Generating Functions, Partitions, and q-Series Modular Forms Applications Generating Functions, Partitions, and q-series: An Introduction to Classical Modular Forms Edray Herber Goins MA 59800-614-27988: Classical Modular Forms Department of Mathematics Purdue University June 28, 2016 MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
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Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Generating Functions, Partitions, and q-series:
An Introduction to Classical Modular Forms
Edray Herber Goins
MA 59800-614-27988: Classical Modular Forms
Department of MathematicsPurdue University
June 28, 2016
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Abstract
It is natural to count the number of objects placed into a geometric shape. Forexample, “triangular numbers” are the number objects that can form an equilateraltriangle, while “pentagonal numbers” are the number of objects that can form aregular pentagon. Similarly, it is natural to ask how to partition a natural number intosmaller parts. In the 1700’s, the German mathematician Leonhard Euler found aremarkable relationship between these two sets of numbers. Euler’s PentagonalNumber Theorem paved the way for the concept of a q-series as a generating functionfor other interesting sequences of numbers.
In 1919, the English mathematician Godfrey Harold Hardy went to visit his ill friendSrinivasa Ramanujan in the hospital. Hardy told Ramanujan that had ridden intaxi-cab No. 1729, and remarked that the number seemed to be rather a dull one.Ramanujan countered the number is very interesting as it is the smallest numberexpressible as the sum of two positive cubes in two different ways. Hardy was workingwith Ramanujan to understand some of his pulchritudinous identities involving thecoefficients of a certain “modular discriminant.” This specific q-series has atransformation property which led to the theory of modular forms.
In this introductory talk, we’ll present various applications of classical modular forms toquestions which naturally arise in combinatorics, algebra, and number theory. Inparticular, we’ll discuss properties of quadratic forms (why is every positive integer thesum of four squares?), cubic forms (what are “taxi-cab” numbers?) and congruencesamong the coefficients (why is Ramanujan’s tau function so closely related to thedivisor function?).
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Outline
1 Generating Functions, Partitions, and q-SeriesFigurate NumbersPartition Functionq-Series
2 Modular FormsRiemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
3 ApplicationsRepresenting Integers as the Sums of SquaresRamanujan Tau FunctionElliptic Curves and the Taniyama-Shimura-Weil Conjecture
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Figurate NumbersPartition Functionq-Series
Triangular Numbers
Definition
A triangular number counts the number of objects that can form an equilateraltriangle. The nth triangular number Tn is the number of dots composing atriangle with n dots on a side.
https://en.wikipedia.org/wiki/Triangular_number
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Figurate NumbersPartition Functionq-Series
Properties
Proposition
Let Tn denote the nth triangular number.
Tn = n + (n − 1) + · · ·+ 2 + 1 = n (n + 1)/2.
The consecutive sum Tn + Tn−1 = n2 is always a perfect square.
The consecutive difference of squares Tn2 − Tn−1
2 = n3 is a perfect cube.∞∑n=0
Tn qn =
q
(1− q)3is a generating function for Tn.
To see the last assertion, differentiate the Geometric Series twice:∞∑n=0
qn+1 =q
1− q
∞∑n=0
(n + 1) qn =1
(1− q)2
∞∑n=0
n (n + 1) qn−1 =2
(1− q)3=⇒
∞∑n=0
n (n + 1)
2qn =
q
(1− q)3
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Figurate NumbersPartition Functionq-Series
Figurate Numbers
Definition
A figurate number is a natural number that can be represented by a regulargeometrical arrangement of equally spaced points. If the arrangement forms aregular polygon, the number is called a polygonal number.
Some examples of polygonal numbers are:
triangular numbers Tn,
square numbers Sn,
pentagonal numbers Gn, and
hexagonal numbers Hn.
http://mathworld.wolfram.com/FigurateNumber.html
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Riemann Zeta Function
Consider the Riemann zeta function
ζ(s) =∞∑n=1
1
ns=
∏primes p
1
1− p−s.
Theorem (Bernhard Riemann)
For s ∈ C satisfying Re(s) > 1, define the functions
Λ(s) =1
πs/2Γ( s
2
)ζ(s), Γ(s) =
∫ ∞0
e−t ts−1 dt = (s − 1)!
