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A Motivated Introduction to Modular Forms
Nathan C. Ryan
May 3, 2006
Outline of talk:
I. Motivating questions
II. Ramanujan’s τ function
III. Theta Series
IV. Congruent Number Problem
V. My Research
Nathan C. Ryan A Motivated Introduction to Modular Forms
Old Questions . . .
I What can you say about the coefficients of products ofbinomials?
I What numbers can be represented as the sum of four squares?For those that can be represented in how many ways can it bedone?
I Given an integer n is there a way to determine if there’s aright triangle with rational sides and area n?
Nathan C. Ryan A Motivated Introduction to Modular Forms
with a very modern answer . . .
Headway can be made in all these questions withsome knowledge of modular forms.
Nathan C. Ryan A Motivated Introduction to Modular Forms
Ingredients to make modular forms
I an upper half-space
I a group acting on the upper half-space
I an arithmetic group
I a functional equation and an automorphy factor
I a Fourier expansion
Nathan C. Ryan A Motivated Introduction to Modular Forms
Classical Modular Forms
I Half-plane: H = {z = x + iy ∈ C : y > 0}I Group: SL2(R)
I Group action: For z ∈ H and γ ∈ SL2(R),
γ.z =
(a bc d
).z =def
az + b
cz + d
I Arithmetic group: Γ := SL2(Z)
Nathan C. Ryan A Motivated Introduction to Modular Forms
Classical Modular Forms (cont.)
Let k ∈ Z and f : H → C be holomorphic on H and at ∞. If fsatisfies the functional equation
f (γ.z) = (cz + d)k f (z) for all γ =
(a bc d
)∈ Γ
we call f a modular form of weight k.
Nathan C. Ryan A Motivated Introduction to Modular Forms
Classical Modular Forms (cont.)
I (cz + d)k is the automorphy factor
I f is periodic (let γ = ( 1 10 1 )) it has a Fourier expansion
f (z) =∑n∈Z
anqn
(q =def e2πiz
)I holomorphic at ∞ means an = 0 for n < 0
I if a0 = 0, we call f a cuspform
I we denote the space of modular forms of weight k by Mk andthe space of cusp forms by Sk
Nathan C. Ryan A Motivated Introduction to Modular Forms
Ramanujan’s τ function
(2π)−12∆(z) = q∏n≥1
(1− qn)24 =def
∑n≥1
τ(n)qn
I ∆ is the unique element of S12 normalized so that τ(1) = 1
I Its definition is similar to that of other arithmetic functions.E.g., the partition function:∑
n≥0
p(n)xn =def
∏n≥1
(1− xn)−1
Nathan C. Ryan A Motivated Introduction to Modular Forms
Ramanujan’s τ function (cont.)
I τ(n) is multiplicative
I τ(n) ≡ σ11(n) (mod 691), τ(n) ≡ nσ9(n) (mod 5), . . .
I Lehmer’s Conjecture: τ(n) > 0 for all n
I Ramanujan’s Conjecture: |τ(n)| < σ0(n)n11/2
I τ(n) = 8000 [(σ3 ◦ σ3) ◦ σ3] (n)− 147 [σ5 ◦ σ5] (n)
Nathan C. Ryan A Motivated Introduction to Modular Forms
Hecke operators
I Hecke defined a family of commuting operators T (i)(i ∈ Z≥0) with the following properties:
I T (i) : Mk →Mk , T (i) : Sk → Sk
I T (m)T (n) = T (mn) if (m, n) = 1I Sk has a basis of Hecke eigenformsI if f is a Hecke eigenform, a(1) = 1, f |T (p) = λpf , then
a(p) = λp
Nathan C. Ryan A Motivated Introduction to Modular Forms
Lagrange’s Four Squares Theorem
In 1770, Lagrange proved
Every integer can be written as the sum of 4 squares.
In 1813, Cauchy proved the more general
Every integer can be written as the sum of n n-gonal numbers.
Nathan C. Ryan A Motivated Introduction to Modular Forms
Jacobi’s Formulae
In 1834 Jacobi proved that the number r4(n) of ways to representn as the sum of four squares was
r4(n) =
{8σ1(n) if n is odd
24σ1(n0) if n = 2rn0 even, 2 6 |n0
I Proof uses
Θ4(z) =∑
(x ,y ,z,w)∈Z4
qx2+y2+z2+w2
which is a modular form of weight 2 and level 4.
Nathan C. Ryan A Motivated Introduction to Modular Forms
Modular Forms with Level
I We will rewrite the functional equation in terms of the slashoperator. Let α =
(a bc d
)∈ GL+
2 (R) then
(f |kα)(z) =def (detα)k/2(cz + d)−k f (α.z)
I We change the arithmetic group in the definition of modularforms:
Γ(N) =def
{(a bc d
)∈ SL2(Z) : a ≡ d ≡ 1, b ≡ c ≡ 0 (mod N)
}
Nathan C. Ryan A Motivated Introduction to Modular Forms
Modular Forms with Level (cont.)
I A congruence subgroup of level N is any Γ′ so thatΓ(N) ≤ Γ′.
