From K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015

Transcription

1 From K3 Surfaces to Noncongruence Modular Forms Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 Winnie Li Pennsylvania State University 1

2 A K3 surface with rank 20 There are thirteen K3 surfaces defined over Q whose NS group has rank 20, generated by algebraic cycles over Q. Elkies-Schütt constructed them from suitable double covers of P 2 branched above 6 lines. Consider such a K3 surface E 2 constructed by Beukers and Stienstra the same way, with the 6 lines positioned as 2

3 The zeta function at a good prime p has the form Z(E 2 /F p, T ) = 1 (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 L(E 2, s) := 1 P (E p 2 ; p; p s ) = L(η(4z)6, s) is modular. 3

4 Elliptic surfaces E 2 has a nonhomogeneous model in the sense of Shioda with parameter t. E 2 : y 2 + (1 t 2 )xy t 2 y = x 3 t 2 x 2 For n 2 consider the elliptic surface in the sense of Shioda E n : y 2 + (1 t n n )xy t n n y = x 3 t n n x 2 parametrized by t n. It is an n-fold cover of P 2 branched above the same configuration of 6 lines. 4

5 The Hodge diamond of E n is of the form (n 1) 10n (n 1) The zeta of E n /F p looks similar, with deg P 2 (T ) = 12n 2. P 2 (T ) is a product of 10n linear factors, from points on algebraic cycles, and P (E n ; p; T ) Z[T ] of degree 2n 2. Similarly define L(E n, s) = p 1 P (E n ;p;p s ). Question: Is L(E n, s) automorphic, i.e., equal to the L-function of an automorphic form? 5

6 Base curves as modular curves Beukers and Stienstra: The elliptic surface E : y 2 + (1 τ)xy τy = x 3 τx 2 parameterized by τ is fibered over the genus 0 modular curve (defined over Q) of Γ 1 (5) = {( a b c d ) SL(2, Z), ( ) } mod 5. 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 matrix A = Γ 0 (5) normalizes Γ (5), A 2 = Id. 6

7 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). With t n = n τ, the curve X n is unramified over X Γ 1 (5) except 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 En j 2 ) 1/n > 1 j n 1 is (n 1)-dimensional, corresponding to holomorphic 2-differentials on E n. 7

8 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. He showed that W n,l can be embedded into H 2 et (E n Q Q, Ql ) 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 conjectured 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 GK } is automorphic. 8

9 Properties of Scholl representations ρ n,l 1. ρ n,l is unramified outside nl; 2. For l large, ρ n,l GQl 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 multiplicity n 1; 4. The actions A(t n ) = ζ 2n t n and ζ(t n ) = ζn 1 t n on X n, where ( ) 1 5 ζ =, induce actions on the space of ρ 0 1 n,l. 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

10 Automorphy of L(E 3, s) This was proved by L-Long-Yang in 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 3iq 2 5q 4 ± 3iq 5 + 5q 7 ± 3iq q 10 ± 15iq 11 10q 13 15iq 14 To show them isomorphic, choose l = 2. The actions of A on ρ 3,2 and the Atkin-Lehner involution on ρ 2 allow both representations 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

11 Automorphy of L(E 4, s) This was proved by Atkin-L-Long in 2008 with conceptual explanation given in Atkin-L-Long-Liu in 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 = 2I = (B 2 ) 2 and B 2 B 2 = B 2 B 2. For each quadratic extension K in the biquadratic extension F := Q( 2, 1), 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

12 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 ρ 4,l = IndG Q G K σ K,l = η K,l Ind G Q G K χ 1 K. Both η K,l and Ind G Q G χ 1 K 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) = L(η K,l Ind G Q G K χ 1 K, s) (GL(2) over three K Q( 2, 1)) (GL(2) GL(2) and GL(4) over Q). Similar argument applies to L(E 6, s), done by Long. 12

13 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 ξr(f i 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

14 Character sums and Galois representations At a place of Q(ζ n ) with residue field k of cardinality ( q, ) n divides q 1. The nth power residue symbol at, denoted is a < ζ n > {0}-valued function defined by ( ) a a (q 1)/n (mod ) for all a Z Q(ζn ). 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(ζn ) such that at each place of Q(ζ n ) where σ n,i,l is unramified, one has Trσ n,i,l (Frob ) = ( ) fn (x, y) i. x,y k n n, 14

