The d and T p operators

Hecke Operators

5.2 The d and T p operators

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.

5.2 ThedandTpoperators 169 df =f[α]k for any α=

a b c δ

∈Γ0(N) withδ≡d(mod N).

This is the first type of Hecke operator, also called adiamond operator. For any character χ : (Z/NZ)^∗ −→ C, the space Mk(N, χ) from Chapter 4 is precisely theχ-eigenspace of the diamond operators,

Mk(N, χ) ={f ∈ Mk(Γ₁(N)) :df =χ(d)f for alld∈(Z/NZ)^∗}. That is, the diamond operatordrespects the decompositionMk(Γ₁(N)) =

χMk(N, χ), operating on the eigenspace associated to each characterχas multiplication byχ(d).

The second type of Hecke operator is also a weight-kdouble coset operator [Γ1αΓ2]k where againΓ1=Γ2=Γ1(N), but now

α= 1 0

0p

, pprime.

This operator is denotedTp. Thus

Tp:Mk(Γ1(N))−→ Mk(Γ1(N)), pprime is given by

Tpf =f[Γ1(N) 1 0

0p

Γ1(N)]k. From (3.16), the double coset here is

Γ₁(N) 1 0

0p

Γ₁(N) =

γ∈M₂(Z) :γ≡ 1∗

0p

(modN), detγ=p , so in fact _{1 0}

0p

can be replaced by any matrix in this double coset in the definition ofTp.

The two kinds of Hecke operator commute. To see this, continue to let α=_{1 0}

0p

and check thatγαγ⁻¹≡_1∗

0p

(mod N) for anyγ∈Γ0(N). If

Γ₁(N)αΓ₁(N) =4

j

Γ₁(N)β_j

then the last sentence of the preceding paragraph and the fact thatΓ₁(N) is normal inΓ₀(N) show that the double coset is also

Γ1(N)αΓ1(N) =Γ1(N)γαγ⁻¹Γ1(N) =γΓ1(N)αΓ1(N)γ⁻¹

=γ4

j

Γ₁(N)β_jγ⁻¹=4

j

Γ₁(N)γβ_jγ⁻¹. Comparing the two decompositions of the double coset gives

jΓ₁(N)γβ_j=

jΓ₁(N)β_jγ, even though it need not be true that Γ₁(N)γβ_j =Γ₁(N)β_jγ

for eachj. Thus for anyf ∈ Mk(Γ₁(N)) and anyγ∈Γ₀(N) with lower right entryδ≡d(modN),

dTpf =

j

f[βjγ]k=

j

f[γβj]k=Tpdf, f ∈ Mk(Γ1(N)).

To find an explicit representation ofTp, recall that it is specified by orbit representatives for Γ₁\Γ₁αΓ₂, and these are coset representatives for Γ₃\Γ₂ left multiplied byα, whereΓ₃=α⁻¹Γ₁α∩Γ₂. For the particularΓ₁,Γ₂, and αin play here, recall from Exercise 1.5.6 the groups

Γ⁰(p) = a b

c d

∈SL₂(Z) : a b

c d

≡ ∗0

∗ ∗

(mod p) and

Γ₁⁰(N, p) =Γ1(N)∩Γ⁰(p).

ThenΓ3=Γ₁⁰(N, p) (Exercise 5.2.1). SinceΓ3isΓ2 subject to the additional conditionb≡0 (modp), the obvious candidates for coset representatives are

γ_2,j = 1j

0 1

, 0≤j < p.

Given γ2 = _{a b}

c d

∈ Γ2, we have γ2 ∈ Γ3γ2,j if γ2γ_2,j⁻¹ ∈ Γ3 = Γ2∩Γ⁰(p).

Certainly γ2γ_2,j⁻¹ ∈Γ2 for any j since γ2 and γ2,j are, but also we need the upper right entryb−jaofγ2γ_2,j⁻¹=_{a b}

c d 1−j 0 1

to be 0 (modp).

If p a then setting j = ba⁻¹ (modp) does the job. But if p | a then b−ja can’t be 0 (modp) for any j, for then p | b and sop | ad−bc= 1.

