The Picard lattice

  • X an algebraic K3 surface; its Picard lattice:

Pic (X) ≃ℤρ, ρ(X) := rk Pic (X)

Pic (Xal) ≃ℤ⟨algebraic curves in X ⟩/ ⟨linear equivalences ⟩⊂H2(X, ℤ)

  • Records the algebraic cycles on X: curves modulo linear, algebraic or numerical equivalence.
  • For K3 surfaces the three agree; Pic 0 = 0, and Pic is finite free.
  • Pic (ℙ2) = ℤ, Pic (ℙ1 ×ℙ1) = ℤ2, Pic (cubic surface) = ℤ7.
  • ρ, and more precisely the Picard lattice, is a coarse invariant.
  • K3 theorems are stated by lattice or rank, not equation; as for abelian varieties.

The lattice structure

  • Pic (Xal) with the intersection pairing (D,D') ↦D ·D'.
  • Even symmetric bilinear form: D ·D = 2pa(D) − 2 for every curve D, by adjunction (KX = 0).
  • Signature (1,ρ− 1) (Hodge index theorem): one positive direction, negative definite complement; hence non-degenerate.
  • disc Pic (Xal) := det of the Gram matrix in any ℤ-basis; a non-zero integer. A basis change multiplies it by det (M)2 = 1.
  • Gal (kal/k) permutes curves, respects linear equivalence and preserves the pairing: an orthogonal action.

Pic inside H2

  • Over ℚal, viewing X also as a complex manifold,

Pic (Xal)≃H1,1(Xℂ) ∩H2(Xℂ, ℤ)⊂H2(Xℂ, ℤ) ≃(−E8)2 ⊕U3 ≃ℤ22

  • Lefschetz (1,1): an integral class is algebraic exactly when it has type (1,1).
  • H2(Xℂ,ℤ) is the unique even unimodular lattice of signature (3,19); the embedding is primitive.
  • dim H1,1(X) = 20, so ρ(Xal) ∈{1,2,...,20}; for a generic K3 surface ρ(Xal) = 1.
  • The degree of "difficulty" is negatively correlated with ρ(X).
  • T(X) := Pic (Xal)⊥ ⊂H2(X,ℤ), the transcendental lattice; equivalently the minimal sub-Hodge structure of H2(X,ℚ) whose complexification contains H2,0(X).

    H2(Xℂ,ℚ) ≃Pic (Xal)ℚ ⊕T(X)ℚ

  • The "new and interesting" Galois representations arise from T(X).
  • In characteristic p the bound 20 fails: ρ(Xpal) can be as large as 22.

Geometric versus ground-field Picard group

  • k a number field, X a K3 surface over k;
  • p a prime of k where X has good reduction Xp;
  • Pic (•), the group of line bundles modulo isomorphism; for K3 surfaces Pic 0(•)=0;
  • ρ(•) := rk Pic (•), the arithmetic or geometric Picard number of •.
  • Pic (X) = Pic (Xal)Gal (kal/k), so ρ(X) ≤ρ(Xal), usually strictly.
  • A class defined only over an extension is invisible over k.
  • "The Picard number" in this lecture means the geometric one, ρ(Xal).

Goal

From the equations of X, compute Pic (Xal) ⊂H2(X,ℤ) as a Gal (kal/k)-module.

"The evaluation of ρ for a given surface presents in general grave difficulties." (Zariski)

Question

  • How are the geometric Picard numbers ρ(Xal) and ρ(Xpal) related?
  • How does the geometric Picard number behave under reduction modulo p?

H1(Gal (kal/k), Pic Xal)≃Br 1(X)/Br 0(X)X(k)⊂X(𝔸k)Br ⊂X(𝔸k)

What the characteristic polynomial gives you

Theorem (many people)

Let X/𝔽q, where q = pn, be an abelian surface or a K3 surface. Then:

  • ρ(Xp) = ord t = q P2(t)
  • ρ(Xpal) = ∑ζ ord t = qζ P2(t), where ζ runs over all roots of unity.
  • For K3 surfaces: ρ(Xpal) is even.
Artin-Tate for K3 surfaces

lim t→q(P2(t))/((t−q)ρ) =(−1)ρ−1q21−ρ#Br (Xp) disc (Pic (Xp))

  • P2(t) ⇝disc (Pic (Xp)) mod ℚ×2

ρ=rk Pic (Xp), #Br (Xp)∈ℚ×2Tate⇒Artin−Tate

Picard lattice, over finite fields

  • P2(t) = det (t − Frob ∣H2); roots αi, |αi| = q.
  • q−22P2(qt) monic; roots ζi := αi/q, |ζi| = 1.
  • Tate classes correspond to roots of unity (Tate, a theorem for K3 surfaces over finite fields).

