The double coset operator

Hecke Operators

5.1 The double coset operator

5

Γ₁αΓ₂={γ₁αγ₂:γ₁∈Γ₁, γ₂∈Γ₂}

is adouble cosetin GL⁺₂(Q). Under a definition to be developed in this section, such double cosets transform modular forms with respect toΓ₁into modular forms with respect toΓ₂.

The group Γ₁ acts on the double cosetΓ₁αΓ₂by left multiplication, par- titioning it into orbits. A typical orbit isΓ1β with representativeβ=γ1αγ2, and the orbit spaceΓ1\Γ1αΓ2is thus a disjoint union

Γ1βj for some choice of representativesβj. The next two lemmas combine to show that this union is finite.

Lemma 5.1.1.Let Γ be a congruence subgroup of SL₂(Z) and let α be an element ofGL⁺₂(Q). Thenα⁻¹Γ α∩ SL₂(Z)is again a congruence subgroup ofSL₂(Z).

Proof. There exists ˜N ∈ Z⁺ satisfying the conditions Γ( ˜N) ⊂ Γ, ˜N α ∈ M2(Z), ˜N α⁻¹∈M2(Z). SetN = ˜N³. The calculation

αΓ(N)α⁻¹⊂α(I+ ˜N³M2(Z))α⁻¹

=I+ ˜N·N α˜ ·M2(Z)·N α˜ ⁻¹⊂I+ ˜NM2(Z) and the observation thatαΓ(N)α⁻¹consists of determinant-1 matrices com- bine to show thatαΓ(N)α⁻¹⊂Γ( ˜N). Thus Γ(N)⊂α⁻¹Γ( ˜N)α⊂α⁻¹Γ α, and intersecting with SL2(Z) completes the proof.

Lemma 5.1.2.Let Γ₁ andΓ₂ be congruence subgroups ofSL₂(Z), and let α be an element ofGL⁺₂(Q). Set Γ₃ =α⁻¹Γ₁α ∩ Γ₂, a subgroup of Γ₂. Then left multiplication byα,

Γ₂−→Γ₁αΓ₂ given by γ₂→αγ₂,

induces a natural bijection from the coset space Γ3\Γ2 to the orbit space Γ₁\Γ₁αΓ₂. In concrete terms,{γ_2,j}is a set of coset representatives forΓ₃\Γ₂ if and only if{β_j}={αγ_2,j} is a set of orbit representatives for Γ₁\Γ₁αΓ₂. Proof. The mapΓ₂−→Γ₁\Γ₁αΓ₂takingγ₂toΓ₁αγ₂clearly surjects. It takes elementsγ2, γ₂ to the same orbit whenΓ1αγ2=Γ1αγ₂, i.e.,γ₂γ₂⁻¹∈α⁻¹Γ1α, and of courseγ₂γ⁻₂¹∈Γ2 as well. So the definitionΓ3 =α⁻¹Γ1α ∩ Γ2 gives a bijection Γ3\Γ2 −→ Γ1\Γ1αΓ2 from cosets Γ3γ2 to orbits Γ1αγ2. The last

statement of the lemma follows immediately.

Any two congruence subgroupsG₁ andG₂ of SL₂(Z) arecommensurable, meaning that the indices [G₁:G₁∩G₂] and [G₂:G₁∩G₂] are finite (Exer- cise 5.1.2). In particular, sinceα⁻¹Γ₁α ∩ SL₂(Z) is a congruence subgroup of SL₂(Z) by Lemma 5.1.1, the coset space Γ₃\Γ₂ in Lemma 5.1.2 is finite and hence so is the orbit spaceΓ₁\Γ₁αΓ₂. With finiteness of the orbit space established, the double cosetΓ₁αΓ₂can act on modular forms. Recall that for

β∈GL⁺₂(Q) andk∈Z, theweight-k β operatoron functionsf :H −→C is given by

(f[β]k)(τ) = (detβ)^k−1j(β, τ)^−kf(β(τ)), τ ∈ H.