Instances of γ2 ∈ Γ2 with p | a occur if and only if p N, and when this happensγ_2,0, . . . , γ_2,p−1 fail to represent Γ₃\Γ₂. To complete the set of coset representatives in this case, set

γ_2,∞= mp n

N 1

wheremp−nN = 1.

Now givenγ2 = _{a b}

c d

∈Γ2 with p| a, it is easy to show thatγ2γ⁻_2,∞¹ ∈ Γ3

as needed (Exercise 5.2.2). Thus γ2,0, . . . , γ2,p−1 are a complete set of coset representatives whenp|N, but γ2,∞ is required as well whenpN. In any case, it is easy to show that theγ2,j represent distinct cosets (Exercise 5.2.3).

The reader may recognize these matrices from Exercise 1.5.6(b).

The corresponding orbit representativesβj=αγ2,j forΓ1\Γ1αΓ2, needed to compute the double coset operator, work out to

β_j = 1j

0p

for 0≤j < p, β_∞= m n

N p p0 0 1

ifpN. (5.2) This proves

5.2 ThedandTpoperators 171 Proposition 5.2.1.Let N ∈ Z⁺, letΓ₁ = Γ₂ =Γ₁(N), and let α= _{1 0}

0p

wherepis prime. The operatorT_p= [Γ1αΓ2]_k onMk(Γ1(N))is given by

Tpf =

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

p−1

j=0

f[ 1j

0p

]k ifp|N,

p−1

j=0

f[ 1j

0p

]k+f[[^{m n}N p]_p0

0 1

]k ifpN, wheremp−nN = 1.

LettingΓ₁ =Γ₂ =Γ₀(N) instead and keeping α=_{1 0}

0p

gives the same orbit representatives forΓ₁\Γ₁αΓ₂ (Exercise 5.2.4), but in this case the last representative can be replaced byβ_∞=_p0

0 1

since [^{m n}_{N p}]∈Γ₁. The next result describes the effect ofT_p on Fourier coefficients.

Proposition 5.2.2.Letf ∈ Mk(Γ1(N)). Since[^{1 1}_{0 1}]∈Γ1(N),f has period 1 and hence has a Fourier expansion

f(τ) = ∞ n=0

an(f)qⁿ, q=e^2πiτ. Then:

(a)Let 1N : (Z/NZ)^∗ −→ C^∗ be the trivial character modulo N. Then Tpf has Fourier expansion

(T_pf)(τ) = ∞ n=0

a_np(f)qⁿ+1_N(p)p^k−1 ∞ n=0

a_n(pf)q^np

= ∞ n=0

5a_np(f) +1_N(p)p^k−1a_n/p(pf)6 qⁿ. That is,

an(Tpf) =anp(f) +1N(p)p^k−1a_n/p(pf) forf ∈ Mk(Γ1(N)). (5.3) (Here a_n/p = 0 when n/p /∈N. As in Chapter 4, 1_N(p) = 1 whenpN and1_N(p) = 0whenp|N.)

(b)Let χ : (Z/NZ)^∗ −→ C^∗ be a character. If f ∈ Mk(N, χ) then also T_pf ∈ Mk(N, χ), and now its Fourier expansion is

(Tpf)(τ) = ∞ n=0

anp(f)qⁿ+χ(p)p^k−1 ∞ n=0

an(f)q^np

= ∞ n=0

5a_np(f) +χ(p)p^k−1a_n/p(f)6 qⁿ. That is,

an(Tpf) =anp(f) +χ(p)p^k−1a_n/p(f) forf ∈ Mk(N, χ). (5.4)

Proof. For part (a), take 0≤j < pand compute f[

1j 0p

]k(τ) =p^k−1(0τ+p)^−kf τ+j

p

=1 p

∞ n=0

an(f)e2πin(τ+j)/p

= 1 p

∞ n=0

a_n(f)qⁿ_pµ^nj_p

whereqp =e^{2πiτ /p} andµp =e^2πi/p. Since the geometric sum p−1

j=0µ^nj_p isp whenp|N and 0 whenpN, summing over j gives

p−1

j=0

f[ 1j

0p

]k(τ) =

n≡0 (p)

an(f)qⁿ_p = ∞ n=0

anp(f)qⁿ.

