Computing Néron-Severi groups

Computing N´

eron-Severi groups

Ronald van Luijk

February 27, 2013

Setting

I k a finitely generated field.

I X a nicek-variety (smooth, projective, geometrically integral).

I ks a separable closure ofk with Galois group Γ = Gal(ks/k).

I Xs = X ×k ks.

I Picard group Pic X ⊂ (Pic Xs)Γ.

I Pic0(X ) subgroup of classes algebraically equivalent to0.

I N´eron-Severi groupNS(X ) = Pic X / Pic0(X ) ⊂ NS(Xs₎Γ_. Goal: ComputeNS(Xs) (or its rank).

Special cases

I Elliptic fibrations: mapNS to the Mordell-Weil group.

I Fibrations into abelian varieties.

I If a finite group G acts on Y andX = Y /G, then

NS(Xs) ⊗ Q → (NS(Ys) ⊗ Q)G

is an isomorphism.

Application (Shioda):

Delsarte surfaces (given by tetranomials inP3) are

Special cases

I Elliptic fibrations: mapNS to the Mordell-Weil group.

I Fibrations into abelian varieties.

I If a finite group G acts on Y andX = Y /G, then

NS(Xs) ⊗ Q → (NS(Ys) ⊗ Q)G

is an isomorphism. Application (Shioda):

Delsarte surfaces (given by tetranomials inP3) are

K3 surfaces of degree 2

Theorem (Hassett, Kresch, Tschinkel)

There is an algorithm that takes as input a K3 surfaceX of degree2 over a number field, and returnsPic Xs_{= NS X}s_. Method: Kuga-Satake correspondence.

Tate conjecture(s)

I Fix0 ≤ p ≤ dim X and prime` 6= char k.

I Zp_{(X )is group of codimension-p cycles onX.}

I V2p= H2p_et(Xs, Q`(p)).

I Vtate⊂ V2p _{is set of Tate classes}

(each fixed by some finite-index open subgroupG ⊂ Γ).

Conjecture (Tp(X , `))

The cycle class mapZp_(Xs_{) ⊗ Q}

`→ Vtate is surjective.

Theorem (Nijgaard, Ogus, Maulik, Charles, Madapusi Pera)

Supposechar k 6= 2. If X is a K3 surface, thenT1(X , `) holds.

Conjecture (Ep_{(X , `))}

An element inZp_(Xs_{, `) is numerically equivalent to0 if and only} if it maps to0in V2p.

Tate conjecture(s)

I Fix0 ≤ p ≤ dim X and prime` 6= char k.

I Zp_{(X )is group of codimension-p cycles onX.}

I V2p= H2p_et(Xs, Q`(p)).

I Vtate⊂ V2p _{is set of Tate classes}

(each fixed by some finite-index open subgroupG ⊂ Γ).

Conjecture (Tp(X , `))

The cycle class mapZp_(Xs_{) ⊗ Q}

`→ Vtate is surjective.

Theorem (Nijgaard, Ogus, Maulik, Charles, Madapusi Pera)

Supposechar k 6= 2. If X is a K3 surface, thenT1_{(X , `) holds.}

Conjecture (Ep_{(X , `))}

An element inZp_(Xs_{, `) is numerically equivalent to0 if and only} if it maps to0in V2p.

Tate conjecture(s)

I Fix0 ≤ p ≤ dim X and prime` 6= char k.

I Zp_{(X )is group of codimension-p cycles onX.}

I V2p= H2p_et(Xs, Q`(p)).

I Vtate⊂ V2p _{is set of Tate classes}

(each fixed by some finite-index open subgroupG ⊂ Γ).

Conjecture (Tp(X , `))

The cycle class mapZp_(Xs_{) ⊗ Q}

`→ Vtate is surjective.

Theorem (Nijgaard, Ogus, Maulik, Charles, Madapusi Pera)

Supposechar k 6= 2. If X is a K3 surface, thenT1_{(X , `) holds.}

Conjecture (Ep_{(X , `))}

An element inZp_(Xs_{, `) is numerically equivalent to0 if and only} if it maps to0in V2p.

Algorithms for general

p

I Nump(X )is group of codimension-p cycle classes up to numerical equivalence.

I Assuming Ep(X , `), the mapNump(X , `) ⊗ Q`,→ Vtate is an injection that is an isomorphism if and only if Tp(X , `) holds.

