Mod Forms Number
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Modular Forms in NumberTheory
Karl Mahlburg (HMC ’01)C.L.E. Moore Instructor (MIT)
December ! "00#
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$u%line
& 'ums o suares
& Di*isor 'ums& Ferma%’s +as% Theorem& ,erec% -o.er Fibonacci numbers
& ,ar%i%ion congruences& Modular Forms
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+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;
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<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
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=@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
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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
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& <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;;;
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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
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& 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;;;
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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>
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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;
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=lli-%ic Cur*es
& +e% E deno%esolu%ions %o %he
eua%ion:
y" + y = x5 - x"
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,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!") }
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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)
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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"
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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"
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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.
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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
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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;
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& 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
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!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>
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,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
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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! >
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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
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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
& There are cubes
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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
& There are cubes& E--ears %o be no more>
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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
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& ,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
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,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)
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=@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
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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;;
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,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
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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;
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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
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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
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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
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$no’s congruences
Theorem ($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*)
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Theorem (M9 "006): For any -rime R 5!%here are .! / so %ha%
( ! ! .n 3 /) ≡ 0 (mod )
Corollary: $no’s congruences;
emar?: The amanuQan congruences are*ery s-ecial in general %he %r!nk isuneual9
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,roo idea: ( ! ! n) and p(n) are rela%ed%hrough %he modular orm
n
n
q
n pn ∑
≥
−
0
)(),,(
Gu% i%’s no% *ery Bnice →
+o%s o .or? beore using %heearlier %ools;
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Modular Forms
& The q2series are ac%ually Fourier series:
q = 0"$ 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)
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& 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
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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
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& 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
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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! >