This is (Tpf)(τ) whenp|N. WhenpN, (Tpf)(τ) also includes the term f[[^{m n}N p]_p0

0 1

]k(τ) = (pf)[_p0

0 1

]k(τ)

=p^k−1(0τ+ 1)^−k(pf)(pτ) =p^k−1 ∞ n=0

an(pf)q^np. In either case, the Fourier series ofTpf is as claimed.

The first statement of (b) follows from the relation d(Tpf) =Tp(df).

Formula (5.4) is immediate from (5.3).

Since double coset operators take cusp forms to cusp forms, the Tp op- erator restricts to the subspaceSk(Γ1(N)) of Mk(Γ1(N)). In particular the weight 12 cusp form∆, the discriminant function from Chapter 1, is an eigen- vector of the Tp operators for SL2(Z) since S12(SL2(Z)) is 1-dimensional, and similarly for the generator of any other 1-dimensional space such as the functionsϕk(τ) =η(τ)^kη(N τ)^kfrom Proposition 3.2.2. Part (b) of the propo- sition here shows thatTp further restricts to the subspaceSk(N, χ) for every characterχmoduloN.

Eisenstein series are also eigenvectors of the Hecke operators. Recall the seriesE_k^ψ,ϕfrom Chapter 4 for any pairψ,ϕof Dirichlet characters modulou andvsuch that uv|N and (ψϕ)(−1) = (−1)^k,

E_k^ψ,ϕ(τ) =δ(ψ)L(1−k, ϕ) + 2 ∞ n=1

σ^ψ,ϕ_k−1(n)qⁿ, q=e^2πiτ,

δ(ψ) =

1 ifψ=11,

0 otherwise, σ^ψ,ϕ_k−1(n) =

m|n m>0

ψ(n/m)ϕ(m)m^k−1.

Recall also the seriesE_k^ψ,ϕ,t(τ) =E_k^ψ,ϕ(tτ) where ψandϕare primitive and t is a positive integer such that tuv | N. From Chapter 4, as (ψ, ϕ, t) runs

5.2 ThedandTpoperators 173 through a set of such triples such thatψϕ = χ at level N these Eisenstein series represent a basis of Ek(N, χ). (At weight k = 2 when ψ = ϕ = 11

the definition is different,E₂^11,11,t(τ) =E₂^11,11(τ)−tE₂^11,11(tτ), so the triple (11,11,1) contributes nothing and is excluded.) By definition ofMk(N, χ) as an eigenspace,dE_k^ψ,ϕ,t=χ(d)E_k^ψ,ϕ,tfor alldrelatively prime toN, but also Proposition 5.2.3.LetχmoduloN,ψ,ϕ, andtbe as above. Letpbe prime.

Excluding the casek= 2,ψ=ϕ=1₁,

T_pE_k^ψ,ϕ,t= (ψ(p) +ϕ(p)p^k−1)E_k^ψ,ϕ,t if uv=N or if pN . Also,

TpE₂^11,11,t= (1 +1N(p)p)E₂^11,11,t if tis prime andN is a power of t or if pN.

This is a direct calculation (Exercise 5.2.5). Whenuv < N (excluding the special casek = 2,ψ =ϕ=1₁) the series E^ψ,ϕ,t_k is “old” at level N in the sense that it comes from a lower level, being the seriesE_k^ψ,ϕat leveluvraised to leveltuv by multiplying the variable bytand then viewed at levelN since tuv | N. On the other hand, when uv =N (or in the special case if also t is prime andN =t) the series is “new” at levelN, not arising from a lower level. The proposition shows that Eisenstein series at levelN are eigenvectors for the Hecke operators away from the level, and new Eisenstein series at level N are eigenvectors for all the Hecke operators. We will see this same phenomenon for cusp forms later in the chapter.

The Hecke operators commute.

Proposition 5.2.4.Let dandebe elements of (Z/NZ)^∗, and letpandq be prime. Then

(a)dT_p=T_pd,

(b)de=ed=de, (c)T_pT_q=T_qT_p.

