challenge: Transorm in%o some%hing Bnice



& In%er-lay be%.een: Combina%orics o coeicien%s

Eri%hme%ic modulo Enaly%ic %ransorma%ions

=@am-le: Modulo 6!

( ) ( )∏∏≥≥

−≡−⋅− 1

24

1

55 111

1

n

n

n

n

nqq

q

+e%2side coeicien%s: rela%ed %o p(n)

- es-ecially i (n!6) 1



Modulari%y E--roach *s9Lenera%ing Func%ions (Coeicien%s)

& ecall %he modulari%y a--roach: Con*er% solu%ions %o an im-ossible E

'-eciic modular orm is Bunim-or%an%

& Coeicien%s can be *ery in%eres%ing%hemsel*es;; 'ums o suares! Di*isor sums! ,ar%i%ions! >



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