I For p = 1we haveNum1(X ) ∼= NS(X )/ NS(X )tors and an injectionNS(X ) ⊗ Q` → Vtate.

Strategy for boundingrk Nump(Xs_).

1. List cycles to find lower bound (also cycles to intersect with).

2. Bound dim_Q`Vtate from above for upper bound. Trivial upper bound: Betti number b2p.

Problem with computingVtate⊂ V2p_{= H}2p

et(Xs, Q`(p)) is that

Algorithms for general

p

I Nump(X )is group of codimension-p cycle classes up to numerical equivalence.

I Assuming Ep(X , `), the mapNump(X , `) ⊗ Q`,→ Vtate is an injection that is an isomorphism if and only if Tp(X , `) holds.

I For p = 1we haveNum1(X ) ∼= NS(X )/ NS(X )tors and an injectionNS(X ) ⊗ Q` → Vtate.

Strategy for boundingrk Nump(Xs_).

1. List cycles to find lower bound (also cycles to intersect with).

2. Bound dim_Q`Vtate from above for upper bound. Trivial upper bound: Betti number b2p.

Problem with computingVtate⊂ V2p_{= H}2p

et(Xs, Q`(p)) is that

Algorithms for general

p

I Nump(X )is group of codimension-p cycle classes up to numerical equivalence.

I Assuming Ep(X , `), the mapNump(X , `) ⊗ Q`,→ Vtate is an injection that is an isomorphism if and only if Tp(X , `) holds.

I For p = 1we haveNum1(X ) ∼= NS(X )/ NS(X )tors and an injectionNS(X ) ⊗ Q` → Vtate.

Strategy for boundingrk Nump(Xs_).

1. List cycles to find lower bound (also cycles to intersect with).

2. Bound dim_Q`Vtate from above for upper bound. Trivial upper bound: Betti number b2p.

Problem with computingVtate⊂ V2p_{= H}2p

et(Xs, Q`(p)) is that

Hypothesis. Can computeTi

`n = Hi<sub>et</sub>(Xs, Z/`nZ)as Γ-module.

Remark. This holds in characteristic0 and for liftable X.

Theorem (Poonen, Testa, vL)

Assume the hypothesis. Then there is an algorithm that takes as input(k, p, X , `) as before, such that, assumingEp(X , `), the algorithm terminates if and only ifTp(X , `)holds, and if the algorithm terminates, it returnsrk Nump(Xs).

Sketch of proof of upper bound for r = dim Vtate.

1. Extend k so thatΓ acts trivially onT_`2p0 (p)with

`0 = `for ` > 2and `0 = 4 for ` = 2.

2. Γ acts trivially onM/Mtors with M = H2pet(Xs, Z`(p))tate (Minkovski’s Lemma on finite-order elements in GLn(Z`)).

3. Computet such that `t killsH2pet(Xs, Z`(p))tors (Wittenberg).

4. `r (n−t)≤ #`t_{M/`</sub>n<sub>M ≤ #(M/`}n_M)Γ_{≤ #T}2p `n (p)Γ= O(`rn). 5. Sequence  log #T<sub>`n</sub>2p(p)Γ log `n−t 

Hypothesis. Can computeTi

`n = Hi<sub>et</sub>(Xs, Z/`nZ)as Γ-module.

Remark. This holds in characteristic0 and for liftable X.

Theorem (Poonen, Testa, vL)

Assume the hypothesis. Then there is an algorithm that takes as input(k, p, X , `) as before, such that, assumingEp(X , `), the algorithm terminates if and only ifTp(X , `)holds, and if the algorithm terminates, it returnsrk Nump(Xs).

Sketch of proof of upper bound for r = dim Vtate.

1. Extend k so thatΓ acts trivially onT_`2p0 (p)with

`0 = `for ` > 2and `0 = 4 for ` = 2.

2. Γ acts trivially onM/Mtors with M = H2pet(Xs, Z`(p))tate (Minkovski’s Lemma on finite-order elements in GLn(Z`)).

3. Computet such that `t killsH2pet(Xs, Z`(p))tors (Wittenberg).

4. `r (n−t)≤ #`t_{M/`</sub>n<sub>M ≤ #(M/`}n_M)Γ_{≤ #T}2p `n (p)Γ= O(`rn). 5. Sequence  log #T<sub>`n</sub>2p(p)Γ log `n−t 

Hypothesis. Can computeTi

`n = Hi<sub>et</sub>(Xs, Z/`nZ)as Γ-module.