Consider those s ∈ C which satisfy 0 < Re(s) < 1.
Λ(1− s) = Λ(s).
The Riemann zeta function satisfies the functional equation
ζ(1− s) =2
(2π)scos
(π s
2
)Γ(s) ζ(s).
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Proof of Theorem
Θ(t) =∞∏m=1
(1− e−2mπt2
)(1 + e(1−2m)πt2
)2
= 1 + 2∞∑n=1
e−πn2 t2
, t > 0.
Lemma 1∫ ∞0
[Θ(t)− 1] ts−1 dt = Λ(s) for Re(s) > 1.
∫ ∞0
[Θ(t)− 1] ts−1 dt =∞∑n=1
2
∫ ∞0
e−πn2t2
ts−1 dt =1
πs/2Γ( s
2
) ∞∑n=1
1
ns= Λ(s).
Lemma 2
Λ(s) = Λ(1− s) when 0 < Re(s) < 1.
Using Lemmas 1 and the identity Θ(1/t) = t Θ(t) we find that
Λ(s) =1
s (1− s)+
∫ ∞1
[Θ(t)− 1][t−s + ts−1
].
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Proof of Theorem (continued)
Lemma 3 (Legendre’s Duplication Formula).
For s ∈ C satisfying 0 < Re(s) < 1,
Γ(s) =2s−1
√π
Γ
(s
2
)Γ
(1 + s
2
), Γ
(1− s
2
)Γ
(1 + s
2
)= π
/cos
(π s
2
)
ζ(1− s)
2
(2π)scos
(π s
2
)Γ(s) ζ(s)
=(2π)s
2 cos
(π s
2
)Γ(s)
· π(1−s)/2 Λ(1− s)
Γ
(1− s
2
)︸ ︷︷ ︸
ζ(1−s)
/πs/2 Λ(s)
Γ
(s
2
)︸ ︷︷ ︸
ζ(s)
=
2s−1
√π
Γ
(s
2
)Γ
(1 + s
2
)/Γ(s)
Γ
(1− s
2
)Γ
(1 + s
2
)· 1
πcos
(π s
2
) · Λ(1− s)
Λ(s)
= 1.
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Special Values
Theorem (Leonhard Euler)
Let Bn denote the nth Bernoulli number, that is, that rational number whichappears in the power series expansion
t
et − 1=∞∑n=1
Bntn
n!= 1 +
(−1
2
)t +
(+
1
6
)t2
2+
(− 1
30
)t4
24+ · · · .
ζ(−n) = − Bn+1
n + 1is a rational number when evaluated at any negative
integer.
ζ(−2 k) = 0 at the negative even integers.
ζ(2k) =(−1)k+1 B2k
2 (2k)!(2π)2k at the positive even integers.
The negative even integers are called the trivial zeros of the Riemann zetafunction. The Riemann Hypothesis asserts that if ζ(s) = 0 then either (1)s = −2 k is a negative even integer, or (2) Re(s) = 1/2.
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
(But ζ(s) = 1 + 1/2s + 1/3s + · · ·+ 1/ns + · · · only makes sense for Re(s) > 1!)
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Motivating Question
Proposition (Carl Gustav Jacob Jacobi, 1829)
Denote the function
Θ(t) =∞∏m=1
(1− e−2mt2
)(1 + e(1−2m)t2
)2
= 1 + 2∞∑n=1
e−n2 t2
.
Θ(1/t) = t Θ(t) for all t > 0.
limt→∞Θ(t) = 1.
Can this be generalized?
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Modular Forms
Γ(1) = SL2(Z) acts on the upper-half plane H = {z ∈ C | Im(z) > 0} by
γ z =a z + b
c z + dwhere γ =
(a bc d
).
Definitions
Let N be a positive integer and k be a positive even integer.
Γ0(N) =
{(a bc d
)∈ SL2(Z)
∣∣∣∣ c ≡ 0 mod N
}.
A modular form of level N and weight k is a function f : H→ C satisfyingf is holomorphic: f analytic on the upper-half plane;f is cuspidal: limz→i∞ |f (z)| <∞; and
f is automorphic: f (γ z) = (c z + d)k f (z) for γ ∈ Γ0(N).
