Atkin-Swinnerton-Dyer Congruences on Modular Forms for Noncongruence Subgroups in memory of A. O. L. Atkin Winnie Li Pennsylvania State University, U.S.A. and National Center for Theoretical Sciences, Taiwan 1
Modular forms • A modular form is a holomorphic function on the Poincar´ e upper half-plane H with a lot of symmetries w.r.t. a finite- index subgroup Γ of SL 2 ( Z ). • Γ is called a congruence subgroup if it contains the group � a b � � a b � � 1 0 � Γ( N ) = { ∈ SL 2 ( Z ) : ≡ mod N } c d c d 0 1 for some positive integer N . Forms for such Γ are called congruence modular forms. • Otherwise Γ is called a noncongruence subgroup, and forms are called noncongruence modular forms. • Congruence forms well-studied; noncongruence forms much less understood. 2
Modular curves • The group Γ acts on H by fractional linear transformations. We compactify the orbit space Γ \ H by joining finitely many cusps to get a Riemann surface, called the modular curve X Γ 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 ( ζ N ). • Belyi: Every smooth projective irreducible curve defined over a number field is isomorphic to a modular curve X Γ (for infinitely many finite-index subgroups Γ of SL 2 ( Z )). • SL 2 ( Z ) has far more noncongruence subgroups than congru- ence subgroups. 3
Modular forms for congruence subgroups n ≥ 1 a n ( g ) q n , where q = e 2 πiτ , be a normalized Let g = � ( a 1 ( g ) = 1) newform of weight k ≥ 2 level N and character χ . Key properties: • The Fourier coefficients are multiplicative, i.e., a mn ( g ) = a m ( g ) a n ( g ) whenever m and n are coprime. • (Hecke) It is an eigenfunction of the Hecke operators T p with eigenvalue a p ( g ) for all primes p ∤ N , i.e., for all n ≥ 1, a np ( g ) − a p ( g ) a n ( g ) + χ ( p ) p k − 1 a n/p ( g ) = 0 . For primes p | N and all n ≥ 1, a np ( g ) = a n ( g ) a p ( g ) . 4
• The Fourier coefficients of a newform are algebraic integers. Given a congruence subgroup and weight, there is a basis of cusp forms with integral Fourier coefficients. An algebraic cusp form has bounded denominators. • (Eichler-Shimura, Deligne) There exists a compatible family of l -adic deg. 2 rep’ns ρ g,l of the Galois group Gal( ¯ Q / Q ) such that Tr( ρ g,l (Frob p )) = a p ( g ) , det( ρ g,l (Frob p )) = χ ( p ) p k − 1 , for all primes p not dividing lN . 5
The char. poly. H p ( T ) = T 2 − A p T + B p of ρ g,l (Frob p ) is indep. of l , and a np ( g ) − A p a n ( g ) + B p a n/p ( g ) = 0 for n ≥ 1 and primes p ∤ lN . • Ramanujan-Petersson conjecture holds for newforms. That is, | a p ( g ) | ≤ 2 p ( k − 1) / 2 for all primes p ∤ N . 6
Modular forms for noncongruence subgroups Γ : a noncongruence subgroup of SL 2 ( Z ) with finite index S k (Γ) : space of cusp forms of weight k ≥ 2 for Γ of dim d A cusp form has an expansion in powers of q 1 /µ . Assume the modular curve X Γ is defined over Q and the cusp at infinity is Q -rational. Key players: Fricke, Kline, Atkin and Swinnerton-Dyer, Scholl Atkin and Swinnerton-Dyer: there exists a positive integer M such that S k (Γ) has a basis consisting of forms with coeffs. in Z [ 1 M ] (called M -integral) : a n ( f ) q n/µ . � f ( τ ) = n ≥ 1 7
No efficient Hecke operators on noncongruence forms • Let Γ ′ be the smallest congruence subgroup containing Γ. Naturally, S k (Γ ′ ) ⊂ S k (Γ). • Tr Γ Γ ′ : S k (Γ) → S k (Γ ′ ) such that its restriction on S k (Γ ′ ) is multiplication by [Γ ′ : Γ]. • The kernel of Tr Γ Γ ′ consists of genuine noncongruence forms in S k (Γ). Conjecture (Atkin ). The Hecke operators on S k (Γ) for p ∤ M defined using double cosets as for congruence forms is zero on genuine noncongruence forms in S k (Γ). This was proved by Serre, Berger. 8
Atkin-Swinnerton-Dyer congruences Let E be an elliptic curve defined over Q with conductor M . By Belyi, E ≃ X Γ for a finite index subgroup Γ of SL 2 ( Z ). Eg. E : x 3 + y 3 = z 3 , Γ is an index-9 noncongruence subgp of Γ(2). Atkin and Swinnerton-Dyer: The normalized holomorphic differ- ential 1-form f dq n ≥ 1 a n q ndq q = � q on E satisfies the congruence relation mod p 1+ord p n a np − [ p + 1 − # E ( F p )] a n + pa n/p ≡ 0 for all primes p ∤ M and all n ≥ 1. Note that f ∈ S 2 (Γ). Taniyama-Shimura modularity theorem: There is a normalized n ≥ 1 b n q n with b p = p + 1 − # E ( F p ). congruence newform g = � This gives congruence relations between f and g . 9