Remark. This holds in characteristic0 and for liftable X.

Theorem (Poonen, Testa, vL)

Assume the hypothesis. Then there is an algorithm that takes as input(k, p, X , `) as before, such that, assumingEp(X , `), the algorithm terminates if and only ifTp(X , `)holds, and if the algorithm terminates, it returnsrk Nump(Xs).

Sketch of proof of upper bound for r = dim Vtate.

1. Extend k so thatΓ acts trivially onT_`2p0 (p)with

`0 = `for ` > 2and `0 = 4 for ` = 2.

2. Γ acts trivially onM/Mtors withM = H2pet(Xs, Z`(p))tate (Minkovski’s Lemma on finite-order elements in GLn(Z`)).

3. Computet such that`t killsH2pet(Xs, Z`(p))tors (Wittenberg).

4. `r (n−t)≤ #`t_{M/`</sub>n<sub>M ≤ #(M/`}n_M)Γ_{≤ #T}2p `n (p)Γ= O(`rn). 5. Sequence  log #T<sub>`n</sub>2p(p)Γ log `n−t 

Finite fields

Supposek is finite. LetVµ⊂ V2p = H2pet(Xs, Q`(p))be the largest subspace on which all eigenvalues of Frobenius are roots of unity.

Nump(Xs) ⊗ Q` → Vtate⊂ Vµ

Theorem

AssumingEp(X , `), the following are equivalent.

1. rk Nump(Xs) = dim Vtate.

2. Conjecture Tp(X , `)holds.

3. rk Nump(Xs) = dim Vtate= dim Vµ.

Proof. 1 ⇔ 2. Under Ep(X , `), the first map is injective, so it is surjective if and only if 1 holds.

2 ⇒ 3. Vtate= Vµ follows asEp(X , `)andTp(X , `) together imply that Frobenius acts semi-simple onVµ. 3 ⇒ 1. Obvious.

Finite fields

Supposek is finite. LetVµ⊂ V2p = H2pet(Xs, Q`(p))be the largest subspace on which all eigenvalues of Frobenius are roots of unity.

Nump(Xs) ⊗ Q` → Vtate⊂ Vµ

Theorem

AssumingEp(X , `), the following are equivalent.

1. rk Nump(Xs) = dim Vtate.

2. Conjecture Tp(X , `)holds.

3. rk Nump(Xs) = dim Vtate= dim Vµ.

Proof. 1 ⇔ 2. Under Ep(X , `), the first map is injective, so it is surjective if and only if 1 holds.

2 ⇒ 3. Vtate= Vµ follows asEp(X , `)andTp(X , `) together imply that Frobenius acts semi-simple onVµ. 3 ⇒ 1. Obvious.

Finite fields

Supposek is finite. LetVµ⊂ V2p = H2pet(Xs, Q`(p))be the largest subspace on which all eigenvalues of Frobenius are roots of unity.

Nump(Xs) ⊗ Q` → Vtate⊂ Vµ

Theorem

AssumingEp(X , `), the following are equivalent.

1. rk Nump(Xs) = dim Vtate.

2. Conjecture Tp(X , `)holds.

3. rk Nump(Xs) = dim Vtate= dim Vµ.

Proof. 1 ⇔ 2. Under Ep(X , `), the first map is injective, so it is surjective if and only if 1 holds.

2 ⇒ 3. Vtate_{= V}

µ follows asEp(X , `) andTp(X , `) together imply that Frobenius acts semi-simple onVµ. 3 ⇒ 1. Obvious.

Finite fields

Theorem

There is an algorithm that takes as input(k, p, X , `), with k a finite field, and that, assumingEp(X , `), terminates if and only if

Tp(X , `)holds, and if it terminates, it returns rk NumpXs.

Proof. By searching for cycles, we get a lower bound for

rk NumpXs that eventually is sharp. To verify that it is, it suffices to computedim Vµ. Say k = Fq. The degree of the zeta-function

ZX(T ) = 2 dim X Y i =0 det 1 − T · Frob∗| Hiet(Xs, Q`)

(−1)i +1

is bounded by the sum of Betti numbers: computable boundB. Computing#X (Fqn) forn = 1, . . . , 2B gives enough information

to determineZX(T ). Thendim Vµ equals the number of poles of

Finite fields

Theorem

There is an algorithm that takes as input(k, p, X , `), with k a finite field, and that, assumingEp(X , `), terminates if and only if

Tp(X , `)holds, and if it terminates, it returns rk NumpXs.

