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