Definition 5.1.3.For congruence subgroups Γ₁ and Γ₂ of SL₂(Z) and α ∈ GL⁺₂(Q), the weight-k Γ₁αΓ₂ operator takes functions f ∈ Mk(Γ₁)to

f[Γ₁αΓ₂]_k=

j

f[β_j]_k where {β_j} are orbit representatives, i.e., Γ₁αΓ₂ =

jΓ₁β_j is a disjoint union.

The double coset operator is well defined, i.e., it is independent of how theβ_j are chosen (Exercise 5.1.3). Seeing that it takes modular forms with respect toΓ₁ to modular forms with respect toΓ₂,

[Γ₁αΓ₂]_k:Mk(Γ₁)−→ Mk(Γ₂),

means showing that for eachf ∈ M^k(Γ1), the transformedf[Γ1αΓ2]k is Γ2- invariant and is holomorphic at the cusps. Seeing that the double coset oper- ator takes cusp forms to cusp forms,

[Γ1αΓ2]k :Sk(Γ1)−→ Sk(Γ2),

means showing that for eachf ∈ Sk(Γ1), the transformedf[Γ1αΓ2]k vanishes at the cusps.

To show invariance, note that any γ2 ∈ Γ2 permutes the orbit space Γ₁\Γ₁αΓ₂ by right multiplication. That is, the map γ₂ : Γ₁\Γ₁αΓ₂ −→

Γ₁\Γ₁αΓ₂ given by Γ₁β → Γ₁βγ₂ is well defined and bijective. So if {β_j} is a set of orbit representatives for Γ₁\Γ₁αΓ₂ then {β_jγ₂} is a set of orbit representatives as well. Thus

(f[Γ₁αΓ₂]_k)[γ₂]_k =

j

f[β_jγ₂]_k =f[Γ₁αΓ₂]_k, andf[Γ₁αΓ₂]_k is weight-kinvariant underΓ₂as claimed.

To show holomorphy at the cusps, first note that for any f ∈ M^k(Γ1) and for anyγ ∈GL⁺₂(Q), the functiong =f[γ]k is holomorphic at infinity, meaning it has a Fourier expansion

g(τ) =

n≥0

an(g)e^{2πinτ /h}

for some period h ∈ Z⁺ (this was Exercise 1.2.11(b)). Second, note that if functionsg₁, ..., g_d :H −→C are holomorphic at infinity, meaning that each g_j has a Fourier expansion

gj(τ) =

n≥0

an(gj)e^{2πinτ /hj},

then so is their sumg1+· · ·+gd (Exercise 5.1.4). For any δ∈ SL2(Z), the function (f[Γ1αΓ2]k)[δ]k is a sum of functions gj =f[γj]k with γj =βjδ ∈ GL⁺₂(Q), so it is holomorphic at infinity by the two facts just noted. Since δ is arbitrary this is the condition for holomorphy at the cusps.

For anyf ∈ Sk(Γ1) and for anyγ∈GL⁺₂(Q), the functiong=f[γ]k van- ishes at infinity (this was also Exercise 1.2.11(b)), and the previous paragraph now shows thatf[Γ₁αΓ₂]_k ∈ Sk(Γ₂), i.e., the double coset operator takes cusp forms to cusp forms as claimed.

Special cases of the double coset operator [Γ₁αΓ₂]_k arise when

(1) Γ₁⊃Γ₂. Takingα=Imakes the double coset operator bef[Γ₁αΓ₂]_k =f, the natural inclusion of the subspaceMk(Γ₁) inMk(Γ₂), an injection.

(2) α⁻¹Γ1α=Γ2. Here the double coset operator is f[Γ1αΓ2]_k =f[α]_k, the natural translation from Mk(Γ1) toMk(Γ2), an isomorphism.

(3) Γ1⊂Γ2. Takingα=Iand letting{γ2,j}be a set of coset representatives for Γ1\Γ2 makes the double coset operator bef[Γ1αΓ2]k =

jf[γ2,j]k, the natural trace map that projects Mk(Γ1) onto its subspace Mk(Γ2) by symmetrizing over the quotient, a surjection.