Proof. Part (a) has already been shown. Since the dandTp operators pre- serve the decompositionMk(Γ₁(N)) =

Mk(N, χ), it suffices to check (b) and (c) on an arbitraryf ∈ Mk(N, χ). Now (b) is immediate (Exercise 5.2.6).

As for (c), applying formula (5.4) twice gives an(Tp(Tqf)) =anp(Tqf) +χ(p)p^k−1a_n/p(Tqf)

=anpq(f) +χ(q)q^k−1a_np/q(f)

+χ(p)p^k−1(anq/p(f) +χ(q)q^k−1an/pq(f))

=a_npq(f) +χ(q)q^k−1a_np/q(f) +χ(p)p^k−1a_nq/p(f) +χ(pq)(pq)^k−1a_n/pq(f),

and this is symmetric inpandq.

The modular curve interpretation ofT_p as at the end of Section 5.1 is T_p: Div(X₁(N))−→Div(X₁(N)), Γ₁(N)τ →

j

Γ₁(N)β_j(τ), (5.5) with the matricesβj from (5.2), excludingβ_∞whenp|N. (Strictly speaking the map on the left is theZ-linear extension of the description on the right, whose domain is onlyX₁(N), but we use this notation from now on without comment.) There is a corresponding interpretation ofT_p in terms of the mod- uli space S₁(N) from Section 4 of Chapter 1. Recall that S₁(N) consists of equivalence classes of enhanced elliptic curves (E, Q) where E is a complex elliptic curve andQis a point of orderN. Let Div(S₁(N)) denote its divisor group. Then the moduli space interpretation is

Tp: Div(S1(N))−→Div(S1(N)), [E, Q]→

C

[E/C, Q+C], (5.6) where the sum is taken over all orderpsubgroupsC⊂Esuch thatC∩ Q= {0_E} and where the square brackets denote equivalence class. To see where this comes from, recall from Chapter 1 the lattice Λ_τ = τZ⊕Z and the complex elliptic curveE_τ=C/Λ_τ forτ ∈ H. Associate to eachβ_j (including β_∞ when pN) a subgroup C=C_j =cΛ_βj(τ) ofE_τ wherec ∈C, working out toC=(τ+j)/p+Λτ for 0≤j < pand toC=1/p+Λτ forj =∞ (Exercise 5.2.7(a)) and therefore satisfying the conditions

C∼=Z/pZinE_τ, C∩(1/N+Λ_τ) ={0}in E_τ. (5.7) The groupsC0, . . . ,Cp−1, C_∞ are subgroups of Eτ[p] and pairwise disjoint except for 0, making their union a subset ofEτ[p] totaling 1 + (p+ 1)(p−1) = p² elements, i.e., their union is all of Eτ[p]. (See Figure 5.1 and Exer- cise 5.2.7(b,c).) Any subgroupC of Eτ satisfying the first condition of (5.7) must lie in Eτ[p], making it one of the Cj. The groupC_∞ fails the second condition of (5.7) whenp|N. Thus the matrices β appearing inTp describe the subgroups C in the moduli space interpretation (5.6). This discussion elaborates on Exercise 1.5.6(a).

The moduli space S₁(N) is in bijective correspondence with the noncom- pact modular curveY₁(N) = Γ₁(N)\H by Theorem 1.5.1. The relation be- tween descriptions (5.5) and (5.6) ofT_p is summarized in the following com- mutative diagram (Exercise 5.2.7(d)):

D(S1(N)) ^Tp //

ψ1

D(S1(N))

ψ1

D(Y1(N)) ^Tp //D(Y1(N)).

(5.8)

Here the vertical map ψ₁ is the bijection from Theorem 1.5.1. Elementwise the mappings are

5.2 ThedandTpoperators 175

N +C_∞] works out to [Epτ,_N^p +Λpτ] (Exer- cise 5.2.7(e)).

