Modular forms, combinatorially and otherwise
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- Jemima Briggs
- 5 years ago
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1 Modular forms, combinatorially and otherwise p. 1/103 Modular forms, combinatorially and otherwise David Penniston
2 Sums of squares Modular forms, combinatorially and otherwise p. 2/103
3 Modular forms, combinatorially and otherwise p. 2/103 Sums of squares Which primes can be written as a sum of two integer squares?
4 Modular forms, combinatorially and otherwise p. 3/103 2 = = = = =
5 A prime p is equal to a sum of two squares if and only p = 2 or p 1 (mod 4) Modular forms, combinatorially and otherwise p. 4/103
6 Modular forms, combinatorially and otherwise p. 5/ = 0 0 (mod 4) 1 2 = 1 1 (mod 4) 2 2 = 4 0 (mod 4) 3 2 = 9 1 (mod 4)
7 Modular forms, combinatorially and otherwise p. 5/ = 0 0 (mod 4) 1 2 = 1 1 (mod 4) 2 2 = 4 0 (mod 4) 3 2 = 9 1 (mod 4) s 2 {0, 1} (mod 4)
8 Modular forms, combinatorially and otherwise p. 5/ = 0 0 (mod 4) 1 2 = 1 1 (mod 4) 2 2 = 4 0 (mod 4) 3 2 = 9 1 (mod 4) s 2 {0, 1} (mod 4) s 2 + t 2 {0, 1, 2} (mod 4)
9 How many squares suffice? Modular forms, combinatorially and otherwise p. 6/103
10 Modular forms, combinatorially and otherwise p. 6/103 How many squares suffice? s 2 {0, 1, 4} (mod 8)
11 Modular forms, combinatorially and otherwise p. 6/103 How many squares suffice? s 2 {0, 1, 4} (mod 8) s 2 + t 2 + u 2 {0, 1, 2, 3, 4, 5, 6} (mod 8)
12 Modular forms, combinatorially and otherwise p. 7/103 Four squares theorem (Legendre) Every positive integer can be written as a sum of four integer squares
13 Modular forms, combinatorially and otherwise p. 8/103 Theta function θ(q) = s Z q s2
14 Modular forms, combinatorially and otherwise p. 8/103 Theta function θ(q) = s Z q s2 = 1 + 2q + 2q 4 + 2q 9 +
15 Modular forms, combinatorially and otherwise p. 9/103 θ(q) 2 = s,t Z q s2 +t 2 = 1 + 4q + 4q 2 + 4q 4 + 8q 5 +
16 Modular forms, combinatorially and otherwise p. 9/103 θ(q) 2 = s,t Z q s2 +t 2 = 1 + 4q + 4q 2 + 4q 4 + 8q 5 + = n=0 r 2 (n)q n r 2 (n) = # of reps of n as a sum of two integer squares
17 Modular forms, combinatorially and otherwise p. 10/ q 4 + 8q = (±2) = (±2) 2 5 = (±1) 2 + (±2) 2 = (±2) 2 + (±1) 2
18 Modular forms, combinatorially and otherwise p. 11/103 Four squares theorem The coefficient of q n in θ(q) 4 is positive for every n 1
19 Modular forms, combinatorially and otherwise p. 11/103 Four squares theorem The coefficient of q n in θ(q) 4 is positive for every n 1 θ(q) 4 = 1 + 8q + 24q q q q 5 +
20 θ analytically Modular forms, combinatorially and otherwise p. 12/103
21 Modular forms, combinatorially and otherwise p. 12/103 θ analytically q = e 2πiz
22 Modular forms, combinatorially and otherwise p. 12/103 θ analytically q = e 2πiz θ(z + 1) = θ(z)
23 Modular forms, combinatorially and otherwise p. 12/103 θ analytically q = e 2πiz θ(z + 1) = θ(z) θ( 1/4z) = 2iz θ(z)
24 Modular forms, combinatorially and otherwise p. 13/103 Partitions A partition of n is a nonincreasing sequence of positive integers whose sum is n
25 Modular forms, combinatorially and otherwise p. 14/103 Partitions of
26 p(n) := number of partitions of n Modular forms, combinatorially and otherwise p. 15/103
27 Modular forms, combinatorially and otherwise p. 15/103 p(n) := number of partitions of n p(4) = 5
28 Modular forms, combinatorially and otherwise p. 16/103 n p(n)
29 p(0) p(1) p(2) p(3) p(4) p(5) p(6) p(7) p(8) p(9) p(10) p(11) p(12) p(13) p(14) p(15) p(16) p(17) p(18) p(19) p(20) p(21) p(22) p(23) p(24) p(25) p(26) p(27) p(28) p(29) p(30) p(31) p(32) p(33) p(34) p(35) p(36) p(37) p(38) p(39) p(40) p(41) p(42) p(43) p(44) p(45) p(46) p(47) p(48) p(49) Modular forms, combinatorially and otherwise p. 17/103
30 Modular forms, combinatorially and otherwise p. 18/103
31 Modular forms, combinatorially and otherwise p. 19/103
32 Modular forms, combinatorially and otherwise p. 20/103 p(4),p(9),p(14),p(19),...,p(49) are all divisible by 5
33 Modular forms, combinatorially and otherwise p. 20/103 p(4),p(9),p(14),p(19),...,p(49) are all divisible by 5 Is p(5n + 4) divisible by 5 for every n 0?