In fact, any double coset operator is a composition of these. Given Γ1, Γ2, andα, setΓ3=α⁻¹Γ1α∩Γ2as usual and setΓ₃ =αΓ3α⁻¹=Γ1 ∩αΓ2α⁻¹. Then Γ₁ ⊃ Γ₃ and α⁻¹Γ₃α =Γ₃ and Γ₃ ⊂Γ₂, giving the three cases. The corresponding composition of double coset operators is

f →f →f[α]_k →

j

f[αγ_2,j]_k, which by Lemma 5.1.2 is the general [Γ₁αΓ₂]_k.

The process of transferring functions forward from Mk(Γ₁) to Mk(Γ₂) by the double coset operator also has a geometric interpretation in terms of transferring points back between the corresponding modular curves. This leads to an algebraic interpretation of the double coset as a homomorphism of divisor groups. Recall that every congruence subgroupΓ has a modular curve X(Γ) =Γ\H^∗consisting of orbitsΓ τ. The configuration of groups is

Γ3 ∼ //

Γ₃

Γ2 Γ1

where the group isomorphism is γ → αγα⁻¹ and the vertical arrows are inclusions. The corresponding configuration of modular curves is

X₃ ^∼ //

π2

X₃

π1

X2 X1

(5.1)

where the modular curve isomorphism is Γ3τ → Γ₃α(τ), denoted α (Exer- cise 5.1.5). Again lettingΓ3\Γ2=

jΓ3γ2,j andβj=αγ2,j for eachjso that Γ1αΓ2 =

jΓ1βj, each point ofX2 is taken back by π1◦α◦π₂⁻¹ to a set of points ofX1,

{Γ₃γ_2,j(τ)} ^α //{Γ₃β_j(τ)}

π1

Γ_2τ

π⁻₂¹

OO

{Γ₁β_j(τ)}.

Here π₂⁻¹ takes each point x ∈ X2 to the multiset (meaning elements can repeat) of overlying points y ∈ X3 each with multiplicity according to its ramification degree, π⁻₂¹(x) = {ey·y : y ∈ X3, π2(y) = x}. To place the composition in the right environment for counting with multiplicity let Div(X) denote the divisor group of any modular curveX, where as in Chapter 3 the divisor group of a set is the free Abelian group on its points. In terms of divisors the composition is

[Γ₁αΓ₂]_k :X₂−→Div(X₁), Γ₂τ→

j

Γ₁β_j(τ),

and this has a uniqueZ-linear extension to a divisor group homomorphism, [Γ₁αΓ₂]_k : Div(X₂)−→Div(X₁).

In this context the special casesΓ1 ⊃Γ2, α⁻¹Γ1α= Γ2, and Γ1 ⊂Γ2 from before lead respectively to a surjection, an isomorphism, and an injection (Exercise 5.1.6).

This chapter will focus on double coset operators acting on modular forms, but the emphasis later in the book will move toward the divisor group inter- pretation.

Exercises

5.1.1.In Lemma 5.1.1, needα⁻¹Γ αlie in SL₂(Z)?

5.1.2.Show that for congruence subgroupsG₁andG₂of SL₂(Z), the indices [G₁:G₁∩G₂] and [G₂:G₁∩G₂] are finite. (A hint for this exercise is at the end of the book.)

5.1.3.Show that the weight-kdouble coset operator [Γ₁αΓ₂]_k is independent of how the orbit representativesβ_j are chosen. (A hint for this exercise is at the end of the book.)

5.1.4.Show that ifg₁, ..., g_d:H −→C are holomorphic at infinity, then so is their sumg1+· · ·+g_d. (A hint for this exercise is at the end of the book.) 5.1.5.Check that the map α : X₃ −→ X₃ given by Γ₃τ → Γ₃α(τ) is well defined.

5.1.6.Show that in the special casesΓ1⊃Γ2,α⁻¹Γ1α=Γ2, andΓ1⊂Γ2the divisor group interpretation of [Γ1αΓ2]k gives a surjection, an isomorphism, and an injection.