Similarly and more easily for the diamond operator, there is a commutative diagram

S1(N) ^d //

ψ1

S1(N)

ψ1

Y1(N) ^d //Y1(N),

(5.9)

where ifα=_{a b}

c δ

∈Γ0(N) withδ≡d(mod N) then the mappings are given by

[E_τ,_N¹ +Λ_τ] //

_

[Eτ,_N^d +Λτ] _

Γ1(N)τ //Γ1(N)α(τ).

This was Exercise 1.5.3. Naturally,dextendsZ-linearly to divisors.

To summarize, we have four compatible notions of the Hecke operatorTp, starting from the double cosetΓ1(N)_{1 0}

0p

Γ1(N). This works out to

γ∈M2(Z) :γ≡ 1∗

0p

(mod N), detγ=p ,

similar to the definitions of congruence subgroups by conditions on integer matrices. The double coset gives the second version ofT_p, a linear operator

Τ

1 0

Figure 5.1.Eτ[5] as a union of six 5-cyclic subgroups

[E_τ,_N¹ +Λ_τ] //

_

C[E_τ/C,_N¹ +C]

_

Γ1(N)τ //

jΓ₁(N)β_j(τ).

WhenpN the term [Eτ/C_∞, ¹

on the space of modular formsMk(Γ₁(N)), Tp:f →

j

f[βj]k, whereΓ1(N)_{1 0}

0p

Γ1(N) =

jΓ1(N)βj. ThusTpf(τ) evaluatesf at a set of points associated toτ. In terms of points themselves, a third version ofTp is the endomorphism of the divisor group of the modular curveX1(N) induced by

Tp:Γ1(N)τ→

j

Γ1(N)βj(τ).

By the general configuration of the double coset operator from Section 5.1, this lifts each point of X1(N) to its overlying points on the modular curve X3=X₁⁰(N, p) of the groupΓ3=Γ₁⁰(N, p) =Γ1(N)∩Γ⁰(p), translates them to another modular curve X₃ over X1(N) by dividing them byp (since the definition uses the matrix_{1 0}

0p

), and then projects them back down. That is, Tp factors as

Tp:Γ1(N)τ →

j

Γ3γ2,j(τ)→

j

Γ1(N)γ2,j(τ)/p, where _{1 0}

0p

γ2,j = βj for each j (see Exercise 5.2.9). The net effect is to take a point at level N to a formal sum of level N points associated to it in a manner depending onp. Section 6.3 will revisit this description of Tp, and Exercise 7.9.3 will show thatX₃ is the modular curveX_1,0(N, p) of the group Γ_1,0(N, p) = Γ₁(N)∩Γ₀(N p). Fourth, T_p is an endomorphism of the divisor group of the moduli space S₁(N), induced by

Tp: [E, Q]→

C

[E/C, Q+C].

Similarly to the third version (recall (1.11) and see Exercise 5.2.9), this factors as

T_p: [E, Q]→

C

[E, C, Q]→

C

[E/C, Q+C].

Exercises

5.2.1.Show that when Γ1 = Γ2 = Γ1(N) and α = _{1 0}

0p

, the group Γ3 = α⁻¹Γ1α ∩Γ2works out to Γ₁⁰(N, p).

5.2.2.Whenγ2=_{a b}

c d

withp|a, show thatγ2γ_2,⁻¹_∞∈Γ3as needed.

5.2.3.Show that theγ_2,j represent distinct cosets.

5.2.4.Show that lettingΓ₁ =Γ₂ =Γ₀(N) and keeping α=_{1 0}

0p

gives the same orbit representatives{β_j}forΓ₁\Γ₁αΓ₂.

5.2 ThedandTpoperators 177 5.2.5.This exercise proves Proposition 5.2.3.

(a) Show that the generalized divisor sum σ_k^ψ,ϕ₋₁ is multiplicative, i.e., σ_k^ψ,ϕ₋₁(nm) =σ_k^ψ,ϕ₋₁(n)σ^ψ,ϕ_k−1(m) when gcd(n, m) = 1.

(b) Let pbe prime and letn≥1. Writen=np^e withpn ande≥0.

Use part (a) to show that

σ^ψ,ϕ_k−1(np) =ψ(p)σ^ψ,ϕ_k−1(n) +ϕ(p^e+1)(p^e+1)^k−1σ_k^ψ,ϕ₋₁(n).