34 Ramanujan congruences Modular forms, combinatorially and otherwise p. 21/103
35 Modular forms, combinatorially and otherwise p. 22/103 Ramanujan congruences For every n 0, p(5n + 4) 0 (mod 5)
36 Modular forms, combinatorially and otherwise p. 22/103 Ramanujan congruences For every n 0, p(5n + 4) 0 (mod 5) p(7n + 5) 0 (mod 7)
37 Modular forms, combinatorially and otherwise p. 22/103 Ramanujan congruences For every n 0, p(5n + 4) 0 (mod 5) p(7n + 5) 0 (mod 7) p(11n + 6) 0 (mod 11)
38 Other congruences for p(n)? Modular forms, combinatorially and otherwise p. 23/103
39 Modular forms, combinatorially and otherwise p. 24/103 Other congruences for p(n)? (Atkin-O Brien) For every n 0, p( n ) 0 (mod 13)
40 Modular forms, combinatorially and otherwise p. 24/103 Other congruences for p(n)? (Atkin-O Brien) For every n 0, p( n ) 0 (mod 13) (p(111247) is a number with well over 300 digits)
41 Modular forms, combinatorially and otherwise p. 25/103 (K. Ono) For every prime m 5, there exist positive integers A and B such that for every n 0, p(an + B) 0 (mod m).
42 Generating functions Modular forms, combinatorially and otherwise p. 26/103
43 Modular forms, combinatorially and otherwise p. 27/103 Generating functions p(0) p(1) p(2) p(3) p(4)
44 Modular forms, combinatorially and otherwise p. 27/103 Generating functions p(0) p(1) p(2) p(3) p(4) p(0) + p(1)q + p(2)q 2 + p(3)q 3 + p(4)q 4 +
45 Modular forms, combinatorially and otherwise p. 27/103 Generating functions p(0) p(1) p(2) p(3) p(4) p(0) + p(1)q + p(2)q 2 + p(3)q 3 + p(4)q 4 + = 1 + q + 2q 2 + 3q 3 + 5q 4 + 7q
46 Modular forms, combinatorially and otherwise p. 27/103 Generating functions p(0) p(1) p(2) p(3) p(4) p(0) + p(1)q + p(2)q 2 + p(3)q 3 + p(4)q 4 + = 1 + q + 2q 2 + 3q 3 + 5q 4 + 7q = n=0 p(n)q n
47 5q 4 Modular forms, combinatorially and otherwise p. 28/103
48 Modular forms, combinatorially and otherwise p. 28/103 5q 4 = q 4 + q 4 + q 4 + q 4 + q 4
49 Modular forms, combinatorially and otherwise p. 28/103 5q 4 = q 4 + q 4 + q 4 + q 4 + q 4 = q 4 + q q q q
50 Modular forms, combinatorially and otherwise p. 28/103 5q 4 = q 4 + q 4 + q 4 + q 4 + q 4 = q 4 + q q q q = q 4 + q 3 q 1 + q q 2 q q
51 Modular forms, combinatorially and otherwise p. 29/103 (1 + q 1 + q q ) (1 + q 2 + q ) (1 + q 3 + q )
52 Modular forms, combinatorially and otherwise p. 29/103 (1 + q 1 + q q ) (1 + q 2 + q ) (1 + q 3 + q ) = n=0 p(n)q n
53 Modular forms, combinatorially and otherwise p. 30/103 p(n)q n = n=0 (1 + q 1 + q q ) (1 + q 2 + q ) (1 + q 3 + q )
54 Modular forms, combinatorially and otherwise p. 30/103 p(n)q n = n=0 (1 + q 1 + q q ) (1 + q 2 + q ) (1 + q 3 + q ) = ( 1 ) ( 1 ) ( 1 ) ( 1 ) 1 q 1 q 2 1 q 3 1 q 4
55 Modular forms, combinatorially and otherwise p. 31/103 Generating function for p(n) n=0 p(n)q n = n=1 ( 1 ) 1 q n
56 Modular forms, combinatorially and otherwise p. 32/103 Dedekind s eta function η(z) = q 1/24 (1 q n ) n=1 (q := e 2πiz )
57 η(z + 1) = e πi/12 η(z) Modular forms, combinatorially and otherwise p. 33/103
58 Modular forms, combinatorially and otherwise p. 33/103 η(z + 1) = e πi/12 η(z) η( 1/z) = z i η(z)
59 Modular forms, combinatorially and otherwise p. 33/103 η(z + 1) = e πi/12 η(z) η( 1/z) = z i η(z) η 24 (z + 1) = η 24 (z)
60 Modular forms, combinatorially and otherwise p. 33/103 η(z + 1) = e πi/12 η(z) η( 1/z) = z i η(z) η 24 (z + 1) = η 24 (z) η 24 ( 1/z) = z 12 η 24 (z)
61 Modular forms, combinatorially and otherwise p. 34/103 z + 1 = 1z + 1 0z + 1 ( )
62 Modular forms, combinatorially and otherwise p. 34/103 z + 1 = 1z + 1 0z z = 0z 1 1z + 0 ( ( ) )
63 Modular forms, combinatorially and otherwise p. 34/103 z + 1 = 1z + 1 0z z = 0z 1 1z + 0 ( ( ) ) ( ) and ( ) generate SL 2 (Z)
64 Modular forms, combinatorially and otherwise p. 35/103 Integer weight modular forms A modular form of weight k on Γ = SL 2 (Z) is a holomorphic function f : H C such that for every ( ) a b Γ, c d ( ) az + b f = (cz + d) k f(z) cz + d and f is holomorphic at the cusp of Γ.
