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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

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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-Sch¨

utt constructed them from suitable double covers of P2 branched above 6 lines.

• Consider such a K3 surface E2 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

Z(E2/Fp, T) = 1 (1 − T)(1 − p2T)P2(T), where P2(T) = char. poly. of Frobp on H2

et(E2 ⊗Q Q, Qℓ) is in

Z[T] of degree 22.

• Beukers and Stienstra computed

P2(T) = (1 − pT)20P(E2; p; T) with P(E2; p; T) ∈ Z[T] of degree 2.

• They further showed that

L(E2, s) :=

• p

1 P(E2; p; p−s) = L(η(4z)6, s) is modular.

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Elliptic surfaces

• E2 has a nonhomogeneous model in the sense of Shioda

E2 : y2 + (1 − t2)xy − t2y = x3 − t2x2 with parameter t.

• For n ≥ 2 consider the elliptic surface in the sense of Shioda

En : y2 + (1 − t n

n )xy − t n n y = x3 − t n n x2

parametrized by tn. It is an n-fold cover of P2 branched above the same configuration of 6 lines.

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• The Hodge diamond of En is of the form

1 (n − 1) 10n (n − 1) 1

• The zeta of En/Fp looks similar, with deg P2(T) = 12n − 2.

P2(T) is a product of 10n linear factors, from points on alge- braic cycles, and P(En; p; T) ∈ Z[T] of degree 2n − 2.

• Similarly define L(En, s) =

p 1 P(En;p;p−s).

Question: Is L(En, s) automorphic, i.e., equal to the L-function

• f an automorphic form?

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Base curves as modular curves

• Beukers and Stienstra: The elliptic surface

E : y2 + (1 − τ)xy − τy = x3 − τx2 parameterized by τ is fibered over the genus 0 modular curve (defined over Q) of Γ1(5) = a b c d

• ∈ SL(2, Z), ≡

1 0 ∗ 1

• mod 5
• .
• En is fibered over a genus zero n-fold cover Xn (defined over

Q) of XΓ1(5) under τ = t n

n .

• XΓ1(5) has no elliptic points, and 4 cusps ∞, 0, −2, −5/2. The

matrix A = −2 −5 1 2

• ∈ Γ0(5) normalizes Γ1(5), A2 = −Id.

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• Let E1 be an Eisenstein series of weight 3 having simple zeros

at all cusps except ∞, and E2 = E1|A. Then τ = E1

E2 is a

Hauptmodul for Γ1(5) with a simple zero at the cusp −2 and a simple pole at the cusp ∞. A(τ) = −1/τ is an involution on XΓ1(5).

• With tn =

n

√τ, the curve Xn is unramified over XΓ1(5) except totally ramified above the cusps ∞ and −2 (with τ-coordinates ∞ and 0, resp.). This describes the index-n normal subgroup Γn of Γ1(5) such that Xn is the modular curve of Γn.

• En is the universal elliptic curve over Xn.
• Γn is noncongruence if n = 5.
• S3(Γn) =< (Ej

1En−j 2

)1/n >1≤j≤n−1 is (n − 1)-dimensional, corresponding to holomorphic 2-differentials on En.

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Galois representations

• To S3(Γn), Scholl has attached a compatible 2(n−1)-dimensional

ℓ-adic representations ρn,ℓ of GQ = Gal( ¯ Q/Q) acting on Wn,ℓ = H1(Xn ⊗Q ¯ Q, ι∗Fℓ), similar to Deligne’s construction for congruence forms.

• He showed that Wn,ℓ can be embedded into H2

et(En ⊗Q ¯

Q, Qℓ) and the L-function attached to the family {ρn,ℓ} is L(En, s).

• According to Langlands philosophy, the family {ρn,ℓ} is conjec-

tured to correspond to an automorphic representation of some reductive group. If so, call {ρn,ℓ} automorphic, and then L(En, s) is an automorphic L-function. Call {ρn,ℓ} potentially automorphic if there is a finite index subgroup GK of GQ such that {ρn,ℓ|GK} is automorphic.

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Properties of Scholl representations ρn,ℓ

• 1. ρn,ℓ is unramified outside nℓ;
• 2. For ℓ large, ρn,ℓ|GQℓ is crystalline with Hodge-Tate weights 0

and −2, each with multiplicity n − 1;

• 3. ρn,ℓ(complex conjugation) has eigenvalues ±1, each with mul-

tiplicity n − 1;

• 4. The actions A(tn) = ζ2n

tn and ζ(tn) = ζ−1 n tn on Xn, where

ζ = 1 5 0 1

• , induce actions on the space of ρn,ℓ.