q−22P2(qt) = h(t)∏iΦki(t)γiΦk the k-th cyclotomic polynomial; h has no cyclotomic factorρ(X𝔽qr) = ∑ki∣rγideg Φki

  • Example: X := Z(y4 − x3z + yz3 + zw3 + w4) ⊂ℙ3, p = 89.

p−22P2(pt) = (t−1)(t+1)(t−1)4(t4+1)h(t), deg h = 12

  • (t−1) = Φ1, degree 1;
  • (t+1) = Φ2, degree 1;
  • (t−1)4 = Φ14, degree 4;
  • (t4+1) = Φ8, degree 4.
  • Over 𝔽89: only k=1 divides r=1, so ρ(X𝔽89) = 1+4 = 5.
  • Over 𝔽89r: Φ2 joins when 2∣r, Φ8 when 8∣r; ρ(X89al) = 1+1+4+4 = 10, reached at r=8.
  • Pic (X89al)ℚ decomposes as Pζ1⊕Pζ2⊕Pζ8.
  • For p > 7, naive point counting is impractical; crystalline methods [Abbott--Kedlaya--Roe, C, C--Harvey--Kedlaya, Tuitman--Pancratz].

Reduction to finite characteristic

Take f ∈ℤ[x,y,z,w] and X := Z(f) ⊂ℙ3ℚ.

We may consider the surface Xp := Z(f mod p) ⊂ℙ3(𝔽p).

Theorem

If X and Xp are smooth then the specialization map is injective

Pic (Xal) ↪Pic (Xpal) and ρ(Xal) ≤ρ(Xpal).

Goal

For a given f and p, improve the inequality ρ(Xal) ≤ρ(Xpal).

Parity reasons might already force the inequality to not be sharp.

Endomorphisms of the transcendental lattice can complicate things even further.

Pic plays the role of End (A)

  • Pic for a K3 surface plays a similar role as End (A) for an abelian variety A.

Pic (A)/Pic 0(A) = NS (A)

(Pic (A)/Pic 0(A))ℚ ≃{φ∈End (A)ℚ : φ† = φ}, † the Rosati involution

  • "Compute the Picard lattice of a K3 surface" is the same kind of question as "compute the endomorphism algebra of an abelian variety".
  • Slides 9 and 10 answer the second one, for elliptic curves, by reduction mod p.
  • For Kummer surfaces: ρ(Km (A)) = ρ(A)+16, the identity used in section 1.4.

End ℚ Eal = ℚ or ℚ(√(−d)) (CM)

  • End ℚ Eal ↪End ℚ Epal ↩ℚ(Frob p).
  • p∤ap ⟺End ℚ Epal is a quadratic field
  • If E has CM by ℚ(√(−d)), then

    ap ≡0 mod p⟺p inert or ramified in ℚ(√(−d))⟺End ℚ Eal ≄ End ℚ Epal

  • If E is non-CM, then End ℚ Epal ∩End ℚ Eqal ≃ℚ with prob. 1;
    and we expect Prob (ap ≡0 mod p) ∼1/√(p)

Examples: 11.a2 and 27.a2

E: y2 + y = x3 − x2 − 10x − 20 (LMFDB label: 11.a2)

  • End ℚ E3al ≃ℚ(√(−11))
  • End ℚ E13al ≃ℚ(√(−1))
  • ⇒End ℚ Eal = ℚ

E: y2 + y = x3 − 7 (LMFDB label: 27.a2)

  • p = 2 mod 3 ⇒ap = 0 ⇒End ℚ Epal is a quaternion algebra
  • p = 1 mod 3 ⇒End ℚ Epal ≃ℚ(√(−3))
  • ⇝End ℚ Eal = ℚ(√(−3))

Improving upper bounds: two specializations

Pic (Xal) ↪Pic (Xpal) and ρ(Xal) ≤ρ(Xpal)

van Luijk

If p and q are two primes of good reduction, and

ρ(Xpal) = ρ(Xqal) = 2r,disc Pic (Xpal) ≠disc Pic (Xqal) in ℚ×/(ℚ×)2.

then

ρ(Xal) < 2r.

van Luijk, used this technique with r = 1, to provide the first known examples of K3 surfaces over ℚ such that ρ(Xal) = 1

Does this always work?

Let's apply it to a K3 surface with a ℤ/5 automorphism

X : x3 z + 3x2 y2 + 5xw3 + y3 w + 3yz3 − 5z2 w2 = 0 ⊂ ℙ3

pρ(Xpal)disc
1118−55
1318−85
Theorem

P2(t) ⇝disc Pic (Xp) mod (ℚ×)2.