65 Modular forms, combinatorially and otherwise p. 35/103 Integer weight modular forms A modular form of weight k on Γ = SL 2 (Z) is a holomorphic function f : H C such that for every ( ) a b Γ, c d ( ) az + b f = (cz + d) k f(z) cz + d and f is holomorphic at the cusp of Γ. We denote the C-vector space of such functions by M k (Γ), and the subset of forms that vanish at the cusp by S k (Γ).
66 Modular forms, combinatorially and otherwise p. 36/103 Examples η 24 (z) M 12 (Γ)
67 Modular forms, combinatorially and otherwise p. 36/103 Examples η 24 (z) M 12 (Γ) E 4 (z) = n=0 σ 3 (n)q n M 4 (Γ)
68 Modular forms, combinatorially and otherwise p. 36/103 Examples η 24 (z) M 12 (Γ) E 4 (z) = E 6 (z) = n=0 n=0 σ 3 (n)q n M 4 (Γ) σ 5 (n)q n M 6 (Γ)
69 Modular forms, combinatorially and otherwise p. 36/103 Examples η 24 (z) M 12 (Γ) E 4 (z) = E 6 (z) = n=0 n=0 σ 3 (n)q n M 4 (Γ) σ 5 (n)q n M 6 (Γ) σ k (n) = d n d k
70 - M k (Γ) is a finite dimensional vector space Modular forms, combinatorially and otherwise p. 37/103
71 Modular forms, combinatorially and otherwise p. 37/103 - M k (Γ) is a finite dimensional vector space - M k (Γ) is spanned by {E i 4E j 6 4i + 6j = k}
72 Modular forms, combinatorially and otherwise p. 38/103 Modular forms with character If f ( ) az + b cz + d = χ(d)(cz + d) k f(z) for all ( a b c d ) Γ with N c, we write f M k (Γ 0 (N),χ).
73 Modular forms, combinatorially and otherwise p. 39/103 Example θ(z) 4 M 2 (Γ 0 (4))
74 Modular forms, combinatorially and otherwise p. 39/103 Example θ(z) 4 M 2 (Γ 0 (4)) dim(m 2 (Γ 0 (4))) = 2
75 Modular forms, combinatorially and otherwise p. 39/103 Example θ(z) 4 M 2 (Γ 0 (4)) dim(m 2 (Γ 0 (4))) = 2 G 2 (z) = n=1 σ 1 (n)q n
76 Modular forms, combinatorially and otherwise p. 39/103 Example θ(z) 4 M 2 (Γ 0 (4)) dim(m 2 (Γ 0 (4))) = 2 G 2 (z) = n=1 σ 1 (n)q n basis: {G 2 (z) 2G 2 (2z),G 2 (2z) 2G 2 (4z)}
77 θ(z) 4 = 8[G 2 (z) 2G 2 (2z)] + 16[G 2 (2z) 2G 2 (4z)] Modular forms, combinatorially and otherwise p. 40/103
78 Modular forms, combinatorially and otherwise p. 40/103 θ(z) 4 = 8[G 2 (z) 2G 2 (2z)] + 16[G 2 (2z) 2G 2 (4z)] = 8G 2 (z) 32G 2 (4z)
79 Modular forms, combinatorially and otherwise p. 40/103 θ(z) 4 = 8[G 2 (z) 2G 2 (2z)] + 16[G 2 (2z) 2G 2 (4z)] = 8G 2 (z) 32G 2 (4z) r 4 (n) = 8 d n,4 d d
80 Modular forms, combinatorially and otherwise p. 40/103 θ(z) 4 = 8[G 2 (z) 2G 2 (2z)] + 16[G 2 (2z) 2G 2 (4z)] = 8G 2 (z) 32G 2 (4z) r 4 (n) = 8 d n,4 d d > 0
81 Modular forms, combinatorially and otherwise p. 41/103 ( ) Γ 0 (N), χ(1) = 1
82 Modular forms, combinatorially and otherwise p. 41/103 ( ) Γ 0 (N), χ(1) = 1 f(z + 1) = f(z)
83 Modular forms, combinatorially and otherwise p. 41/103 ( ) Γ 0 (N), χ(1) = 1 f(z + 1) = f(z) f(z) = n=0 a(n)q n
84 Restricted partition functions Modular forms, combinatorially and otherwise p. 42/103
85 Modular forms, combinatorially and otherwise p. 43/103 Restricted partition functions Suppose I am only interested in partitions into distinct parts:
86 Modular forms, combinatorially and otherwise p. 43/103 Restricted partition functions Suppose I am only interested in partitions into distinct parts: (1 + q 1 +q q ) (1 + q 2 +q ) (1 + q 3 +q )
87 Modular forms, combinatorially and otherwise p. 44/103 The generating function in this case is the infinite product (1 + q 1 )(1 + q 2 )(1 + q 3 )(1 + q 4 )