Proof. By searching for cycles, we get a lower bound for

rk NumpXs _{that eventually is sharp. To verify that it is, it suffices} to computedim Vµ. Say k = Fq. The degree of the zeta-function

ZX(T ) = 2 dim X Y i =0 det 1 − T · Frob∗| Hiet(Xs, Q`)

(−1)i +1

is bounded by the sum of Betti numbers: computable boundB.

Computing#X (Fqn) forn = 1, . . . , 2B gives enough information

to determineZX(T ). Thendim Vµ equals the number of poles of

Finite fields

Theorem

There is an algorithm that takes as input(k, p, X , `), with k a finite field, and that, assumingEp(X , `), terminates if and only if

Tp(X , `)holds, and if it terminates, it returns rk NumpXs.

Proof. By searching for cycles, we get a lower bound for

rk NumpXs _{that eventually is sharp. To verify that it is, it suffices} to computedim Vµ. Say k = Fq. The degree of the zeta-function

ZX(T ) = 2 dim X Y i =0 det 1 − T · Frob∗| Hiet(Xs, Q`)

(−1)i +1

is bounded by the sum of Betti numbers: computable boundB. Computing#X (Fqn)for n = 1, . . . , 2B gives enough information

to determineZX(T ). Thendim Vµ equals the number of poles of

Finite fields

Example. Let X ⊂ P3 overF2 be given bydet M = 0 with M =

    x0 x2 x1+ x2 x2+ x3 x1 x2+ x3 x0+ x1+ x2+ x3 x0+ x3 x0+ x2 x0+ x1+ x2+ x3 x0+ x1 x2 x0+ x1+ x3 x0+ x2 x3 x2     . Φ = Frob∗| H2_(Xs_{, Q`(1)) t} n= Tr Φn #X (F2n) = 1 + 2nt_n+ 22n n 1 2 3 4 5 6 7 8 9 10 #X (F2n) 6 26 90 258 1146 4178 17002 64962 260442 1044786 tn 12 94 258 161 12132 8164 617128 −575256 −1703512 −37911024 fΦ= palindromic or antipalindromic = t22− 1 2t 21_{− t}20₋ 1 2t 19_{+ t}18₋1 2t 15_{+ t}14₊1 2t 13_{− 2t}11_{+ . . .} = (t − 1)2(t20+3₂t19+ t18−1 2t 13_{+ t}11_{+ 2t}10_{+ . . .).} Conclusion. We haverk NS(Xs) = 2.

Finite fields

Example. Let X ⊂ P3 overF2 be given bydet M = 0 with M =

    x0 x2 x1+ x2 x2+ x3 x1 x2+ x3 x0+ x1+ x2+ x3 x0+ x3 x0+ x2 x0+ x1+ x2+ x3 x0+ x1 x2 x0+ x1+ x3 x0+ x2 x3 x2     . Φ = Frob∗| H2_(Xs_{, Q`(1)) t} n= Tr Φn #X (F2n) = 1 + 2nt_n+ 22n n 1 2 3 4 5 6 7 8 9 10 #X (F2n) 6 26 90 258 1146 4178 17002 64962 260442 1044786 tn 12 94 258 161 12132 8164 617128 −575256 −1703512 −37911024 fΦ= palindromic or antipalindromic = t22− 1 2t 21_{− t}20₋ 1 2t 19_{+ t}18₋1 2t 15_{+ t}14₊1 2t 13_{− 2t}11_{+ . . .} = (t − 1)2(t20+3₂t19+ t18−1 2t 13_{+ t}11_{+ 2t}10_{+ . . .).} Conclusion. We haverk NS(Xs) = 2.