Since Serre’s modularity conjecture is proved by Kahre-Wintenberger and Kisin in 2007, all degree 2 Scholl representations are modular. So L(E2, s) is modular, as proved by Beukers-Stienstra.

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Automorphy of L(E3, s) This was proved by L-Long-Yang in 2005. We computed the char. poly. of ρ3,ℓ(Frobp) for small primes p and found them agree with those of ˜ ρℓ := ρg+,ℓ ⊕ ρg−,ℓ, where ρg±,ℓ are the ℓ-adic Deligne representations attached to the wt 3 newforms g± of level 27 quad. char. χ−3: g±(z) = q ∓ 3iq2 − 5q4 ± 3iq5 + 5q7 ± 3iq8 + +9q10 ± 15iq11 − 10q13 ∓ 15iq14 − · · · To show them isomorphic, choose ℓ = 2. The actions of A on ρ3,2 and the Atkin-Lehner involution on ˜ ρ2 allow both representations to be viewed as 2-dimensional representations over Q(i)1+i. Then Faltings-Serre was applied to prove ρ3,2 ≃ ˜ ρ2, only used char.

• polys. at primes 5 ≤ p ≤ 19.

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Automorphy of L(E4, s) This was proved by Atkin-L-Long in 2008 with conceptual ex- planation given in Atkin-L-Long-Liu in 2013. The repn ρ4,ℓ = ρ2,ℓ ⊕ ρ−

4,ℓ as eigenspaces with eigenvalues ±1

• f ζ2, 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 B2 := A(1 − ζ) defined over Q(√∓2) resp., satisfying (B−2)2 = −2I = (B2)2 and B−2B2 = −B2B−2. For each quadratic extension K in the biquadratic extension F := Q( √ 2, √−1), ρ−

4,ℓ|GK = σK,ℓ ⊕ (σK,ℓ ⊗ δF/K),

where δF/K is the quadratic char. of F/K.

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There is a finite character χK of GK so that σK,ℓ ⊗χK extends to a degree-2 representation ηK,ℓ of GQ and ρ−

4,ℓ = IndGQ GKσK,ℓ = ηK,ℓ ⊗ IndGQ GKχ−1 K .

Both ηK,ℓ and IndGQ

GKχ−1 K are automorphic, and so is σK,ℓ.

Now L(E4, s) = L(E2, 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(σK,ℓ, s)

(GL(2) over three K ⊂ Q( √ 2, √ −1)) = L(ηK,ℓ ⊗ IndGQ

GKχ−1 K , s)

(GL(2) × GL(2) and GL(4) over Q). Similar argument applies to L(E6, s), done by Long.

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Computing 1/P(En; p; T) Let p ∤ n. To compute 1/P(En; p; T), we use a model birational to En over Q defined by the nonhomogeneous equation sn = (xy)n−1(1 − y)(1 − x)(1 − xy)n−1 =: fn(x, y). The points with s = 0 lie on algebraic cycles. Let q be a power of p. The number of solutions to sn = fn(x, y)

• ver Fq with s = 0 is given by

r

• i=1
• x,y∈Fq, fn(x,y)=0

ξi

r(fn(x, y)),

where r = gcd(n, q − 1) and ξr is a character of F×

q of order r.

The sums with i = r contribute to 1/P(En; p; T) and the sum with i = r contributes to other factors of Z(En/Fp, T).

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Character sums and Galois representations At a place ℘ of Q(ζn) with residue field k℘ of cardinality q, n divides q −1. The nth power residue symbol at ℘, denoted

• ℘
• n,

is a < ζn > ∪{0}-valued function defined by a ℘

• n

≡ a(q−1)/n (mod ℘) for all a ∈ ZQ(ζn). 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 σn,i,ℓ of GQ(ζn) such that at each place ℘ of Q(ζn) where σn,i,ℓ is unramified, one has Trσn,i,ℓ(Frob℘) =

• x,y∈k℘

fn(x, y) ℘ i

n

.

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This gives the decomposition ρn,ℓ|GQ(ζn) = σn,1,ℓ ⊕ σn,2,ℓ ⊕ · · · ⊕ σn,n−1,ℓ. ζ preserves each σn,i,ℓ, while A sends σn,i,ℓ to σn,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., Trσn,i,ℓ(Frob℘) = −1 ℘ i

n

Trσn,n−i,ℓ(Frob℘). Therefore, either σn,i,ℓ ≃ σn,n−i,ℓ, or they differ by a quadratic twist.

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Automorphy of L(En, s) revisited (I) n = 2. Q(ζ2) = Q. In this case σ2,1,ℓ = ρ2,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 ρ2,ℓ is invariant under the quadratic twist by χ−1, hence it is induced from a character of GQ(√−1). It is modular and the corresponding weight 3 cusp form η(4z)6 has CM, as observed by Beukers-Stienstra.