88 Modular forms, combinatorially and otherwise p. 44/103 The generating function in this case is the infinite product (1 + q 1 )(1 + q 2 )(1 + q 3 )(1 + q 4 ) = (1 + q n ) n=1
89 Suppose I am only interested in partitions where no summand exceeds 3: Modular forms, combinatorially and otherwise p. 45/103
90 Modular forms, combinatorially and otherwise p. 45/103 Suppose I am only interested in partitions where no summand exceeds 3: ( 1 ) ( 1 )( 1 ) ( 1 ) 1 q 1 q 2 1 q 3 1 q 4
91 Modular forms, combinatorially and otherwise p. 46/103 The generating function in this case is the finite product ( 1 1 q ) ( 1 1 q 2 )( 1 1 q 3 )
92 Modular forms, combinatorially and otherwise p. 47/103 l-regular partitions A partition is called l-regular provided that none of its summands is divisible by l
93 Modular forms, combinatorially and otherwise p. 47/103 l-regular partitions A partition is called l-regular provided that none of its summands is divisible by l b l (n) := number of l-regular partitions of n
94 Modular forms, combinatorially and otherwise p. 48/
95 Modular forms, combinatorially and otherwise p. 49/
96 Modular forms, combinatorially and otherwise p. 49/ b 2 (5) = 3
97 Generating function for b 2 (n)? Modular forms, combinatorially and otherwise p. 50/103
98 Modular forms, combinatorially and otherwise p. 50/103 Generating function for b 2 (n)? ( 1 ) ( 1 )( 1 ) ( 1 ) 1 q 1 q 2 1 q 3 1 q 4
99 Modular forms, combinatorially and otherwise p. 51/103 b 2 (n)q n = n=0 ( 1 ) ( 1 1 q 1 q 3 ) ( 1 ) 1 q 5
100 Modular forms, combinatorially and otherwise p. 51/103 b 2 (n)q n = n=0 ( ) ( ) ( ) q 1 q 3 1 q 5 ( ) 1 q 2n = 1 q n n=1
101 (Euler) The number of 2-regular partitions of n is equal to the number of partitions of n into distinct parts Modular forms, combinatorially and otherwise p. 52/103
102 Modular forms, combinatorially and otherwise p. 52/103 (Euler) The number of 2-regular partitions of n is equal to the number of partitions of n into distinct parts n=0 b 2 (n)q n = n=1 ( ) 1 q 2n 1 q n
103 Modular forms, combinatorially and otherwise p. 52/103 (Euler) The number of 2-regular partitions of n is equal to the number of partitions of n into distinct parts n=0 b 2 (n)q n = n=1 ( ) 1 q 2n 1 q n = (1 + q n ) n=1
104 Modular forms, combinatorially and otherwise p. 53/103 b l (n)q n = n=0 n=1 ( ) 1 q ln 1 q n
105 Given positive integers l and m, exactly when is b l (n) divisible by m? How often is b l (n) divisible by m? Modular forms, combinatorially and otherwise p. 54/103
106 Modular forms, combinatorially and otherwise p. 54/103 Given positive integers l and m, exactly when is b l (n) divisible by m? How often is b l (n) divisible by m? δ l (m,x) := #{1 n X b l(n) 0 X (mod m)}
107 Modular forms, combinatorially and otherwise p. 55/103 δ l (m, 10 6 ) m = l =
108 Modular forms, combinatorially and otherwise p. 55/103 δ l (m, 10 6 ) m = l = m = l =
109 (P.) Suppose 3 l 23 and p 5 are distinct primes. Then for every j 1 we have: Modular forms, combinatorially and otherwise p. 56/103
110 Modular forms, combinatorially and otherwise p. 56/103 (P.) Suppose 3 l 23 and p 5 are distinct primes. Then for every j 1 we have: If p l 1, then lim inf X δ l(p j,x) p + 1 2p.
