Atkin and Swinnerton-Dyer Congruences onNoncongruence Modular Forms

ICERMApril 18, 2013

Winnie Li

Pennsylvania State University, U.S.A.and

National Center for Theoretical Sciences, Taiwan1

Noncongruence subgroups

• Bass-Lazard-Serre: All finite index subgroups of SLn(Z) forn ≥ 3 are congruence subgroups.

• SL2(Z) contains far more noncongruence subgroups than con-gruence subgroups.

• Let Γ be a finite index subgroup of SL2(Z). The orbit spaceΓ\H∗ is a Riemann surface, called the modular curve XΓ forΓ. It has a model defined over a number field.

• The modular curves for congruence subgroups are defined overQ or cyclotomic fields Q(ζN ).

• Belyi: Every smooth projective irreducible curve defined over anumber field is isomorphic to a modular curve XΓ (for infinitelymany finite-index subgroups Γ of SL2(Z)).

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Modular forms for congruence subgroups

Let g =∑n≥1 an(g)qn, where q = e2πiz, be a normalized

(a1(g) = 1) newform of weight k ≥ 2 level N and characterχ.

I. Hecke theory

• It is an eigenfunction of the Hecke operators Tp with eigenvalueap(g) for all primes p - N , i.e., for all n ≥ 1,

anp(g)− ap(g)an(g) + χ(p)pk−1an/p(g) = 0.

• The space of weight k cusp forms for a congruence subgroupcontains a basis of forms with algebraically integral Fourier co-efficients. An algebraic cusp form has bounded denominators.

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II. Galois representations

• (Eichler-Shimura, Deligne) There exists a compatible family ofdegree two l-adic rep’ns ρg,l of GQ := Gal(Q̄/Q) such that atprimes p - lN , the char. poly.

Hp(T ) = T 2 − ApT + Bp = T 2 − ap(g)T + χ(p)pk−1

of ρg,l(Frobp) is indep. of l, and

anp(g)− Ap an(g) + Bp an/p(g) = 0

for n ≥ 1 and primes p - lN .

• Ramanujan-Petersson conjecture holds for newforms. That is,|ap(g)| ≤ 2p(k−1)/2 for all primes p - N .

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Modular forms for noncongruence subgroups

Γ : a noncongruence subgroup of SL2(Z) with finite index

Sk(Γ) : space of cusp forms of weight k ≥ 2 for Γ of dim d

A cusp form has an expansion in powers of q1/µ.

Assume the modular curve XΓ is defined over Q and the cuspat infinity is Q-rational.

Atkin and Swinnerton-Dyer: there exists a positive integer Msuch that Sk(Γ) has a basis consisting of forms with coeffs. integraloutside M (called M -integral) :

f (z) =∑n≥1

an(f )qn/µ.

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No efficient Hecke operators on noncongruence forms

• Let Γc be the smallest congruence subgroup containing Γ.

Naturally, Sk(Γc) ⊂ Sk(Γ).

• TrΓcΓ : Sk(Γ)→ Sk(Γc) such that Sk(Γ) = Sk(Γc)⊕ker(TrΓc

Γ ).

• ker(TrΓcΓ ) consists of genuinely noncongruence forms in Sk(Γ).

Conjecture (Atkin). The Hecke operators on Sk(Γ) for p -M

defined using double cosets Γ

(1 00 p

)Γ as for congruence forms is

zero on genuinely noncongruence forms in Sk(Γ).

This was proved by Serre, Berger.So the progress has been led by computational data.

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Atkin and Swinnerton-Dyer congruences for ellipticcurves

Let E be an elliptic curve defined over Q with conductor M . ByBelyi, E ' XΓ for a finite index subgroup Γ of SL2(Z).Ex. E : x3 + y3 = z3, Γ is an index-9 noncong. subgp of Γ(2).

Atkin and Swinnerton-Dyer: The normalized holomorphic differ-

ential 1-form f dqq =∑n≥1 anq

ndqq on E satisfies the congruence

relation

anp − [p + 1−#E(Fp)]an + pan/p ≡ 0 mod p1+ordpn (1)

for all primes p -M and all n ≥ 1.

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Sketch of a proof using formal group laws

• The formal power series `(x) :=∑n≥1 anx

n is the formal log ofa formal group law G, which is isomorphic to the group law onthe elliptic curve E in a neighborhood of the identity element.

