From K 3 Surfaces to Noncongruence Modular Forms Explicit Methods for Modularity of K 3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 Winnie Li Pennsylvania State University 1
A K 3 surface with rank 20 • There are thirteen K 3 surfaces defined over Q whose NS group has rank 20, generated by algebraic cycles over Q . • Elkies-Sch¨ utt constructed them from suitable double covers of P 2 branched above 6 lines. • Consider such a K 3 surface E 2 constructed by Beukers and Stienstra the same way, with the 6 lines positioned as 2
• The zeta function at a good prime p has the form 1 Z ( E 2 / F p , T ) = (1 − T )(1 − p 2 T ) P 2 ( T ) , where P 2 ( T ) = char. poly. of Frob p on H 2 et ( E 2 ⊗ Q Q , Q ℓ ) is in Z [ T ] of degree 22. • Beukers and Stienstra computed P 2 ( T ) = (1 − pT ) 20 P ( E 2 ; p ; T ) with P ( E 2 ; p ; T ) ∈ Z [ T ] of degree 2. • They further showed that 1 P ( E 2 ; p ; p − s ) = L ( η (4 z ) 6 , s ) � L ( E 2 , s ) := p is modular. 3
Elliptic surfaces • E 2 has a nonhomogeneous model in the sense of Shioda y 2 + (1 − t 2 ) xy − t 2 y = x 3 − t 2 x 2 E 2 : with parameter t . • For n ≥ 2 consider the elliptic surface in the sense of Shioda y 2 + (1 − t n n y = x 3 − t n n ) xy − t n n x 2 E n : parametrized by t n . It is an n -fold cover of P 2 branched above the same configuration of 6 lines. 4
• The Hodge diamond of E n is of the form 1 0 0 ( n − 1) 10 n ( n − 1) 0 0 1 • The zeta of E n / F p looks similar, with deg P 2 ( T ) = 12 n − 2. P 2 ( T ) is a product of 10 n linear factors, from points on alge- braic cycles, and P ( E n ; p ; T ) ∈ Z [ T ] of degree 2 n − 2. 1 • Similarly define L ( E n , s ) = � P ( E n ; p ; p − s ) . p Question: Is L ( E n , s ) automorphic, i.e., equal to the L -function of an automorphic form? 5
Base curves as modular curves • Beukers and Stienstra: The elliptic surface y 2 + (1 − τ ) xy − τy = x 3 − τx 2 E : parameterized by τ is fibered over the genus 0 modular curve (defined over Q ) of �� a b � � 1 0 � � Γ 1 (5) = ∈ SL (2 , Z ) , ≡ mod 5 . c d ∗ 1 • E n is fibered over a genus zero n -fold cover X n (defined over Q ) of X Γ 1 (5) under τ = t n n . • X Γ 1 (5) has no elliptic points, and 4 cusps ∞ , 0, − 2, − 5 / 2. The � − 2 − 5 � ∈ Γ 0 (5) normalizes Γ 1 (5), A 2 = − Id . matrix A = 1 2 6
• Let E 1 be an Eisenstein series of weight 3 having simple zeros at all cusps except ∞ , and E 2 = E 1 | A . Then τ = E 1 E 2 is a Hauptmodul for Γ 1 (5) with a simple zero at the cusp − 2 and a simple pole at the cusp ∞ . A ( τ ) = − 1 /τ is an involution on X Γ 1 (5) . √ τ , the curve X n is unramified over X Γ 1 (5) except n • With t n = totally ramified above the cusps ∞ and − 2 (with τ -coordinates ∞ and 0, resp.). This describes the index- n normal subgroup Γ n of Γ 1 (5) such that X n is the modular curve of Γ n . • E n is the universal elliptic curve over X n . • Γ n is noncongruence if n � = 5. • S 3 (Γ n ) = < ( E j 1 E n − j ) 1 /n > 1 ≤ j ≤ n − 1 is ( n − 1)-dimensional, 2 corresponding to holomorphic 2-differentials on E n . 7