Finite fields

Example. Let X ⊂ P3 overF2 be given bydet M = 0 with M =

    x0 x2 x1+ x2 x2+ x3 x1 x2+ x3 x0+ x1+ x2+ x3 x0+ x3 x0+ x2 x0+ x1+ x2+ x3 x0+ x1 x2 x0+ x1+ x3 x0+ x2 x3 x2     . Φ = Frob∗| H2_(Xs_{, Q`(1)) t} n= Tr Φn #X (F2n) = 1 + 2nt_n+ 22n n 1 2 3 4 5 6 7 8 9 10 #X (F2n) 6 26 90 258 1146 4178 17002 64962 260442 1044786 tn 12 94 258 161 12132 8164 617128 −575256 −1703512 −37911024 fΦ= palindromic or antipalindromic = t22− 1 2t 21_{− t}20₋ 1 2t 19_{+ t}18₋ 1 2t 15_{+ t}14₊ 1 2t 13_{− 2t}11_{+ . . .} = (t − 1)2(t20+3₂t19+ t18−1 2t 13_{+ t}11_{+ 2t}10_{+ . . .).} Conclusion. We haverk NS(Xs) = 2.

Surfaces over global fields

I Global field K, discrete valuation ring R ⊂ K, residue fieldk.

I X a nice surface over K, integral modelX overR with good reduction.

NS(Xs) ⊗ Q`,→ NS(Xks) ⊗ Q<sub>`</sub>

rk NS(Xs) ≤ rk NS(Xks)

Problem. IfT1_(X

ks, `) holds, then rk NS(X_ks) ≡ b₂(X ) (mod 2).

Proof.

The roots offΦ that are not roots of unity come in conjugate pairs.

Question.

Surfaces over global fields

I Global field K, discrete valuation ring R ⊂ K, residue fieldk.

I X a nice surface over K, integral modelX overR with good reduction.

NS(Xs) ⊗ Q`,→ NS(Xks) ⊗ Q<sub>`</sub>

rk NS(Xs) ≤ rk NS(Xks)

Problem. IfT1_(X

ks, `) holds, then rk NS(X_ks) ≡ b₂(X ) (mod 2).

Proof.

The roots offΦ that are not roots of unity come in conjugate pairs.

Question.

The injection

Num1(Xs) ,→ Num1(Xks)

respects the intersection pairing.

Lemma. IfΛ0 is a sublattice of finite index of Λ, then we have

disc Λ0 = [Λ : Λ0]2disc Λ.

Hence,disc Λ = disc Λ0 in Q∗/(Q∗)2.

Corollary(vL). Ifv , w are two places of good reduction with 1)rk Num1(X_{k(v )}s) = r = rk Num1(X_{k(w )}s), and

2)disc Num1(X_{k(v )}s) 6= disc Num1(X_{k(w )}s) in_Q∗/(Q∗)2,

thenrk Num1(Xs) < r.

The injection

Num1(Xs) ,→ Num1(Xks)

respects the intersection pairing.

Lemma. IfΛ0 is a sublattice of finite index of Λ, then we have

disc Λ0 = [Λ : Λ0]2disc Λ.

Hence,disc Λ = disc Λ0 in Q∗/(Q∗)2.

Corollary(vL). Ifv , w are two places of good reduction with 1)rk Num1(X_{k(v )}s) = r = rk Num1(X_{k(w )}s), and

2)disc Num1(X_{k(v )}s) 6= disc Num1(X_{k(w )}s)in _Q∗/(Q∗)2,

thenrk Num1(Xs_{) < r.}

Example. Let X ⊂ P3

Q be given by

wf = 3pq − 2zg

withf ∈ Z[x, y , z, w ]and g , p, q ∈ Z[x, y , z]equal to

f = x3− x2y − x2z + x2w − xy2− xyz + 2xyw + xz2+ 2xzw + y3 + y2z − y2w + yz2+ yzw − yw2+ z2w + zw2+ 2w3, g = xy2+ xyz − xz2− yz2_{+ z}3_, p = z2+ xy + yz, q = z2+ xy . Thenrk NS(Xs) = 1.

Proof. Two primes of good reduction: 2and3. For both we obtaindim Vµ= 2 as before.

ReductionX_F2 contains conic C given byw = p = 0. ReductionX_F3 contains line L given byw = z = 0.

ThenNum1(X_F2) enNum

1_(X

F3) contain finite-index sublattices

4 2 2 −2  and 4 1 1 −2 

with discriminants−12 and−9in Q∗/(Q∗)2.

Example. Let X ⊂ P3

Q be given by

wf = 3pq − 2zg

with . . . Thenrk NS(Xs) = 1.

Proof. Two primes of good reduction: 2and3. For both we obtaindim Vµ= 2 as before.

ReductionX_F2 contains conic C given byw = p = 0.

ReductionX_F3 contains line Lgiven by w = z = 0.