111 Modular forms, combinatorially and otherwise p. 56/103 (P.) Suppose 3 l 23 and p 5 are distinct primes. Then for every j 1 we have: If p l 1, then If p l 1, then lim inf X δ l(p j,x) p + 1 2p. lim inf X δ l(p j,x) p 1 p.
112 Modular forms, combinatorially and otherwise p. 57/103 (Serre) Suppose m and k are positive integers and f(z) = a(n)q n S k (Γ 0 (N),χ) Z[[q]]. Then for almost all n, a(n) 0 (mod m).
113 Modular forms, combinatorially and otherwise p. 58/103 η(lz) η(z) = ql/24 n=1 (1 qln ) q 1/24 n=1 (1 qn )
114 Modular forms, combinatorially and otherwise p. 58/103 η(lz) η(z) = ql/24 n=1 (1 qln ) q 1/24 n=1 (1 qn ) l 1 n+ = b l (n)q 24 n=0
115 Modular forms, combinatorially and otherwise p. 58/103 η(lz) η(z) = ql/24 n=1 (1 qln ) q 1/24 n=1 (1 qn ) l 1 n+ = b l (n)q 24 n=0 is NOT a modular form
116 Modular forms, combinatorially and otherwise p. 59/103 Given l and p distinct odd primes, there exists a positive integer α such that f l,p (z) = η(lz) η(z) η(24α 1)p (lpz)η (24α+1)p (pz) is a modular form.
117 Modular forms, combinatorially and otherwise p. 60/103 Twisting forms If f(z) = a(n)q n M k (Γ 0 (N),χ) and ψ is a character modulo M, then
118 Modular forms, combinatorially and otherwise p. 60/103 Twisting forms If f(z) = a(n)q n M k (Γ 0 (N),χ) and ψ is a character modulo M, then (f ψ)(z) := ψ(n)a(n)q n
119 Modular forms, combinatorially and otherwise p. 60/103 Twisting forms If f(z) = a(n)q n M k (Γ 0 (N),χ) and ψ is a character modulo M, then (f ψ)(z) := ψ(n)a(n)q n M k (Γ 0 (NM 2 ),χψ 2 ).
120 Modular forms, combinatorially and otherwise p. 61/103 Legendre symbol Given an odd prime p and d Z, ψ p (d) = 1 if d is a nonzero square mod p 1 if d is not a square mod p 0 if p d
121 d ψ 7 (d) Modular forms, combinatorially and otherwise p. 62/103
122 Modular forms, combinatorially and otherwise p. 63/103 If p l 1, let F l,p,t (z) = (f l,p (z) ± (f l,p ψ p )(z)) E p,t (z)
123 Modular forms, combinatorially and otherwise p. 63/103 If p l 1, let F l,p,t (z) = (f l,p (z) ± (f l,p ψ p )(z)) E p,t (z) If p l 1, let F l,p,t (z) = (f l,p ψ p )(z) E p,t (z)
124 Modular forms, combinatorially and otherwise p. 63/103 If p l 1, let F l,p,t (z) = (f l,p (z) ± (f l,p ψ p )(z)) E p,t (z) If p l 1, let F l,p,t (z) = (f l,p ψ p )(z) E p,t (z) E p,t (z) = ( η p3 (z) η(p 3 z) ) 2p t 1 (mod p t )
125 F l,p,t (z) is a cusp form that for large t vanishes to high enough order at the cusps that on dividing out by the auxiliary eta product, one obtains a modular form Modular forms, combinatorially and otherwise p. 64/103
126 Durfee squares Modular forms, combinatorially and otherwise p. 65/103
127 Modular forms, combinatorially and otherwise p. 66/103 Durfee squares Represent the partition by
128 Modular forms, combinatorially and otherwise p. 67/103 The Durfee square for this partition is
129 Modular forms, combinatorially and otherwise p. 68/103 The Durfee square for this partition is
130 Modular forms, combinatorially and otherwise p. 69/103 The Durfee square for this partition is
131 Modular forms, combinatorially and otherwise p. 69/103 The Durfee square for this partition is The partition breaks down into the 3 3 Durfee square and two partitions (2 + 1 and 3 + 1) with summands not exceeding 3
132 Modular forms, combinatorially and otherwise p. 70/103 The generating function for partitions with 3 3 Durfee square is therefore q 3 3 (( 1 ) ( 1 ) ( 1 )) 2 1 q 1 q 2 1 q 3