Galois representations • To S 3 (Γ n ), Scholl has attached a compatible 2( n − 1)-dimensional ℓ -adic representations ρ n,ℓ of G Q = Gal ( ¯ Q / Q ) acting on W n,ℓ = H 1 ( X n ⊗ Q ¯ Q , ι ∗ F ℓ ), similar to Deligne’s construction for congruence forms. et ( E n ⊗ Q ¯ • He showed that W n,ℓ can be embedded into H 2 Q , Q ℓ ) and the L -function attached to the family { ρ n,ℓ } is L ( E n , s ). • According to Langlands philosophy, the family { ρ n,ℓ } is conjec- tured to correspond to an automorphic representation of some reductive group. If so, call { ρ n,ℓ } automorphic , and then L ( E n , s ) is an automorphic L -function. Call { ρ n,ℓ } potentially automorphic if there is a finite index subgroup G K of G Q such that { ρ n,ℓ | G K } is automorphic. 8
Properties of Scholl representations ρ n,ℓ 1. ρ n,ℓ is unramified outside nℓ ; 2. For ℓ large, ρ n,ℓ | G Q ℓ is crystalline with Hodge-Tate weights 0 and − 2, each with multiplicity n − 1; 3. ρ n,ℓ (complex conjugation) has eigenvalues ± 1, each with mul- tiplicity n − 1; 4. The actions A ( t n ) = ζ 2 n t n and ζ ( t n ) = ζ − 1 n t n on X n , where � 1 5 � ζ = , induce actions on the space of ρ n,ℓ . 0 1 Since Serre’s modularity conjecture is proved by Kahre-Wintenberger and Kisin in 2007, all degree 2 Scholl representations are modular. So L ( E 2 , s ) is modular, as proved by Beukers-Stienstra. 9
Automorphy of L ( E 3 , s ) This was proved by L-Long-Yang in 2005. We computed the char. poly. of ρ 3 ,ℓ (Frob p ) for small primes p and found them agree with those of ˜ ρ ℓ := ρ g + ,ℓ ⊕ ρ g − ,ℓ , where ρ g ± ,ℓ are the ℓ -adic Deligne representations attached to the wt 3 newforms g ± of level 27 quad. char. χ − 3 : 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 − · · · To show them isomorphic, choose ℓ = 2. The actions of A on ρ 3 , 2 ρ 2 allow both representations and the Atkin-Lehner involution on ˜ to be viewed as 2-dimensional representations over Q ( i ) 1+ i . Then Faltings-Serre was applied to prove ρ 3 , 2 ≃ ˜ ρ 2 , only used char. polys. at primes 5 ≤ p ≤ 19. 10
Automorphy of L ( E 4 , s ) This was proved by Atkin-L-Long in 2008 with conceptual ex- planation given in Atkin-L-Long-Liu in 2013. The repn ρ 4 ,ℓ = ρ 2 ,ℓ ⊕ ρ − 4 ,ℓ as eigenspaces with eigenvalues ± 1 of ζ 2 , where ρ − 4 ,ℓ is 4-dim’l and want to prove it automorphic. Its space admits quaternion multiplication by B − 2 := A (1 + ζ ) and B 2 := A (1 − ζ ) defined over Q ( √∓ 2) resp., satisfying ( B − 2 ) 2 = − 2 I = ( B 2 ) 2 and B − 2 B 2 = − B 2 B − 2 . For each quadratic extension K in the biquadratic extension √ 2 , √− 1), F := Q ( ρ − 4 ,ℓ | G K = σ K,ℓ ⊕ ( σ K,ℓ ⊗ δ F/K ) , where δ F/K is the quadratic char. of F/K . 11