ThenNum1(XF2) enNum

1_(X

F3) contain finite-index sublattices

4 2 2 −2  and 4 1 1 −2 

with discriminants−12 and−9in Q∗/(Q∗)2.

Example. Let X ⊂ P3

Q be given by

wf = 3pq − 2zg

with . . . Thenrk NS(Xs) = 1.

Proof. Two primes of good reduction: 2and3. For both we obtaindim Vµ= 2 as before.

ReductionX_F2 contains conic C given byw = p = 0.

ReductionX_F3 contains line Lgiven by w = z = 0. ThenNum1(X_F2) enNum

1_(X

F3) contain finite-index sublattices

4 2 2 −2  and 4 1 1 −2 

with discriminants−12 and−9in Q∗/(Q∗)2.

Extension by Kloosterman

I ρ = rk Num1(X ) and∆ = disc Num1(X ).

I b2= b2(X )and α = χ(X , OX) − 1 − dim Pic0(X ).

Conjecture(Artin–Tate). lim T →q−1 fΦ(T ) (1 − qT )ρ = (−1)ρ−1· # Br X · ∆ qα_{(# NS(X )} tors)2 . Facts. T1(X , `)⇒ Artin–Tate. T1(X , `)⇒ _{# Br X ∈ (Q}∗)2 (Liu–Lorenzini–Raynaud).

Extension by Kloosterman

I ρ = rk Num1(X ) and∆ = disc Num1(X ).

I b2= b2(X )and α = χ(X , OX) − 1 − dim Pic0(X ).

Conjecture(Artin–Tate). lim T →q−1 fΦ(T ) (1 − qT )ρ = (−1)ρ−1· # Br X · ∆ qα_{(# NS(X )} tors)2 . Facts. T1(X , `)⇒ Artin–Tate. T1(X , `)⇒ _{# Br X ∈ (Q}∗)2 (Liu–Lorenzini–Raynaud).

Application

Theorem (Kloosterman)

The elliptic K3 surfaceπ : X → P1 overQgiven by y2 = x3+ 2(t8+ 14t4+ 1)x + 4t2(t8+ 6t4+ 1)

Extension by Elsenhans-Jahnel, I

Main idea. If you consider Galois action, you may not need

r = rk Num1(Xs) + 1.

Example. (Elsenhans, Jahnel)

LetX : w2 = f (x , y , z)be a K3 surface of degree2 overQwith f ≡ y6+ x4y2− 2x2_y4_{+ 2x}5_{z + 3xz}5_{+ z}6 _{(mod 5)} and

f ≡ 2x6+ x4y2+ 2x3y2z + x2y2z2+ x2yz3+ 2x2z4+ xy4z + xy3z2+ xy2z3+ 2xz5+ 2y6+ y4z2+ y3z3 (mod 3).

Extension by Elsenhans-Jahnel, I

Main idea. If you consider Galois action, you may not need

r = rk Num1(Xs) + 1.

Example. (Elsenhans, Jahnel)

LetX : w2 = f (x , y , z)be a K3 surface of degree2 overQwith f ≡ y6+ x4y2− 2x2_y4_{+ 2x}5_{z + 3xz}5_{+ z}6 _{(mod 5)} and

f ≡ 2x6+ x4y2+ 2x3y2z + x2y2z2+ x2yz3+ 2x2z4+ xy4z + xy3z2+ xy2z3+ 2xz5+ 2y6+ y4z2+ y3z3 (mod 3).

Extension by Elsenhans-Jahnel, I

LetLdenote the pull-back of a line inP2(x , y , z).

The characteristic polynomial of Frobenius acting on the space

(NS X

F3⊗ Q)/hLi

equals(t − 1)(t2+ t + 1), so only finitely many Galois-invariant subspaces ofNS X_F

3⊗ Q containingL; dimensions are1, 2, 3, 4.

The characteristic polynomial of Frobenius acting on the space

(NS X

F5⊗ Q)/hLi

equals(t − 1)Φ5(t)Φ15(t), whereΦn denotes then-th cyclotomic polynomial. So only finitely many Galois-invariant subspaces of

NS X

F5⊗ QcontainingL; dimensions are 1, 2, 5, 6, 9, 10, 13, 14.

Only common dimensions are1and 2. Comparing discriminants up to squares of the subspaces of dimension2 yields rk NS(Xs) = 1.

