The Impact of Computing on Noncongruence Modular Forms ANTS X, San Diego July 10, 2012 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 ). • It is called a congruence modular form if Γ is a congruence subgroup, otherwise it is called a noncongruence modular form. • Congruence forms well-studied; noncongruence forms much less understood. 2
Modular curves • The orbit space Γ \ H ∗ is 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 πiz , be a normalized Let g = � ( a 1 ( g ) = 1) newform of weight k ≥ 2 level N and character χ . I. Hecke theory • 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 . • 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. 4
II. Galois representations • (Eichler-Shimura, Deligne) There exists a compatible family of l -adic deg. 2 rep’ns ρ g,l of Gal( ¯ Q / Q ) such that at primes p ∤ lN , the char. poly. H p ( T ) = T 2 − A p T + B p = T 2 − a p ( g ) T + χ ( p ) p k − 1 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 . 5
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. Atkin and Swinnerton-Dyer: there exists a positive integer M such that S k (Γ) has a basis consisting of forms with coeffs. integral outside M (called M -integral) : a n ( f ) q n/µ . � f ( z ) = n ≥ 1 6
No efficient Hecke operators on noncongruence forms • Let Γ c be the smallest congruence subgroup containing Γ. Naturally, S k (Γ c ) ⊂ S k (Γ). • Tr Γ c Γ : S k (Γ) → S k (Γ c ) such that S k (Γ) = S k (Γ c ) ⊕ ker ( Tr Γ c Γ ). • ker ( Tr Γ c Γ ) consists of genuinely 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 genuinely noncongruence forms in S k (Γ). This was proved by Serre, Berger. 7
Atkin and 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 . 8
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 (1971) : 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. The basis varies with p in general. 9
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 Gal( ¯ Q / Q ) unramified outside lM such that for primes p > k + 1 not dividing Ml , the following hold. (i) The char. polynomial H p ( T ) = T 2 d + C 1 ( p ) T 2 d − 1 + · · · + C 2 d − 1 ( p ) T + C 2 d ( p ) of ρ l (Frob p ) lies in Z [ T ] , is indep. of l , and its roots are alge- braic integers with complex absolute value p ( k − 1) / 2 ; 10
(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 ) + C 1 ( p ) a np d − 1 ( f ) + · · · + + C 2 d − 1 ( p ) a n/p d − 1 ( f ) + C 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. 11
In general, to go from Scholl congruences to ASD congruences, ideally one hopes to factor ( 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 subgroup Γ, this is achieved by using Hecke operators 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. 12
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 established Serre’s conjecture 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. 13
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 S k (Γ) with algebraic Fourier coefficients has bounded denominators if and only if it is a con- gruence modular form, i.e., lies in S k (Γ c ). Kurth-Long: quantitative confirmation for certain families of noncongruence groups. Theorem [L-Long 2012] The conjecture holds when X Γ is de- fined over Q , S k (Γ) is 1 -dim’l, and forms with Fourier coeffi- cients in Q . 14
Explicit examples of noncongruence groups and forms Consider �� a b � � 1 0 � � Γ 1 (5) = ⊳ Γ 0 (5) . ≡ mod 5 c d ∗ 1 cusps of ± Γ 1 (5) generators of stabilizers � 1 5 � ∞ γ = 0 1 � 1 0 � 0 δ = − 1 1 � 11 20 � AγA − 1 = − 2 − 5 − 9 � 11 25 � AδA − 1 = − 5 2 − 4 − 9 15
� − 2 − 5 � ∈ Γ 0 (5), A 2 = − I . Here A = 1 2 Γ 1 (5) is generated by γ , δ , AγA − 1 , AδA − 1 with one relation ( AδA − 1 )( AγA − 1 ) δγ = I. Let φ n be the character of Γ 1 (5) given by • φ n ( γ ) = ζ n , a primitive n -th root of unity, • φ n ( AγA − 1 ) = ζ − 1 n , and • φ n ( δ ) = φ n ( AδA − 1 ) = 1. Γ n = the kernel of φ n is a normal subgroup of Γ 1 (5) of index n , noncongruence if n � = 5. 16
The modular curve X Γ 1 (5) has a model over Q , of genus zero and contains no elliptic points. Same for X Γ n . It is a degree n cover over X Γ 1 (5) unramified everywhere except totally ramified above the cusps ∞ and − 2. Take two weight 3 Eisenstein series for Γ 1 (5) E 1 ( z ) = 1 − 2 q 1 / 5 − 6 q 2 / 5 + 7 q 3 / 5 + 26 q 4 / 5 + · · · , E 2 ( z ) = q 1 / 5 − 7 q 2 / 5 + 19 q 3 / 5 − 23 q 4 / 5 + q + · · · , which vanish at all cusps except at the cusps ∞ and − 2, resp. Then S 3 (Γ n ) = < ( E 1 ( z ) j E 2 ( z ) n − j ) 1 /n > 1 ≤ j ≤ n − 1 is ( n − 1)-dimensional. Let ρ n,l be the attached l -adic Scholl representation. 17
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