133 Modular forms, combinatorially and otherwise p. 70/103 The generating function for partitions with 3 3 Durfee square is therefore q 3 3 (( 1 ) ( 1 ) ( 1 )) 2 1 q 1 q 2 1 q 3 = q 9 (1 q) 2 (1 q 2 ) 2 (1 q 3 ) 2
134 Modular forms, combinatorially and otherwise p. 71/103 The generating function for partitions with n n Durfee square is q n2 (1 q) 2 (1 q 2 ) 2 (1 q n ) 2
135 Modular forms, combinatorially and otherwise p. 72/103 Since every partition has a Durfee square of some size, the generating function for p(n) can be written as 1 + n=1 P(q) := n=0 p(n)q n = q n2 (1 q) 2 (1 q 2 ) 2 (1 q n ) 2
136 Modular forms, combinatorially and otherwise p. 73/103 Modularity q = e 2πiz ρ(z) := q 1 P(q 24 )
137 Modular forms, combinatorially and otherwise p. 73/103 Modularity q = e 2πiz ρ(z) := q 1 P(q 24 ) ρ( 1/z) = i z ρ(z)
138 Modular forms, combinatorially and otherwise p. 74/103 Half integer wt modular forms - Theory developed by Shimura in the 1970s
139 Modular forms, combinatorially and otherwise p. 74/103 Half integer wt modular forms - Theory developed by Shimura in the 1970s - η(24z) S 1/2 (Γ 0 (576),χ 12 )
140 Modular forms, combinatorially and otherwise p. 74/103 Half integer wt modular forms - Theory developed by Shimura in the 1970s - η(24z) S 1/2 (Γ 0 (576),χ 12 ) - Shimura lift S t,k : S k+ 1 2 (Γ 0(4N),χ) M 2k (Γ 0 (2N),χ 2 ) (t a squarefree positive integer)
141 Mock theta functions Modular forms, combinatorially and otherwise p. 75/103
142 Modular forms, combinatorially and otherwise p. 75/103 Mock theta functions Two examples:
143 Modular forms, combinatorially and otherwise p. 75/103 Mock theta functions Two examples: f(q) := 1 + n=1 q n2 (1 + q) 2 (1 + q 2 ) 2 (1 + q n ) 2
144 Modular forms, combinatorially and otherwise p. 75/103 Mock theta functions Two examples: f(q) := 1 + n=1 q n2 (1 + q) 2 (1 + q 2 ) 2 (1 + q n ) 2 ω(q) := n=0 q 2n2 +2n (1 q) 2 (1 q 3 ) 2 (1 q 2n+1 ) 2
145 Φ(z) := q 1/24 f(q) Modular forms, combinatorially and otherwise p. 76/103
146 Modular forms, combinatorially and otherwise p. 76/103 Φ(z) := q 1/24 f(q) +2 3 i z g f (τ) i(τ + z) dτ
147 Modular forms, combinatorially and otherwise p. 76/103 Φ(z) := q 1/24 f(q) +2 3 i z g f (τ) i(τ + z) dτ g f (τ) := n= ( ( 1) n n + 1 ) 6 e 3πi (n+ 1 6) 2 τ
148 Ω(z) := 2q 1/3 ω(q 1/2 ) Modular forms, combinatorially and otherwise p. 77/103
149 Modular forms, combinatorially and otherwise p. 77/103 Ω(z) := 2q 1/3 ω(q 1/2 ) 2 3 i z g ω (τ) i(τ + z) dτ
150 Modular forms, combinatorially and otherwise p. 77/103 Ω(z) := 2q 1/3 ω(q 1/2 ) 2 3 i z g ω (τ) i(τ + z) dτ g ω (τ) := n= ( ( 1) n n + 1 ) 3 e 3πi (n+ 1 3) 2 τ
151 Modular forms, combinatorially and otherwise p. 78/103 Mock theta modularity (S. Zwegers) Φ( 1/z) = iz Ω(z)
152 Modular forms, combinatorially and otherwise p. 78/103 Mock theta modularity (S. Zwegers) Φ( 1/z) = iz Ω(z) Ω( 1/z) = iz Φ(z)
153 Modular forms, combinatorially and otherwise p. 78/103 Mock theta modularity (S. Zwegers) (K. Bringmann, K. Ono) Φ( 1/z) = iz Ω(z) Ω( 1/z) = iz Φ(z) Φ(24z) and Ω(6z) are harmonic weak Maass forms of weight 1/2
154 Modular forms, combinatorially and otherwise p. 79/103 Mock theta congruences ω(q) := n=0 α ω (n)q n = 1 + 2q + 3q 2 + 4q 3 + 6q 4 + 8q q 6 +
155 Modular forms, combinatorially and otherwise p. 79/103 Mock theta congruences ω(q) := n=0 α ω (n)q n = 1 + 2q + 3q 2 + 4q 3 + 6q 4 + 8q q 6 + (S. Garthwaite-P.) For every prime m 5, there exist positive integers A and B such that for every n 0, α ω (An + B) 0 (mod m).