• The Hasse-Weil L-function of E is

L(s, E) =∏p-M

1

1− bpp−s + p1−2s

∏p|M

1

1− bpp−s=∑n≥1

bnn−s,

in which bp = p + 1−#E(Fp) for p -M .

• Honda: ˜̀(x) :=∑n≥1 bnx

n is the formal log of a formal grouplaw which is strictly isomorphic to G.

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• Hence for p -M , the sequences {an} and {bn} satisfy the samecongruence relation of ASD type:

cnp−Apcn +Bp,1cn/p +Bp,2cn/p2 + · · · ≡ 0 mod p1+ordpn.

• The bn’s satisfy the three term recursion

bnp − bpbn + pbn/p = 0 for all p -M and n ≥ 1.

So Ap = bp, Bp,1 = p, and other Bp,i = 0. This proves (1).

• For p - M , Hp(T ) = T 2 − bpT + p is the characteristic poly.of the Frobp acting on the Tate module Tl(E) for l 6= p.

• Taniyama-Shimura modularity theorem: g =∑n≥1 bnq

n is anormalized congruence newform. Note that f ∈ S2(Γ). Thus(1) gives congruence relations between f and g.

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ASD congruences in general

Back to general case where XΓ has a model over Q, and thed-dim’l space Sk(Γ) has a basis of M -integral forms.

ASD congruences (1971): for each prime p - M , Sk(Γ,Zp)has a p-adic basis {hj}1≤j≤d such that the Fourier coefficients ofhj satisfy a three-term congruence relation

anp(hj)− Ap(j)an(hj) + Bp(j)an/p(hj) ≡ 0 mod p(k−1)(1+ordpn)

for all n ≥ 1. Here

• Ap(j) is an algebraic integer with |Ap(j)| ≤ 2p(k−1)/2, and

• Bp(j) is equal to pk−1 times a root of unity.

This is proved to hold for k = 2 and d = 1 by ASD.

The basis varies with p in general.

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Galois representations attached to Sk(Γ) and Schollcongruences

Theorem[Scholl] Suppose that the modular curve XΓ has amodel over Q such that the cusp at infinity is Q-rational. At-tached to Sk(Γ) is a compatible family of 2d-dim’l l-adic rep’nsρl of GQ unramified outside lM such that for primes p > k+1not dividing Ml, the following hold.

(i) The char. polynomial

Hp(T ) = T 2d + C1(p)T 2d−1 + · · · + C2d−1(p)T + C2d(p)

of ρl(Frobp) lies in Z[T ], is indep. of l, and its roots are alge-

braic integers with complex absolute value p(k−1)/2;

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(ii) For any form f in Sk(Γ) integral outside M , its Fouriercoeffs satisfy the (2d + 1)-term congruence relation

anpd(f ) + C1(p)anpd−1(f ) + · · ·++ C2d−1(p)an/pd−1(f ) + C2d(p)an/pd(f )

≡ 0 mod p(k−1)(1+ordpn)

for n ≥ 1.

Theorem If Sk(Γ) is 1-dimensional, then

• the ASD congruences hold for almost all p;

• the degree two l-adic Scholl rep’ns of GQ are modular.

The 2nd statement follows from Kahre-Wintenberger’s work onSerre’s conjecture on modular rep’ns, and various modularity lift-ing theorems.

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Application: Characterizing noncongruence modularforms

The following conjecture, supported by all known examples,gives a simple characterization for noncongruence forms. If true,it has wide applications.

Conjecture. A modular form in Sk(Γ) with algebraic Fouriercoefficients has bounded denominators if and only if it is a con-gruence modular form, i.e., lies in Sk(Γc).

Theorem[L-Long] The conjecture holds when Sk(Γ) is 1-dim’l,containing a basis with Fourier coefficients in Q.

The proof uses ASD congruences, modularity of the Scholl rep-resentations, and the Selberg bound on Fourier coefficients of a wtk cusp form f : an(f ) = O(nk/2−1/5).

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From Scholl congruences to ASD congruences

Ideally one hopes to factor

Hp(T ) =∏

1≤j≤d(T 2 − Ap(j)T + Bp(j))

and find a p-adic basis {hj}1≤j≤d, depending on p, for Sk(Γ,Zp)such that each hj satisfies the three-term ASD congruence rela-tions given by Ap(j) and Bp(j).

