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 1
Noncongruence subgroups • Bass-Lazard-Serre: All finite index subgroups of SL n ( Z ) for n ≥ 3 are congruence subgroups. • SL 2 ( Z ) contains far more noncongruence subgroups than con- gruence subgroups. • Let Γ be a finite index subgroup of SL 2 ( Z ). 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 )). 2
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. 3
II. Galois representations • (Eichler-Shimura, Deligne) There exists a compatible family of degree two l -adic rep’ns ρ g,l of G Q := 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 . 4
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 5
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 � 1 0 � defined using double cosets Γ Γ as for congruence forms is 0 p zero on genuinely noncongruence forms in S k (Γ). This was proved by Serre, Berger. So the progress has been led by computational data. 6
Atkin and Swinnerton-Dyer congruences for elliptic curves 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 ). Ex. E : x 3 + y 3 = z 3 , Γ is an index-9 noncong. 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 (1) for all primes p ∤ M and all n ≥ 1. 7
Sketch of a proof using formal group laws n ≥ 1 a n x n is the formal log of • The formal power series ℓ ( x ) := � 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 1 1 b n n − s , � � � L ( s, E ) = 1 − b p p − s = 1 − b p p − s + p 1 − 2 s n ≥ 1 p ∤ M p | M in which b p = p + 1 − # E ( F p ) for p ∤ M . n ≥ 1 b n x n is the formal log of a formal group • Honda: ˜ ℓ ( x ) := � law which is strictly isomorphic to G . 8
• Hence for p ∤ M , the sequences { a n } and { b n } satisfy the same congruence relation of ASD type: mod p 1+ord p n . c np − A p c n + B p, 1 c n/p + B p, 2 c n/p 2 + · · · ≡ 0 • The b n ’s satisfy the three term recursion b np − b p b n + pb n/p = 0 for all p ∤ M and n ≥ 1 . So A p = b p , B p, 1 = p , and other B p,i = 0. This proves (1). • For p ∤ M , H p ( T ) = T 2 − b p T + p is the characteristic poly. of the Frob p acting on the Tate module T l ( E ) for l � = p . n ≥ 1 b n q n is a • Taniyama-Shimura modularity theorem: g = � normalized congruence newform. Note that f ∈ S 2 (Γ). Thus (1) gives congruence relations between f and g . 9
ASD congruences in general 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. 10
Galois representations attached to S k (Γ) and Scholl congruences Theorem [Scholl] Suppose that the modular curve X Γ has a model over Q such that the cusp at infinity is Q -rational. At- tached to S k (Γ) is a compatible family of 2 d -dim’l l -adic rep’ns ρ l of G 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 ; 11
(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. Theorem If S k (Γ) is 1 -dimensional, then • the ASD congruences hold for almost all p ; • the degree two l -adic Scholl rep’ns of G Q 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. 12
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 ). Theorem [L-Long] The conjecture holds when S k (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 : a n ( f ) = O ( n k/ 2 − 1 / 5 ). 13
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 space and S k (Γ) into pieces. For a noncongruence Γ, no such tools are available. Theorem [Scholl] If ρ l ( Frob p ) is diagonalizable and ordinary (i.e. half of the eigenvalues are p -adic units), then ASD con- gruences at p hold. 14
Examples of ASD congruences The group Γ 1 (5) has genus 0, no elliptic elements, and 4 cusps. The wt 3 Eisenstein series 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 + · · · . have simple zeros at all cusps except ∞ and − 2, resp., and non- vanishing elsewhere. So X Γ 1 (5) is defined over Q with t = E 2 E 1 as √ n a Hauptmodul, and t n = t is a Hauptmodul of a smooth irred. modular curve X Γ n over Q . Let ρ n,l be the l -adic Scholl representation attached to S 3 (Γ n ). Ex 1. When n = 2, S 3 (Γ 2 ) = < E 1 t 2 > is 1-dim’l, ASD congru- ences hold for odd p , and ρ 2 ,l are isom. to ρ η (4 z ) 6 ,l . 15
Ex 2. (L-Long-Yang) (1) When n = 3, the space S 3 (Γ 3 ) = < E 1 t 3 , E 1 t 2 3 > has a basis f ± ( z ) = 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 quadratic character χ − 3 given by g ± ( z ) = q ∓ 3 iq 2 − 5 q 4 ± 3 iq 5 + 5 q 7 ± 3 iq 8 + +9 q 10 ± 15 iq 11 − 10 q 13 ∓ 15 iq 14 − · · · such that ρ 3 ,l = ρ g + ,l ⊕ ρ g − ,l over Q l ( √− 1). (3) f ± satisfy the 3-term ASD congruences with A p = a p ( g ± ) and B p = χ − 3 ( p ) p 2 for all primes p ≥ 5. 16
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