156 Ω(6z) = a(n)q n + β n (y)q dn2 Modular forms, combinatorially and otherwise p. 80/103
157 Modular forms, combinatorially and otherwise p. 81/103 (B. Dandurand-P.) Given a nonnegative integer n, write 6n + 1 = r i=1 p e i i.
158 Modular forms, combinatorially and otherwise p. 81/103 (B. Dandurand-P.) Given a nonnegative integer n, write 6n + 1 = r i=1 p e i i. For each i with p i 1 (mod 3) write p i = x 2 i + 3y 2 i.
159 Modular forms, combinatorially and otherwise p. 81/103 (B. Dandurand-P.) Given a nonnegative integer n, write 6n + 1 = r i=1 p e i i. For each i with p i 1 (mod 3) write p i = x 2 i + 3y 2 i. Then b 5 (n) is divisible by 5 if and only if at least one of the following holds:
160 p i = 5 Modular forms, combinatorially and otherwise p. 82/103
161 Modular forms, combinatorially and otherwise p. 83/103 p i 2 (mod 3), p i 5 and e i is odd
162 Modular forms, combinatorially and otherwise p. 84/103 p i 1 (mod 3), 5 x i and e i is odd
163 p i 1 (mod 3), 5 y i and e i 4 (mod 5) Modular forms, combinatorially and otherwise p. 85/103
164 p i 1 (mod 3), 5 (x 2 i y2 i ) and e i 2 (mod 3) Modular forms, combinatorially and otherwise p. 86/103
165 p i 1 (mod 3), 5 x i y i (x 2 i y2 i ) and e i 5 (mod 6) Modular forms, combinatorially and otherwise p. 87/103
166 Modular forms, combinatorially and otherwise p. 88/103 p(n)x n = n=0 ( 1 ) ( 1 )( 1 ) ( 1 ) 1 x 1 x 2 1 x 3 1 x 4
167 Modular forms, combinatorially and otherwise p. 88/103 p(n)x n = n=0 ( 1 ) ( 1 )( 1 ) ( 1 ) 1 x 1 x 2 1 x 3 1 x 4 Consider the reciprocal (1 x)(1 x 2 )(1 x 3 )(1 x 4 )
168 Modular forms, combinatorially and otherwise p. 88/103 p(n)x n = n=0 ( 1 ) ( 1 )( 1 ) ( 1 ) 1 x 1 x 2 1 x 3 1 x 4 Consider the reciprocal (1 x)(1 x 2 )(1 x 3 )(1 x 4 ) = (1 x n ) n=1
169 Modular forms, combinatorially and otherwise p. 89/103 (1 x n ) n=1
170 Modular forms, combinatorially and otherwise p. 89/103 (1 x n ) n=1 = 1 x 1 x 2 + x 5 + x 7 x 12 x 15 + x 22 + x 26
171 Modular forms, combinatorially and otherwise p. 89/103 (1 x n ) n=1 = 1 x 1 x 2 + x 5 + x 7 x 12 x 15 + x 22 + x 26 (1, 2) (5, 7) (12, 15) (22, 26)
172 Modular forms, combinatorially and otherwise p. 89/103 (1 x n ) n=1 = 1 x 1 x 2 + x 5 + x 7 x 12 x 15 + x 22 + x 26 (1, 2) (5, 7) (12, 15) (22, 26) k k(3k + 1)/
173 Modular forms, combinatorially and otherwise p. 90/103 (1 x n ) 3 n=1
174 Modular forms, combinatorially and otherwise p. 90/103 (1 x n ) 3 n=1 = 1 3x 1 + 5x 3 7x 6 + 9x 10 11x 15 +
175 Modular forms, combinatorially and otherwise p. 90/103 (1 x n ) 3 n=1 = 1 3x 1 + 5x 3 7x 6 + 9x 10 11x 15 + (1, 3, 6, 10, 15)
176 Modular forms, combinatorially and otherwise p. 90/103 (1 x n ) 3 n=1 = 1 3x 1 + 5x 3 7x 6 + 9x 10 11x 15 + (1, 3, 6, 10, 15) l l l(l + 1)/
177 Modular forms, combinatorially and otherwise p. 91/103 (1 x n ) 5 n=1
178 Modular forms, combinatorially and otherwise p. 91/103 (1 x n ) 5 n=1 (1 x n ) 5 = 1 5x n + 10x 2n 10x 3n + 5x 4n x 5n
179 Modular forms, combinatorially and otherwise p. 91/103 (1 x n ) 5 n=1 (1 x n ) 5 = 1 5x n + 10x 2n 10x 3n + 5x 4n x 5n 1 x 5n (mod 5)
180 Modular forms, combinatorially and otherwise p. 92/103 (1 x n ) 4 n=1