There is a finite character χ K of G K so that σ K,ℓ ⊗ χ K extends to a degree-2 representation η K,ℓ of G Q and 4 ,ℓ = Ind G Q G K σ K,ℓ = η K,ℓ ⊗ Ind G Q ρ − G K χ − 1 K . Both η K,ℓ and Ind G Q G K χ − 1 K are automorphic, and so is σ K,ℓ . Now L ( E 4 , s ) = L ( E 2 , s ) L ( ρ − 4 ,ℓ , s ), and there are 5 ways to see the automorphicity of L ( ρ − 4 ,ℓ , s ): √ √ L ( ρ − 4 ,ℓ , s ) = L ( σ K,ℓ , s ) ( GL (2) over three K ⊂ Q ( 2 , − 1)) = L ( η K,ℓ ⊗ Ind G Q G K χ − 1 K , s ) ( GL (2) × GL (2) and GL (4) over Q ) . Similar argument applies to L ( E 6 , s ), done by Long. 12
Computing 1 /P ( E n ; p ; T ) Let p ∤ n . To compute 1 /P ( E n ; p ; T ), we use a model birational to E n over Q defined by the nonhomogeneous equation s n = ( xy ) n − 1 (1 − y )(1 − x )(1 − xy ) n − 1 =: f n ( x, y ) . The points with s = 0 lie on algebraic cycles. Let q be a power of p . The number of solutions to s n = f n ( x, y ) over F q with s � = 0 is given by r ξ i � � r ( f n ( x, y )) , i =1 x,y ∈ F q , f n ( x,y ) � =0 where r = gcd ( n, q − 1) and ξ r is a character of F × q of order r . The sums with i � = r contribute to 1 /P ( E n ; p ; T ) and the sum with i = r contributes to other factors of Z ( E n / F p , T ). 13
Character sums and Galois representations At a place ℘ of Q ( ζ n ) with residue field k ℘ of cardinality q , n � � divides q − 1. The n th power residue symbol at ℘ , denoted n , ℘ is a < ζ n > ∪{ 0 } -valued function defined by � a � ≡ a ( q − 1) /n (mod ℘ ) for all a ∈ Z Q ( ζ n ) . ℘ n It induces a character of k × ℘ with order n . Fuselier-Long-Ramakrishna-Swisher-Tu show that, for 1 ≤ i ≤ n − 1 there exists a degree-2 representation σ n,i,ℓ of G Q ( ζ n ) such that at each place ℘ of Q ( ζ n ) where σ n,i,ℓ is unramified, one has � i � f n ( x, y ) � Tr σ n,i,ℓ (Frob ℘ ) = . ℘ n x,y ∈ k ℘ 14
This gives the decomposition ρ n,ℓ | G Q ( ζn ) = σ n, 1 ,ℓ ⊕ σ n, 2 ,ℓ ⊕ · · · ⊕ σ n,n − 1 ,ℓ . ζ preserves each σ n,i,ℓ , while A sends σ n,i,ℓ to σ n,n − i,ℓ . Further, the character sum can be expressed as a finite field analogue of hypergeometric series, which was shown by Greene to equal to its complex conjugation up to sign, i.e., � i � − 1 Tr σ n,i,ℓ (Frob ℘ ) = Tr σ n,n − i,ℓ (Frob ℘ ) . ℘ n Therefore, either σ n,i,ℓ ≃ σ n,n − i,ℓ , or they differ by a quadratic twist. 15
Automorphy of L ( E n , s ) revisited (I) n = 2. Q ( ζ 2 ) = Q . In this case σ 2 , 1 ,ℓ = ρ 2 ,ℓ is the only representation. The character is the Legendre symbol, which is the quadratic character χ − 1 of Q ( √− 1) over Q . This shows that ρ 2 ,ℓ is invariant under the quadratic twist by χ − 1 , hence it is induced from a character of G Q ( √− 1) . It is modular and the corresponding weight 3 cusp form η (4 z ) 6 has CM, as observed by Beukers-Stienstra. 16
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