I A holomorphic modular form of weight k and level N isan f : H → C so that
1. f (γ.z) = (cz + d)k f (z) for all γ ∈ Γ′
2. if γ0 ∈ SL2(Z), then (f |kγ0)(z) has a Fourier expansion of theform
∑n≥nγ0
aγ0(n)qnN where qN =def e2πiz/N
3. f is holomorphic on H and all its cusps
Nathan C. Ryan A Motivated Introduction to Modular Forms
Congruent Numbers
n ∈ Z≥0 is congruent if there is a right triangle with rational sideand area n.
1. Can you give an example of a congruent number?
2. How many congruent numbers are there?
3. What is the smallest congruent number?
Nathan C. Ryan A Motivated Introduction to Modular Forms
Congruent Numbers (cont.)
I 6 is the area of a 3-4-5 right triangle, so it is a congruentnumber.
I There are infinitely many congruent numbers since there areinfinitely many Pythagorean triples.
I 5 is congruent since it is the area of a triangle with sides 32 , 20
3and 41
6 .
Nathan C. Ryan A Motivated Introduction to Modular Forms
A (somewhat) open problem
The only open problem left from antiquity is the congruent numberproblem:
Given an n, how can you determine if n is congruent.
The problem is equivalent to
Given an n, can you determine the number of rational points onthe elliptic curve
En : y2 = x(x − n)(x + n)
Nathan C. Ryan A Motivated Introduction to Modular Forms
The problem is hard (picture is Karl Rubin’s)
Nathan C. Ryan A Motivated Introduction to Modular Forms
More on En : y 2 = x3 − n2x
I By Taniyama-Shimura, En is associated to a modular form fEn
of weight 2 and level NEn .
I ap = p + 1−#En(Fp)
I L(En, s) =∏
p-NEn
11−app−s+p1−2s
Nathan C. Ryan A Motivated Introduction to Modular Forms
Connection to Modular Forms
n is congruent ⇔ #En(Q) = ∞⇒ L(En, 1) = 0⇔ an = 0 for a particular
modular form
Nathan C. Ryan A Motivated Introduction to Modular Forms
n is congruent ⇔ #E (Q) = ∞
I There is a 1-1 correspondence between rational solutionsX ,Y ,Z to
1
2XY = n and X 2 + Y 2 = Z 2
and rational numbers x , x + n, x − n that are squares ofrational numbers. The map is (X ,Y ,Z ) 7→ (Z/2)2
I (X ± Y )2 = Z 2 ± 4nI Divide both sides by four:(
X
2± Y
2
)2
± n =
(Z
2
)2
I Take-home Show that the map is injective
I Theorem En has positive rank
Nathan C. Ryan A Motivated Introduction to Modular Forms
#En(Q) = ∞ ⇒ L(En, 1) = 0
I A deep result of Coates and Wiles
I The converse is a deep ($1 000 000) conjecture of Birch andSwinnerton-Dyer
Nathan C. Ryan A Motivated Introduction to Modular Forms
L(En, 1) = 0 ⇔ an = 0 for a particular modularform
I Let k ≥ 3 be odd. Shimura showed there is a “nice” injectivemap from Mk/2 of level N to Mk−1 for some level N ′.
I Tunnell constructed two forms of weight 32 , f =
∑anq
n andf ′ =
∑a′nq
n so that
L(E , 1) = 0 iff an = 0 (n odd) or a′n/2 = 0 (n even)
Moreover, Shimura(f ) = Shimura(f ′).
Nathan C. Ryan A Motivated Introduction to Modular Forms
Congruent Number Problem
If n is squarefree and odd and n is the area of a right triangle withrational sides, then
#{x , y , z ∈ Z : n = 2x2 + y2 + 32z2
}=
#{x , y , z ∈ Z : n = 2x2 + y2 + 8z2
}I Converse holds if BSD is true
I Proof follows from matching up images under Shimura
I Similar formula holds for even n.I SAGE implementation of solution online
I http://modular.math.washington.edu/sage/apps/2005-10-18-congruent/cong.sage
Nathan C. Ryan A Motivated Introduction to Modular Forms
Integer vs half integer weight
I Half-space: H
I Group: Let
G = {(α, φ(z)) : α ∈ GL2(Q), φ : H → C hol.}
where(α, φ(z))(β, ψ(z)) = (αβ, φ(βz)ψ(z))
I Arithmetic group: Let Γ′ ⊂ Γ0(4)
Γ̃ ={(γ, j(γ, z)) : γ ∈ Γ′
}
Nathan C. Ryan A Motivated Introduction to Modular Forms
Integer vs half integer weight (cont.)
I Functional Equation: Let ξ = (α, φ(z)) ∈ G
f (z)|[ξ]k/2 = f (αz)φ(z)−k
I Fourier Expansion: Complicated, but do-able
Nathan C. Ryan A Motivated Introduction to Modular Forms
Siegel Modular Forms
I ComparisonI Spg (Q) vs SL2(Q), Hg vs H, clean Fourier expansion indexed
by quadratic formsI Θ series count number of ways of representing one quadratic
form by another
I My interestsI Hecke theory computations for g = 2I Compute forms for g = 3
Nathan C. Ryan A Motivated Introduction to Modular Forms
Function Field Modular Forms
I ComparisonI H = PSL2(K∞)/PSL2(O∞), transform like Θ functions, finite
Fourier expansion index by integersI Θ series count number of ways of representing one polynomial
over Fq by another
I My interestsI I have developed a Hecke TheoryI Asymptotics for representation numbers
Nathan C. Ryan A Motivated Introduction to Modular Forms