Back to general case where X Γ has a model over Q , and the d -dim’l space S k (Γ) has a basis of M -integral forms. ASD congruences : for each prime p ∤ M , S k (Γ , Z p ) has a p -adic basis { h j } 1 ≤ j ≤ d such that the Fourier coefficients of h j satisfy a three-term congruence relation mod p ( k − 1)(1+ord p n ) a np ( h j ) − A p ( j ) a n ( h j ) + B p ( j ) a n/p ( h j ) ≡ 0 for all n ≥ 1. Here • A p ( j ) is an algebraic integer with | A p ( j ) | ≤ 2 p ( k − 1) / 2 , and • B p ( j ) is equal to p k − 1 times a root of unity. This is proved to hold for k = 2 and d = 1 by ASD. Their data also show that the A p ’s satisfy the Sato-Tate distri- bution. 10
Remarks. (1) The basis varies with p . (2) The three-term congruence relations for noncongruence forms capture the spirit of the Hecke operators in essence. (3) From where do A p ( j ) and B p ( j ) arise? Any modularity interpretations? Belyi’s theorem tells us that, viewed simply as algebraic curves, noncongruence modular curves are very general, and that we should not expect them to have any special arithmetic properties. On the other hand, the uniformization of noncongruence curves by the upper half-plane is quite special, and leads to surprising con- sequences. 11
Galois representations attached to S k (Γ) and congru- ences Theorem [Scholl] Suppose that the modular curve X Γ has a model over Q . Attached to S k (Γ) is a compatible family of 2 d - dim’l l -adic rep’ns ρ l of the Galois group Gal( ¯ Q / Q ) unramified outside lM such that for primes p > k + 1 not dividing Ml , the following hold. (i) The char. polynomial B r ( p ) T 2 d − r � H p ( T ) = 0 ≤ r ≤ 2 d of ρ l (Frob p ) lies in Z [ T ] , is indep. of l , and its roots are alge- braic integers with absolute value p ( k − 1) / 2 ; 12
(ii) For any form f in S k (Γ) integral outside M , its Fourier coeffs satisfy the (2 d + 1) -term congruence relation a np d ( f ) + B 1 ( p ) a np d − 1 ( f ) + · · · + + B 2 d − 1 ( p ) a n/p d − 1 ( f ) + B 2 d ( p ) a n/p d ( f ) mod p ( k − 1)(1+ord p n ) ≡ 0 for n ≥ 1. The Scholl rep’ns ρ l are generalizations of Deligne’s construction to the noncongruence case. The congruence in (ii) follows from comparing l -adic theory to an analogous p -adic de Rham/crystalline theory; the action of Frob p on both sides have the same charac- teristic polynomials. Scholl’s theorem establishes the ASD congruences if d = 1. 13
In general, to go from Scholl congruences to ASD congruences, ideally one hopes to factorize ( T 2 − A p ( j ) T + B p ( j )) � H p ( T ) = 1 ≤ j ≤ d and find a p -adic basis { h j } 1 ≤ j ≤ d , depending on p , for S k (Γ , Z p ) such that each h j satisfies the three-term ASD congruence rela- tions given by A p ( j ) and B p ( j ). For a congruence group Γ, this is achieved by using Hecke oper- ators to further break the l -adic and p -adic spaces into pieces. For a noncongruence Γ, no such tools are available. Scholl representations, being motivic, should correspond to au- tomorphic forms for reductive groups according to Langlands phi- losophy. They are the link between the noncongruence and con- gruence worlds. 14
Modularity of Scholl representations when d = 1 Scholl: the rep’n attached to S 4 (Γ 7 , 1 , 1 ) is modular, coming from a newform of wt 4 for Γ 0 (14); ditto for S 4 (Γ 4 , 3 ) and S 4 (Γ 5 , 2 ). Li-Long-Yang: True for wt 3 noncongruence forms assoc. with K3 surfaces defined over Q . In 2006 Kahre, Wintenberger and Kisin established Serre’s con- jecture on modular representations. This leads to Theorem If S k (Γ) is 1 -dimensional, then the degree two l - adic Scholl representations of Gal( ¯ Q / Q ) are modular. Therefore for S k (Γ) with dimension one, we have both ASD congruences and modularity. Consequently, every f ∈ S k (Γ) with algebraic Fourier coefficients satisfies three-term congruence rela- tions with a wt k congruence form. 15
ASD congruences and modularity for d ≥ 2 For each n ≥ 1, there is an index- n subgroup Γ n of Γ 1 (5) whose modular curve is defined over Q and S 3 (Γ n ) is ( n − 1)-dim’l with explicit basis and attached Scholl rep’n ρ n,l . Case d = 2. Theorem [L-Long-Yang, 2005, for Γ 3 ] (1) The space S 3 (Γ 3 ) has a basis consisting of 3 -integral forms f ± ( τ ) = q 1 / 15 ± iq 2 / 15 − 11 3 q 4 / 15 ∓ i 16 3 q 5 / 15 − − 4 9 q 7 / 15 ± i 71 9 q 8 / 15 + 932 81 q 10 / 15 + · · · . (2) (Modularity) There are two cuspidal newforms of weight 3 level 27 and character χ − 3 given by 16
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