Also use part (a) to show that whene≥1 andpN,

χ(p)p^k−1σ^ψ,ϕ_k−1(n/p) =ϕ(p)p^k−1σ_k^ψ,ϕ₋₁(n)−ϕ(p^e+1)(p^e+1)^k−1σ_k^ψ,ϕ₋₁(n).

Use the two formulas to show that for alle≥0 andpN,

σ^ψ,ϕ_k−1(np) +χ(p)p^k−1σ^ψ,ϕ_k−1(n/p) = (ψ(p) +ϕ(p)p^k−1)σ^ψ,ϕ_k−1(n).

Show that this formula also holds for alle≥0 andp|uv.

(c) Excluding the casek= 2,ψ=ϕ=11, use part (b) and formula (5.4) to show that ifuv=N orpN then

a_n(T_pE_k^ψ,ϕ,t) = (ψ(p) +ϕ(p)p^k−1)a_n(E^ψ,ϕ,t_k ), n≥1.

Show the same result forn= 0. (Hints for this exercise are at the end of the book.)

(d) Complete the proof by computing a_n(E₂^11,11,t) anda_n(T_pE¹₂^1,11,t) for alln.

5.2.6.Verify part (b) of Proposition 5.2.4.

5.2.7.(a) Show that for each τ ∈ H and each matrix β_j = 1j

0p

where 0 ≤j < p, the groupC =Λ_βj(τ) is(τ+j)/p+Λτ. Show that this group satisfies conditions (5.7). IfpN show that for eachτ ∈ H and the matrix β_∞ = [^{m n}N p]_p0

0 1

, the groupC =C_∞ = (N τ + 1)Λβ_∞(τ) is 1/p+Λτ and this group satisfies conditions (5.7). (A hint for this exercise is at the end of the book.)

(b) How do the subgroups ofE_τ[5] in Figure 5.1 correspond to the matrices β₀, . . . , β₅, β_∞?

(c) As an alternative to the counting argument in the section, consider any subgroupC⊂E_τ satisfying (5.7). By the first condition [E_τ[p] :C] =pand so a basis ofC is

β

τ /p+Λτ

1/p+Λ_τ

, β ∈M2(Z), det(β) =p.

Two such matricesβ andβ specify the same group exactly whenβ =γβfor someγ∈SL₂(Z). Ifβ=_{a b}

c d

then gcd(a, c) is 1 orp. Show that repeated left multiplication by elements of SL₂(Z) to carry out the Euclidean algorithm on

aand c and then further left multiplication if necessary reduceβ to exactly one ofβ0, . . . ,β_p−1,β_∞.

(d) Show diagram (5.9) commutes by showing thatψ1([E/Cj,1/N+Cj]) = Γ1(N)βj(τ) for allj, includingj=∞whenpN.

(e) Show that [Eτ/C_∞,_N¹ +C_∞] = [Epτ,_N^p +Λpτ] whenpN.

5.2.8.Evaluate T3(SL2(Z)i) ∈ Div(SL2(Z)\H^∗), viewing T3 in the sense of (5.5) and keeping an eye out for ramification. (A hint for this exercise is at the end of the book.)

5.2.9.Specializing the general double coset operator to the data forTp, i.e., Γ1 =Γ2 =Γ1(N) andα=_{1 0}

0p

, so that Γ3 =Γ₁⁰(N, p), the corresponding modular curveX3 is denotedX₁⁰(N, p). Show that in the configuration (5.1) the mapsπ2 andπ1◦αfromX₁⁰(N, p) toX1(N) are

π2(Γ₁⁰(N, p)τ) =Γ1(N)τ, (π1◦α)(Γ₁⁰(N, p)τ) =Γ1(N)τ /p.

The moduli space versions of these maps were given in (1.11) and derived in Exercise 1.5.6. Show that their descriptions and the commutative diagram

S1(N)

ψ1

S⁰₁(N, p)

oo //

ψ⁰₁

S1(N)

ψ1

X₁(N)oo ^π2 X₁⁰(N, p) ^π1◦α//X₁(N) give another derivation of (5.6).