181 Modular forms, combinatorially and otherwise p. 93/103 (1 x n ) 4 = n=1 n=1 (1 x n ) 5 1 x n
182 Modular forms, combinatorially and otherwise p. 93/103 (1 x n ) 4 = n=1 n=1 n=1 (1 x 5n ) 1 x n (1 x n ) 5 1 x n
183 Modular forms, combinatorially and otherwise p. 93/103 (1 x n ) 4 = n=1 n=1 (1 x 5n ) n=1 n=1 (1 x 5n ) 1 x n n=1 (1 x n ) 5 1 x n ( 1 ) 1 x n
184 Modular forms, combinatorially and otherwise p. 93/103 (1 x n ) 4 = n=1 n=1 (1 x 5n ) n=1 (1 x 5n ) n=1 n=1 (1 x 5n ) 1 x n n=0 n=1 (1 x n ) 5 1 x n ( 1 ) 1 x n p(n)x n (mod 5)
185 Modular forms, combinatorially and otherwise p. 94/103 p(m)x m m=0 (1 x m ) 4 m=1 m=1 ( 1 ) 1 x 5m (mod 5)
186 Modular forms, combinatorially and otherwise p. 94/103 p(m)x m m=0 (1 x m ) 4 m=1 m=1 ( 1 ) 1 x 5m (mod 5) The first Ramanujan congruence is equivalent to showing that for any term of the form cx 5n+4 in the product (1 x m ) 4 m=1 we have c 0 (mod 5). m=1 ( 1 ) 1 x 5m,
187 Modular forms, combinatorially and otherwise p. 95/103 The first Ramanujan congruence is equivalent to showing that for any term of the form cx 5n+4 in the product (1 x m ) 4 m=1 we have c 0 (mod 5). m=1 ( 1 ) 1 x 5m
188 Modular forms, combinatorially and otherwise p. 96/103 The first Ramanujan congruence is implied by showing that for any term of the form cx 5n+4 in the product (1 x m ) 4 m=1 we have c 0 (mod 5).
189 Modular forms, combinatorially and otherwise p. 97/103 The first Ramanujan congruence is implied by showing that for any term of the form cx 5n+4 in the product (1 x m ) m=1 (1 x m ) 3 m=1 we have c 0 (mod 5).
190 Modular forms, combinatorially and otherwise p. 98/103 (1 x m ) m=1 k(3k + 1) 2
191 Modular forms, combinatorially and otherwise p. 98/103 (1 x m ) m=1 k(3k + 1) 2 (1 x m ) 3 l(l + 1) 2 m=1
192 k k(3k + 1)/2 l(l + 1)/2 l Modular forms, combinatorially and otherwise p. 99/103
193 k k(3k + 1)/2 l(l + 1)/2 l Modular forms, combinatorially and otherwise p. 100/103
194 Modular forms, combinatorially and otherwise p. 100/103 k k(3k + 1)/2 l(l + 1)/2 l k 4 (mod 5) l 2 (mod 5)
195 k 4 (mod 5) l 2 (mod 5) Modular forms, combinatorially and otherwise p. 101/103
196 Modular forms, combinatorially and otherwise p. 101/103 k 4 (mod 5) l 2 (mod 5) l l l(l + 1)/
197 Modular forms, combinatorially and otherwise p. 102/103 k 4 (mod 5) l 2 (mod 5) l l l(l + 1)/
198 Modular forms, combinatorially and otherwise p. 102/103 k 4 (mod 5) l 2 (mod 5) l l l(l + 1)/ ±x k(3k+1)/2 (2l + 1)x l(l+1)/2 0 (mod 5)
199 Modular forms, combinatorially and otherwise p. 103/103 Here is a mock theta function: f(q) := 1 + n=1 q n2 (1 + q) 2 (1 + q 2 ) 2 (1 + q n ) 2
200 Modular forms, combinatorially and otherwise p. 103/103 Here is a mock theta function: f(q) := 1 + n=1 q n2 (1 + q) 2 (1 + q 2 ) 2 (1 + q n ) 2 = 1 + q (1 + q) 2 + q 4 (1 + q) 2 (1 + q 2 ) 2 +
201 Modular forms, combinatorially and otherwise p. 103/103 Here is a mock theta function: f(q) := 1 + n=1 q n2 (1 + q) 2 (1 + q 2 ) 2 (1 + q n ) 2 = 1 + q (1 + q) 2 + q 4 (1 + q) 2 (1 + q 2 ) 2 + = 1 + q 2q 2 + 3q 3 3q 4 +
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