  • disc: disc Pic (Xpal), the geometric discriminant.
  • Base-field ranks: 1 at 11, 5 at 13.
  • Artin-Tate over 𝔽1130 and 𝔽134: ρ(Xp) = ρ(Xpal) = 18.
  • Equal ranks; −55 : −85 = 11/17 ∉(ℚ×)2.
  • Van Luijk: ρ(Xal) < 18, hence ≤17.
  • Symplectic order-5 action: ρ(Xal) ≥17 [Garbagnati-Sarti 2007, Prop. 1.1].

Torsion-free cokernel

Theorem (Elsenhans-Jahnel)

The specialization map

Pic (Xal) ↪Pic (Xpal)

has torsion-free cokernel for p ≠2.

Thus, if ρ(Xpal) = ρ(Xal) every invertible sheaf lifts.

For example, if ρ(Xpal) = 2,

  1. compute Pic (Xpal)
  2. estimate the degree of a hypothetical effective divisor of the lift
  3. use Gröbner bases to verify that such a divisor does or does not exist

This approach is only practical if one can compute Pic (Xpal) and if the obtained estimates are low.

Why the reduction rank is even

  • Xp/𝔽q K3; P2(t)=det (t−Frob ∣H2)

q−22P2(qt)=h(t)∏iΦki(t)γi

  • h∈ℚ[t]: no cyclotomic factor
  • |z|=1: conj (z)=z−1
  • Real roots: +1,−1, already in Φ1,Φ2
  • Roots of h: nonreal pairs; deg h even
Weil + Tate

ρ(Xpal)=∑iγideg Φki=22−deg h∈2ℤ

Endomorphisms of the transcendental Hodge structure

  • X/k K3; k⊂ℂ a number field

T:=T(X)ℚ=c1(Pic (Xal))ℚ⊥⊂H2(Xℂ,ℚ)

E:=End Hdg(T)={a∈End ℚ(T):aℂ(Ti,j)⊂Ti,j}

T minimal rational sub-Hodge structure with H2,0⊂Tℂ: 0≠α∈E⇒α(H2,0)=H2,0⇒im α=T⇒α−1∈E

Theorem (Zarhin)
  • E: a totally real field or a CM field
  • Totally real: every embedding E↪ℂ lands in ℝ
  • CM: totally imaginary quadratic extension of a totally real field
  • d:=[E:ℚ], m:=dim E T; dm=22−ρ(Xal)
  • E totally real ⇒m≥3 [van Geemen]
  • E totally real; V:=Tℓ(1); g=Frob pa in connected monodromy

V⊗ℚℓal=⨁σ:E↪ℚℓalVσ, dim Vσ=m, g|Vσ∈SO(Vσ)

m odd⇒dim ker (g−1)≥d⇒ρ(Xpal)≥ρ(Xal)+d

Jumping Picard ranks

η(Xal):=min p good(ρ(Xpal)−ρ(Xal))

Consider

Πjump(X):={p good:ρ(Xpal)>ρ(Xal)+η(Xal)}

Is this set infinite? What is its density?

What about

X/ℚ: γ(X,B):=(#{p≤B:p∈Πjump(X)})/(#{p≤B:p prime}) as B→∞ ?

K3 surfaces

So far we have been trying to improve the inequality ρ(Xal)≤ρ(Xpal).
Can we use the inequality to our advantage?

Theorem (Li-Liedtke)

If there are infinitely many p primes such that

ρ(Xal)<ρ(Xpal) and ρ(Xpal)≠22,

then Xal contains infinitely many rational curves.

Theorem (Bogomolov-Zarhin)

The set {p:ρ(Xpal)≠22} has positive density (density 1 after finite extension).

Corollary (after Li-Liedtke; C-Elsenhans-Jahnel)
  • X/k K3; k a number field; e:=[L:k]∈{1,2}

J(X):={p good:ρ(Xpal)>ρ(Xal)}

SL:={{p good}L=k,{p good, inert in L/k}e=2.