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(II) n = 3. Q(ζ3) = Q(√−3). There are two representations σ3,1,ℓ and σ3,2,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 σ3,1,ℓ ≃ σ3,2,ℓ. On the other hand, σ3,2,ℓ is the conjugate of σ3,1,ℓ by the nontrivial element in Gal(Q(√−3)/Q), this means that σ3,1,ℓ extends to a degree 2 representation of GQ, denoted by ρ+

ℓ . Similarly σ3,2,ℓ extends to a representation ρ− ℓ of GQ so that

ρ3,ℓ = ρ+

ℓ ⊕ ρ− ℓ .

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Since ρ±

ℓ have the same restrictions to GQ(√−3), they either

agree of differ by the quadratic twist χ−3. To determine which one,

• ne computes Trρ3,ℓ(Frobp) at primes p ≡ 2 (mod 3) by counting

solutions to s3 = f3(x, y) (mod p) with s = 0. Since p ≡ 2 (mod 3), we have r = gcd(3, p − 1) = 1. Thus Trρ3,ℓ(Frobp) = 0 and ρ−

ℓ = ρ+ ℓ ⊗ χ−3. This explains why g± differ by twist by

χ−3.

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(III) n = 4. Q(ζ4) = Q(√−1). There are 3 representations: σ4,2,ℓ = ρ2,ℓ studied before, and σ4,1,ℓ and σ4,3,ℓ summing to ρ−

4,ℓ|GQ(√−1).

Since n is even, the character

• −1

℘

• 4 of GQ(√−1) has order 2 and

kernel GQ(√−1,

√ 2). In other words, it is the quadratic character

• f Q(√−1,

√ 2) over Q(√−1). So σ4,1,ℓ and its conjugate σ4,3,ℓ are not isomorphic, and ρ−

4,ℓ = IndGQ GQ(√−1)σ4,1,ℓ

as discussed before. Similar discussion applies to n = 6 case.

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Potential automorphy of L(En, s) For each proper divisor d of n, ρn,ℓ naturally contains ρd,ℓ as a GQ-invariant direct summand. After removing the ”old” part from d|n and d < n, the remaining ”new” part is denoted ρprim

n,ℓ ,

which has dimension 2φ(n). Thus ρn,ℓ =

• d|n, d=1

ρprim

d,ℓ

. {ρprim

n,ℓ } remains a compatible family.

Assume n ≥ 7. Then φ(n) is even. Denote by Q(ζn)+ the totally real subfield of Q(ζn). We have the decomposition ρprim

n,ℓ |GQ(ζn) =

• 1≤i≤n−1, (i,n)=1

σn,i,ℓ.

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Recall that σn,n−i,ℓ is the conjugate of σn,i,ℓ under the nontrivial element in Gal(Q(ζn)/Q(ζn)+). First assume n odd. We have σn,i,ℓ ≃ σn,n−i,ℓ and they both extend to degree-2 representations ηn,i,ℓ and ηn,n−i,ℓ of GQ(ζn)+ so that ρprim

n,ℓ |GQ(ζn)+ =

• 1≤i≤n−1, (n,i)=1

ηn,i,ℓ. Next assume n even. In this case σn,n−i,ℓ and σn,i,ℓ differ by a quadratic twist. Then σn,i,ℓ ⊕ σn,n−i,ℓ = ηn,i,ℓ ⊗ Ind

GQ(ζn)+ GQ(ζn) χn,i,ℓ

for a degree-2 representation ηn,i,ℓ of GQ(ζn)+ and a finite char- acter χn,i,ℓ of GQ(ζn). Hence

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ρprim

n,ℓ |GQ(ζn)+ =

• 1≤i≤φ(n)/2, (n,i)=1

ηn,i,ℓ ⊗ Ind

GQ(ζn)+ GQ(ζn) χn,i,ℓ.

In both cases, {ηn,i,ℓ} is a compatible family for each i. In an

• n-going work L-Liu-Long, it is shown that ηn,i,ℓ is potentially

automorphic, hence so is ρprim

n,ℓ .

Theorem [L-Liu-Long] For n ≥ 2, {ρn,ℓ} (hence L(En, s)) is potentially automorphic, and automorphic for n ≤ 6.

• Remark. Scholl has shown that, for p large, forms in S3(Γn)

with p-adically integral Fourier coefficients satisfy a congruence relation with the coefficients of the characteristic polynomial of ρn,ℓ(Frobp). If ρn,ℓ were automorphic, then this would be a con- gruence relation between Fourier coefficients of forms for a non- congruence subgroup with those of a congruence subgroup.

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