For a congruence subgroup Γ, this is achieved by using Heckeoperators to further break the l-adic space and Sk(Γ) into pieces.For a noncongruence Γ, no such tools are available.

Theorem[Scholl] If ρl(Frobp) is diagonalizable and ordinary(i.e. half of the eigenvalues are p-adic units), then ASD con-gruences at p hold.

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Examples of ASD congruences

The group Γ1(5) has genus 0, no elliptic elements, and 4 cusps.The wt 3 Eisenstein series

E1(z) = 1− 2q1/5 − 6q2/5 + 7q3/5 + 26q4/5 + · · · ,E2(z) = q1/5 − 7q2/5 + 19q3/5 − 23q4/5 + · · · .

have simple zeros at all cusps except ∞ and −2, resp., and non-

vanishing elsewhere. So XΓ1(5) is defined over Q with t = E2E1

as

a Hauptmodul, and tn = n√t is a Hauptmodul of a smooth irred.

modular curve XΓn over Q.

Let ρn,l be the l-adic Scholl representation attached to S3(Γn).

Ex 1. When n = 2, S3(Γ2) =< E1t2 > is 1-dim’l, ASD congru-ences hold for odd p, and ρ2,l are isom. to ρη(4z)6,l.

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Ex 2. (L-Long-Yang)

(1) When n = 3, the space S3(Γ3) =< E1t3, E1t23 > has a basis

f±(z) = q1/15 ± iq2/15 − 11

3q4/15 ∓ i16

3q5/15 −

−4

9q7/15 ± i71

9q8/15 +

932

81q10/15 + · · · .

(2) (Modularity) There are two cuspidal newforms of weight 3 level27 and quadratic character χ−3 given by

g±(z) = q ∓ 3iq2 − 5q4 ± 3iq5 + 5q7 ± 3iq8 +

+9q10 ± 15iq11 − 10q13 ∓ 15iq14 − · · ·such that ρ3,l = ρg+,l ⊕ ρg−,l over Ql(

√−1).

(3) f± satisfy the 3-term ASD congruences with Ap = ap(g±) and

Bp = χ−3(p)p2 for all primes p ≥ 5.

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Ex 3. (Atkin, Li, Long) S3(Γ4) =< h1, h2, h3 >, where hi = E1ti4

and h2 = E1t2 satisfies ASD congruences for all odd p.

The space < h1, h3 > has a basis satisfying the ASD congruencedepending on the residue of odd p mod 8:

1. If p ≡ 1 mod 8, then both h1 and h3 satisfy ASD with Ap =

sgn(p)a1(p) and Bp = p2, where sgn(p) = ±1 ≡ 2(p−1)/4

mod p;

2. If p ≡ 5 mod 8, then h1 (resp. h3) satisfies ASD with Ap =

4ia5(p) (resp. −4ia5(p)) and Bp = −p2;

3. If p ≡ 3 mod 8, then h1±h3 satisfy ASD withAp = ±2√−2a3(p)

and Bp = −p2;

4. If p ≡ 7 mod 8, then h1±ih3 satisfy ASD withAp = ∓8√−2a7(p)

and Bp = −p2.

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Here a1(p), a3(p), a5(p), a7(p) are the Fourier coefficients of thewt 3 congruence forms f1, f3, f5, f7 given below:

f1(z) =η(2z)12

η(z)η(4z)5= q1/8(1 + q − 10q2 + · · · ) =

∑n≥1

a1(n)qn/8,

f3(z) = η(z)5η(4z) = q3/8(1− 5q + 5q2 + · · · ) =∑n≥1

a3(n)qn/8,

f5(z) =η(2z)12

η(z)5η(4z)= q5/8(1 + 5q + 8q2 + · · · ) =

∑n≥1

a5(n)qn/8,

f7(z) = η(z)η(4z)5 = q7/8(1− q − q2 + · · · ) =∑n≥1

a7(n)qn/8.

For the last two examples, both l-adic space and S3(Γn) admitquaternion multiplications, which are used to break the spaces.