Extension by Elsenhans-Jahnel, I

LetLdenote the pull-back of a line inP2(x , y , z).

The characteristic polynomial of Frobenius acting on the space

(NS X

F3⊗ Q)/hLi

equals(t − 1)(t2+ t + 1), so only finitely many Galois-invariant subspaces ofNS X_F

3⊗ Q containingL; dimensions are1, 2, 3, 4.

The characteristic polynomial of Frobenius acting on the space

(NS X

F5⊗ Q)/hLi

equals(t − 1)Φ5(t)Φ15(t), whereΦn denotes then-th cyclotomic polynomial. So only finitely many Galois-invariant subspaces of

NS X

F5⊗ QcontainingL; dimensions are 1, 2, 5, 6, 9, 10, 13, 14.

Only common dimensions are1and 2. Comparing discriminants up to squares of the subspaces of dimension2 yields rk NS(Xs) = 1.

Extension by Elsenhans-Jahnel, I

LetLdenote the pull-back of a line inP2(x , y , z).

The characteristic polynomial of Frobenius acting on the space

(NS X

F3⊗ Q)/hLi

equals(t − 1)(t2+ t + 1), so only finitely many Galois-invariant subspaces ofNS X_F

3⊗ Q containingL; dimensions are1, 2, 3, 4.

The characteristic polynomial of Frobenius acting on the space

(NS X

F5⊗ Q)/hLi

equals(t − 1)Φ5(t)Φ15(t), whereΦn denotes then-th cyclotomic polynomial. So only finitely many Galois-invariant subspaces of

NS X

F5⊗ QcontainingL; dimensions are 1, 2, 5, 6, 9, 10, 13, 14.

Only common dimensions are1and2. Comparing discriminants up to squares of the subspaces of dimension2 yieldsrk NS(Xs) = 1.

Extension by Elsenhans-Jahnel, II

I p 6= 2 prime.

I X a scheme that is proper and flat over Z.

Theorem(Elsenhans-Jahnel). If the special fiber Xp is nonsingular, then the cokernel of the specialization homomorphism

sp_Q: Pic(X_Q) → Pic(Xp) is torsion-free.

Corollary(E-J, Hassett–V´arilly-Alvarado).

LetX be a double cover ofP2, ramified over a smooth plane

sexticC. Letp, p0 denote two odd primes of good reduction. Assume that there is a tritangent line`to the curveCp.

SupposePic(X_ps) has rank2 and is generated by the components in the pull-back of`. If there are no tritangent lines toCp0, then

Extension by Elsenhans-Jahnel, II

I p 6= 2 prime.

I X a scheme that is proper and flat over Z.

Theorem(Elsenhans-Jahnel). If the special fiber Xp is nonsingular, then the cokernel of the specialization homomorphism

sp_Q: Pic(X_Q) → Pic(Xp) is torsion-free.

Corollary(E-J, Hassett–V´arilly-Alvarado).

LetX be a double cover ofP2, ramified over a smooth plane

sexticC. Letp, p0 denote two odd primes of good reduction. Assume that there is a tritangent line`to the curveCp.

SupposePic(X_ps) has rank2 and is generated by the components in the pull-back of`. If there are no tritangent lines toCp0, then

“It works” by Charles

Question.

1) Given a nice surfaceX over a number field k, is there always a primep of good reduction with rk Num1(X_ps) ≤ rk Num1(Xs) + 1? 2) Are there two so that the discriminant trick works?

Answer(Charles).

Not always, but if not, then the minimal jumps are still controllable!

Consequence(Charles).

There is an algorithm with input a K3 surfaceX over a number field that either returnsrk NS(Xs_{) or does not terminate.}

IfX × X satisfies the Hodge conjecture for codimension2 cycles, then the algorithm applied to X terminates.

“It works” by Charles

Question.

1) Given a nice surfaceX over a number field k, is there always a primep of good reduction with rk Num1(X_ps) ≤ rk Num1(Xs) + 1? 2) Are there two so that the discriminant trick works?

Answer(Charles).

Not always, but if not, then the minimal jumps are still controllable!

Consequence(Charles).

There is an algorithm with input a K3 surfaceX over a number field that either returnsrk NS(Xs_{) or does not terminate.}

IfX × X satisfies the Hodge conjecture for codimension2 cycles, then the algorithm applied to X terminates.

“It works” by Charles

Question.