15 This gives the decomposition ρ n,l GQ(ζ n) = 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., Trσ n,i,l (Frob ) = ( 1 ) i n Trσ n,n i,l (Frob ). Therefore, either σ n,i,l σ n,n i,l, or they differ by a quadratic twist. 15

16 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(4z) 6 has CM, as observed by Beukers-Stienstra. 16

17 (II) n = 3. Q(ζ 3 ) = Q( 3). There are two representations σ 3,1,l and σ 3,2,l. Since n is odd, at a place of Q( 3) not above 2, q = #k is odd so that (q 1)/3 is even and hence the sign is always 1. Thus σ 3,1,l σ 3,2,l. On the other hand, σ 3,2,l is the conjugate of σ 3,1,l by the nontrivial element in Gal(Q( 3)/Q), this means that σ 3,1,l extends to a degree 2 representation of G Q, denoted by ρ + l. Similarly σ 3,2,l extends to a representation ρ l of G Q so that ρ 3,l = ρ + l ρ l. 17

18 Since ρ ± l have the same restrictions to G Q( 3), they either agree of differ by the quadratic twist χ 3. To determine which one, one computes Trρ 3,l (Frob p ) at primes p 2 (mod 3) by counting solutions to s 3 = f 3 (x, y) (mod p) with s 0. Since p 2 (mod 3), we have r = gcd(3, p 1) = 1. Thus Trρ 3,l (Frob p ) = 0 and ρ l = ρ + l χ 3. This explains why g ± differ by twist by χ 3. 18

19 (III) n = 4. Q(ζ 4 ) = Q( 1). There are 3 representations: σ 4,2,l = ρ 2,l studied before, and σ 4,1,l and σ 4,3,l summing to ρ 4,l G. ( ) Q( 1) Since n is even, the character 1 of G 4 Q( 1) has order 2 and kernel G Q( 1, 2). In other words, it is the quadratic character of Q( 1, 2) over Q( 1). So σ 4,1,l and its conjugate σ 4,3,l are not isomorphic, and ρ 4,l = IndG Q G Q( 1) σ 4,1,l as discussed before. Similar discussion applies to n = 6 case. 19

20 Potential automorphy of L(E n, s) For each proper divisor d of n, ρ n,l naturally contains ρ d,l as a G Q -invariant direct summand. After removing the old part from d n and d < n, the remaining new part is denoted ρ prim n,l, which has dimension 2φ(n). Thus {ρ prim n,l ρ n,l = d n, d 1 } remains a compatible family. ρ prim d,l. Assume n 7. Then φ(n) is even. Denote by Q(ζ n ) + the totally real subfield of Q(ζ n ). We have the decomposition ρ prim n,l GQ(ζ n) = 1 i n 1, (i,n)=1 20 σ n,i,l.

21 Recall that σ n,n i,l is the conjugate of σ n,i,l under the nontrivial element in Gal(Q(ζ n )/Q(ζ n ) + ). First assume n odd. We have σ n,i,l σ n,n i,l and they both extend to degree-2 representations η n,i,l and η n,n i,l of G Q(ζn ) + so that ρ prim n,l GQ(ζ n) + = 1 i n 1, (n,i)=1 η n,i,l. Next assume n even. In this case σ n,n i,l and σ n,i,l differ by a quadratic twist. Then σ n,i,l σ n,n i,l = η n,i,l Ind G Q(ζn) + G Q(ζ n) χ n,i,l for a degree-2 representation η n,i,l of G Q(ζn ) + and a finite character χ n,i,l of G Q(ζn ). Hence 21

22 ρ prim n,l GQ(ζ n) + = 1 i φ(n)/2, (n,i)=1 η n,i,l Ind G Q(ζn) + G χ Q(ζ n,i,l. n) In both cases, {η n,i,l } is a compatible family for each i. In an on-going work L-Liu-Long, it is shown that η n,i,l is potentially automorphic, hence so is ρ prim n,l. Theorem [L-Liu-Long] For n 2, {ρ n,l } (hence L(E n, s)) is potentially automorphic, and automorphic for n 6. Remark. Scholl has shown that, for p large, forms in S 3 (Γ n ) with p-adically integral Fourier coefficients satisfy a congruence relation with the coefficients of the characteristic polynomial of ρ n,l (Frob p ). If ρ n,l were automorphic, then this would be a congruence relation between Fourier coefficients of forms for a noncongruence subgroup with those of a congruence subgroup. 22