Atkin and Swinnerton-Dyer Congruences on Noncongruence Modular Forms - - PowerPoint PPT Presentation

Atkin and Swinnerton-Dyer Congruences on Noncongruence Modular Forms ICERM April 18, 2013 Winnie Li Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan

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Noncongruence subgroups

• Bass-Lazard-Serre: All finite index subgroups of SLn(Z) for

n ≥ 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 space

Γ\H∗ is a Riemann surface, called the modular curve XΓ for Γ. It has a model defined over a number field.

• The modular curves for congruence subgroups are defined over

Q or cyclotomic fields Q(ζN).

• Belyi: Every smooth projective irreducible curve defined over a

number field is isomorphic to a modular curve XΓ (for infinitely many 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 character χ.

• I. Hecke theory
• It is an eigenfunction of the Hecke operators Tp with eigenvalue

ap(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 subgroup

contains 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 of

degree two l-adic rep’ns ρg,l of GQ := Gal( ¯ Q/Q) such that at primes p ∤ lN, the char. poly. Hp(T) = T 2 − ApT + Bp = T 2 − ap(g)T + χ(p)pk−1

• f 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 cusp at infinity is Q-rational. Atkin and Swinnerton-Dyer: there exists a positive integer M such that Sk(Γ) has a basis consisting of forms with coeffs. integral

• utside 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).

• TrΓc

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

• ker(TrΓc

Γ ) 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 0 0 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 elliptic curves Let E be an elliptic curve defined over Q with conductor M. By Belyi, 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 dq

q = n≥1 anqndq q 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 ℓ(x) :=

n≥1 anxn is the formal log of

a formal group law G, which is isomorphic to the group law on the 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: ˜

ℓ(x) :=

n≥1 bnxn is the formal log of a formal group

law which is strictly isomorphic to G.

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• Hence for p ∤ M, the sequences {an} and {bn} satisfy the same

congruence 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.
• f the Frobp acting on the Tate module Tl(E) for l = p.
• Taniyama-Shimura modularity theorem: g =

n≥1 bnqn is a

normalized 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 the d-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 of hj 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 Scholl congruences Theorem[Scholl] Suppose that the modular curve XΓ has a model 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’ns ρl of GQ unramified outside lM such that for primes p > k+1 not dividing Ml, the following hold. (i) The char. polynomial Hp(T) = T 2d + C1(p)T 2d−1 + · · · + C2d−1(p)T + C2d(p)

• f 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 Fourier coeffs 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 on Serre’s conjecture on modular rep’ns, and various modularity lift- ing theorems.

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Application: Characterizing noncongruence modular forms 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 Fourier

coefficients 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 wt k 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 Hecke

• perators 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 = E2

E1 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, E1t2

3 > has a basis

f±(z) = q1/15 ± iq2/15 − 11 3 q4/15 ∓ i16 3 q5/15 − −4 9q7/15 ± i71 9 q8/15 + 932 81 q10/15 + · · · . (2) (Modularity) There are two cuspidal newforms of weight 3 level 27 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 = E1ti

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and h2 = E1t2 satisfies ASD congruences for all odd p. The space < h1, h3 > has a basis satisfying the ASD congruence depending 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 with Ap = ±2√−2a3(p)

and Bp = −p2;

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

and Bp = −p2.

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Here a1(p), a3(p), a5(p), a7(p) are the Fourier coefficients of the wt 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) admit quaternion 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 over

• Q. By Belyi, X ≃ XΓ for a finite index subgroup Γ of SL2(Z).

Put ω1 = dx 2y = f1 dq1/10 q1/10 , ω2 = xdx 2y = f2 dq1/10 q1/10 . Then S2(Γ) =< f1, f2 >, where f1 = q1/10 − 8 5q6/10 − 108 52 q11/10 + 768 53 q16/10 + 3374 54 q21/10 + · · · , f2 = q2/10 − 16 5 q7/10 + 48 52q12/10 + 64 53q17/10 + 724 54 q22/10 + · · · .

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The l-adic representations attached to S2(Γ) are the dual of the Tate modules on the Jacobian of XΓ. They are strongly ordinary at 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 for p ≡ ±2 mod 5.

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

• The 2d-dim’l Scholl rep’ns ρl of GQ are generalizations of Deligne’s

construction to the noncongruence case.

• For p ∤ M, Scholl constructed p-adic de Rham space DR(Γ, k, Zp)
• f 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 is

transported to the de Rham space.

• The action of Frobp on l-adic side (l = p) and φ on the p-adic

side 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 modular forms Mwk

k (Γ, R) ⊃ Mwk−ex k

(Γ, R) ⊃ Swk−ex

k

(Γ, R) ⊃ Dk−1Mwk

2−k(Γ, R)

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

at cusps.

• f ∈ Mwk

k (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−ex

k

(Γ, R) is a cusp form if it has vanishing constant term at all cusps.

• Dk−1 :

n cnqn/µ → n nk−1cnqn/µ maps Mwk 2−k(Γ, R) to

Swk−ex

k

(Γ, R).

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• Using geometric interpretation of weakly holo. modular forms,

Kazalicki-Scholl proved DR(Γ, k, Zp) = Swk−ex

k

(Γ, Zp) Dk−1(Mwk

2−k(Γ, Zp))

.

• n anqn/µ and

n bnqn/µ in Swk−ex k

are equal in DR(Γ, k, Zp) ⇔ an ≡ bn mod p(k−1)ordpn.

• φ :

n anqn/µ → pk−1 n anqnp/µ.

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

n anqn/µ in

Swk−ex

k

(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 integers n ≥ 1, its coefficients satisfy the congruence anp − τ(p)an + p11an/p ≡ 0 (mod p11ordpn).

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