1) Given a nice surfaceX over a number field k, is there always a primep of good reduction with rk Num1(X_ps) ≤ rk Num1(Xs) + 1? 2) Are there two so that the discriminant trick works?

Answer(Charles).

Not always, but if not, then the minimal jumps are still controllable!

Consequence(Charles).

There is an algorithm with input a K3 surfaceX over a number field that either returnsrk NS(Xs_{) or does not terminate.}

IfX × X satisfies the Hodge conjecture for codimension2 cycles, then the algorithm applied toX terminates.

Saturation

Theorem (Poonen, Testa, vL)

There is an algorithm that takesk, X, and a finite setD of divisors as input, and computes the saturation insideNS(Xs_{) of the}

Γ-submodule generated by the classes of divisors in D.

Saturation

Goal. Given a surface X over a global fieldK and a sublattice

G ⊂ Num1(Xs), show that G is primitive.

If not primitive, thenG has nontrivial index in its saturation G˜, so there is a primer | [ ˜G : G ]with r2| [ ˜G : G ]2· disc ˜G = disc G. ThenG ⊗ Fr → Num1(Xs) ⊗ Fr is not injective.

For all primesp of good reduction and H ⊂ Num1(X_ps)the map

Num1(Xs) → Hom(H, Z)

induces a non-injective composition

G ⊗ Fr → Num1(Xs) ⊗ Fr → Hom(H ⊗ Fr, Fr).

Sufficient for primitivity. Find for each r with r2| disc G a prime p

and a subgroupH ⊂ Num1(X_ps)for which the composition is injective (linear algebra overFr).

Saturation

Goal. Given a surface X over a global fieldK and a sublattice

G ⊂ Num1(Xs), show that G is primitive.

If not primitive, thenG has nontrivial index in its saturation G˜, so there is a primer | [ ˜G : G ]with r2| [ ˜G : G ]2· disc ˜G = disc G. ThenG ⊗ Fr → Num1(Xs) ⊗ Fr is not injective.

For all primesp of good reduction and H ⊂ Num1(X_ps)the map

Num1(Xs) → Hom(H, Z)

induces a non-injective composition

G ⊗ Fr → Num1(Xs) ⊗ Fr → Hom(H ⊗ Fr, Fr).

Sufficient for primitivity. Find for each r with r2| disc G a prime p

and a subgroupH ⊂ Num1(X_ps)for which the composition is injective (linear algebra overFr).

Saturation

Goal. Given a surface X over a global fieldK and a sublattice

G ⊂ Num1(Xs), show that G is primitive.

If not primitive, thenG has nontrivial index in its saturation G˜, so there is a primer | [ ˜G : G ]with r2| [ ˜G : G ]2· disc ˜G = disc G. ThenG ⊗ Fr → Num1(Xs) ⊗ Fr is not injective.

For all primesp of good reduction and H ⊂ Num1(X_ps)the map

Num1(Xs) → Hom(H, Z)

induces a non-injective composition

G ⊗ Fr → Num1(Xs) ⊗ Fr → Hom(H ⊗ Fr, Fr).

Sufficient for primitivity. Find for each r with r2| disc G a prime p

and a subgroupH ⊂ Num1(X_ps)for which the composition is injective (linear algebra overFr).

Saturation

Goal. Given a surface X over a global fieldK and a sublattice

G ⊂ Num1(Xs), show that G is primitive.

If not primitive, thenG has nontrivial index in its saturation G˜, so there is a primer | [ ˜G : G ]with r2| [ ˜G : G ]2· disc ˜G = disc G. ThenG ⊗ Fr → Num1(Xs) ⊗ Fr is not injective.

For all primesp of good reduction and H ⊂ Num1(X_ps)the map

Num1(Xs) → Hom(H, Z)

induces a non-injective composition

G ⊗ Fr → Num1(Xs) ⊗ Fr → Hom(H ⊗ Fr, Fr).

Sufficient for primitivity. Find for each r with r2| disc G a primep

and a subgroupH ⊂ Num1(X_ps)for which the composition is injective (linear algebra overFr).

Application

Theorem (Mizukami (m = 4), Sch¨utt–Shioda–vL (m ≤ 100))

For any integer1 ≤ m ≤ 100the N´eron-Severi group of the Fermat surfaceSm⊂ P3 overCgiven by

xm+ ym+ zm+ wm = 0