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An example of failure of ASD congruences

Ex 4. (Kibelbek) X : y2 = x5 + 1 is a genus 2 curve defined overQ. By Belyi, X ' XΓ for a finite index subgroup Γ of SL2(Z).Put

ω1 =dx

2y= f1

dq1/10

q1/10, ω2 = x

dx

2y= f2

dq1/10

q1/10.

Then S2(Γ) =< f1, f2 >, where

f1 = q1/10 − 8

5q6/10 − 108

52q11/10 +

768

53q16/10 +

3374

54q21/10 + · · · ,

f2 = q2/10 − 16

5q7/10 +

48

52q12/10 +

64

53q17/10 +

724

54q22/10 + · · · .

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The l-adic representations attached to S2(Γ) are the dual of theTate modules on the Jacobian of XΓ. They are strongly ordinaryat p ≡ ±1 mod 5, so ASD congruences hold for such p.

For primes p ≡ ±2 mod 5, Hp(T ) = T 4 + p2 (not ordinary).

S2(Γ) has no nonzero forms satisfying the ASD congruences forp ≡ ±2 mod 5.

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Scholl congruences

Back to the finite index subgroup Γ of SL2(Z); Sk(Γ) is d-dim’lwith an M -integral basis.

• The 2d-dim’l Scholl rep’ns ρl ofGQ are generalizations of Deligne’sconstruction to the noncongruence case.

• For p -M , Scholl constructed p-adic de Rham spaceDR(Γ, k,Zp)of rank 2d, which contains Sk(Γ,Zp).• Scholl: de Rham cohomology is isomorphic to the crystalline

cohomology. Thus the φ operator in the crystalline theory istransported to the de Rham space.

• The action of Frobp on l-adic side (l 6= p) and φ on the p-adicside have the same characteristic poly.

Hp(T ) = T 2d + C1(p)T 2d−1 + · · · + C2d(p).

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Scholl congruences for weakly holomorphic modularforms

Mwkk (Γ, R) ⊃Mwk−ex

k (Γ, R) ⊃ Swk−exk (Γ, R) ⊃ Dk−1Mwk2−k(Γ, R)

• A modular form is weakly holo. if it is holo. on H and mero.at cusps.

• f ∈ Mwkk (Γ, R) is called weakly exact if at each cusp c of Γ

the Fourier coefficients an(f, c) of f at c is divisible by nk−1

in R for each n < 0.

• f ∈Mwk−exk (Γ, R) is a cusp form if it has vanishing constant

term at all cusps.

•Dk−1 :∑n cnq

n/µ 7→∑n n

k−1cnqn/µ maps Mwk

2−k(Γ, R) to

Swk−exk (Γ, R).

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• Using geometric interpretation of weakly holo. modular forms,Kazalicki-Scholl proved

DR(Γ, k,Zp) =Swk−exk (Γ,Zp)

Dk−1(Mwk2−k(Γ,Zp))

.

∑n anq

n/µ and∑n bnq

n/µ in Swk−exk are equal inDR(Γ, k,Zp)⇔ an ≡ bn mod p(k−1)ordpn.

• φ :∑n anq

n/µ 7→ pk−1∑n anq

np/µ.

• Since Hp(φ) = 0 on DR(Γ, k,Zp), any f =∑n anq

n/µ in

Swk−exk (Γ,Zp) satisfies the congruence

anpd(f ) + C1(p)anpd−1(f ) + · · · + C2d(p)an/pd(f )

≡ 0 mod p(k−1)ordpn for all n ≥ 1.

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• Since φ(Sk(Γ,Zp)) ⊂ pk−1DR(Γ, k,Zp), for f ∈ Sk(Γ,Zp)the above congruence holds mod p(k−1)(1+ordpn).

Ex 5. S12(SL2(Z)) is 1-dim’l spanned by ∆(z) = η(z)24 =∑n≥1 τ (n)qn. The char. poly Hp(T ) = T 2 − τ (p)T + p11.

E4(z)6/∆(z)− 1464E4(z)3 = q−1 +

∞∑n=1

anqn

= q−1 − 142236q + 51123200q2 + 39826861650q3 + · · ·lies in DR(SL2(Z), k,Z). For every prime p ≥ 11 and integersn ≥ 1, its coefficients satisfy the congruence

anp − τ (p)an + p11an/p ≡ 0 (mod p11ordpn).

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