γ =
(1 10 1
)∈ Γ0(N) =⇒ f (z + 1) = f (z)
so that each modular form has a q-series expansion:
f (z) =∑n∈Z
an e2πinz =
∞∑n=0
an qn where q = e2πiz .
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Eisenstein Series
Definitions
Let the divisor function σk(n) =∑
d|n dk be the sum over the kth powers
of the divisors d of a positive integer n.
Let the Bernoulli numbers Bn be defined by the power series expansion
t
et − 1=∞∑n=1
Bntn
n!= 1 +
(−1
2
)t +
(+
1
6
)t2
2+
(− 1
30
)t4
24+ · · · .
For every even integer k, denote Ek(z) = 1− 2 k
Bk
∞∑n=1
σk−1(n) qn.
Proposition
Ek is a modular form of level N = 1 and weight k. In particular,
Ek(γ z) = (c z + d)k Ek(z) for γ ∈ Γ(1) = SL2(Z).
Ek(−1/z) = zk Ek(z)
limz→i∞ Ek(z) = 1
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Riemann Zeta FunctionSpecial Values of the Riemann Zeta FunctionModular Forms, Eisenstein Series, and Cusp Forms
Modular Discriminant
Proposition
Define
E4(z) = 1 + 240∞∑n=1
σ3(n) qn E12(z) = 1 +65520
691
∞∑n=1
σ11(n) qn
E6(z) = 1− 504∞∑n=1
σ5(n) qn ∆(z) = q∞∏n=1
(1− qn)24
=∞∑n=1
τ(n) qn
691E12(z) = 441E4(z)3 + 250E6(z)2.
1728 ∆(z) = E4(z)3 − E6(z)2.
∆(z) is a modular form of level N = 1 and weight k = 12. In particular,
∆(−1/z) = z12 ∆(z)limz→i∞∆(z) = 0.
∆ is called the modular discriminant. Any modular form f : H→ C satisfyinglimz→i∞ f (z) = 0 is called a cusp form. Denote the collection by Sk
(Γ0(N)
).
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Representing Integers as the Sums of SquaresRamanujan Tau FunctionElliptic Curves and the Taniyama-Shimura-Weil Conjecture
Application #1:
Representing Integers
as the Sums of Squares
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Representing Integers as the Sums of SquaresRamanujan Tau FunctionElliptic Curves and the Taniyama-Shimura-Weil Conjecture
Adrien-Marie LegendreSeptember 18, 1752 – January 10, 1833
Generating Functions, Partitions, and q-SeriesModular Forms
Applications
Representing Integers as the Sums of SquaresRamanujan Tau FunctionElliptic Curves and the Taniyama-Shimura-Weil Conjecture
Taxi Cab Number
I remember once going to see [Srinivasa Ramanujan] when he was lying illat Putney. I had ridden in taxi-cab No. 1729, and remarked that thenumber seemed to be rather a dull one, and that I hoped it was not anunfavourable omen. “No”, he replied, “it is a very interesting number; it isthe smallest number expressible as the sum of two [positive] cubes in twodifferent ways.” – Godfrey Harold Hardy, 1919
Definition
The nth taxicab number N = Taxicab(n) is defined as the least natural numberwhich can be expressed as a sum of two positive cubes in n distinct ways.
Assume that there are nonzero integers a, b, and c such that an + bn = cn.
Say that n has no odd prime divisors. Then x = cn/4, y = bn/4, andz = an/2 are nonzero integers such that x4 − y 4 = z2. In 1670, Pierre deFermat showed no such integers exist.
Say that n has an odd prime divisor `. Then A = an/`, B = bn/`, andC = cn/` are nonzero integers such that A` + B` = C `. Gerhard Freysuggested in 1984 to consider the curve E : y 2 = x (x − A`) (x + B`) withdiscriminant ∆(E) = 16 (AB C)2`. In 1993, Andrew Wiles proved thatsuch an elliptic curve E must be modular. In 1986, Kenneth Ribet showedthat its level must be N = 2. A simple computation shows that there areno cusp forms of level N = 2 and weight k = 2.
MA 59800-614-27988: Classical Modular Forms An Introduction to Classical Modular Forms