  • SL⊂J(X), up to finitely many primes: lower density ≥1/e
  • Additionally L=k or E=ℚ: infinitely many integral rational curves on Xal

Product of elliptic curves

  • X=Km (A); A/ℚ an abelian surface
  • ρ(Aal):=rk (Pic (Aal)/Pic 0(Aal))
  • ρ(Xal)=16+ρ(Aal)
  • ρ(Xpal)=16+ρ(Apal); p>2 good
  • η(Xal)=η(Aal)=ρ(Aal) mod 2
  • Πjump(X)=Πjump(A)
  • Fix a polarization on A; † the Rosati involution

(Pic (Aal)/Pic 0(Aal))ℚ≃{φ∈End (Aal)ℚ:φ†=φ}

  • A=E1×E2; Ei/ℚ

ρ(Xal)=18+rk Hom (E1al,E2al)

X ρ(Xal) γ(X,B), predicted What is known
square of CM 20 1/2 1/2+o(1), CM theory
square of non-CM 19 ∼cX/√(B) infinitely many [Elkies 1987]
CM times CM 18 1/4 1/4+o(1), CM theory
CM times non-CM 18 ∼cX/√(B) infinitely many [Charles 2018]
non-CM times non-CM 18 ∼cX/√(B) infinitely many [Charles 2018]
  • Product rows: geometrically non-isogenous factors
  • Non-CM rates: Lang-Trotter heuristics; per-prime scale 1/√(p)

Remark

p∈Πjump(X) depends uniquely on the pair (aE1(p),aE2(p)).

Jumping Picard ranks for Kummer surfaces

  • ρ(Apal)≥4⟺Apal∼E2, E an elliptic curve
  • ρ(Apal)=6⟺Apal∼E2, E a supersingular elliptic curve
  • If Aal∼E2, then p∈Πjump(A) iff p is supersingular for E.
  • If Aal∼E1×E2 with E1al≁ E2al, then p∈Πjump(A) iff E1,pal∼E2,pal.
  • If End (Aal)=ℤ, then p∈Πjump(A) iff Apal∼E2.

O or SO?

  • V:=Tℓ(1); cup-product pairing

τ:Gal (kal/k)⟶O(V)

  • det τ=1⟺im τ⊂SO(V)
  • det τ≠1: nontrivial quadratic character
  • An easy way to explain some jumps: O vs SO.

What det =−1 costs you

  • ρ(Xal) even; φ:=Frob p|Tℓ(1); det φ=−1
  1. Orthogonality: λ and λ−1, with equal multiplicities.
  2. Other pairs: determinant +1; multiplicity of −1 odd.
  3. dim Tℓ(1) even: multiplicity of +1 odd.
  4. Tate: +1,−1 give two new geometric divisor classes.

    ρ(Xpal)≥ρ(Xal)+2

Discriminant of a K3 surface

  • X/ℚ quartic K3
  • DX:=ΔH2(X)∈ℚ×/(ℚ×)2: determinant-character square class
  • DX∈ℤ∖{0} a representative; p good, p∤2DX
Theorem (Deligne; C-Elsenhans-Jahnel 2020)

The functional equation of the Frobenius action on H2(X) has the plus sign if and only if DX is square mod p.

εp=det (−Frob p∣H2et(Xal,ℚℓ(1)))=((DX)/(p))

  • Gal (ℚal/ℚ) fixes Pic (Xal): ΔPic(X)=1
Theorem (C-Elsenhans-Jahnel)

ρ(Xal)=2r, ((DX)/(p))=−1 ⇒ ρ(Xpal)≥2r+2

We can explain the 1/2

  • X/ℚ K3; r:=ρ(Xal) even; η(Xal)=0
  • dX:=ΔH2(X)ΔPic(X) modulo squares; dX∈ℤ∖{0}
Theorem (C-Elsenhans-Jahnel)

p good, p∤2dX: det (Frob p∣Tℓ(1))=((dX)/(p))=−1⇒ρ(Xpal)≥r+2

Corollary
  • dX nonsquare: L=ℚ(√(dX)), [L:ℚ]=2
  • SL⊂J(X)=Πjump(X), up to finitely many primes
  • liminf B→∞γ(X,B)≥1/2
  • E=ℚ: infinitely many integral rational curves on Xal
  • Example: Costa-Tschinkel 2014, Ex. 3.3

dX=−1·5·151·22490817357414371041·387308497430149337233666358807996260780875056740850984213276970343278935342068889706146733313789

Computing ρ(Xal)

  • T=T(X)ℚ; E=End Hdg(T); d=[E:ℚ]; m=dim E T
Theorem (Charles 2014)

ρ(Xpal)≥{ρ(Xal)if E is CM or m is even,ρ(Xal)+dif E is totally real and m is odd.

  • Equality occurs infinitely often (density 1 after some finite extension).

Further, assume that we are in the second case, then exist infinitely many pairs (p,q) such that the equality holds and

disc Pic (Xpal)≢disc Pic (Xqal) mod (ℚ×)2

When every prime overshoots

  • r:=ρ(Xal); d:=[E:ℚ]; m:=dim E T
  • E=ℚ, m odd: η=1; van Luijk succeeds [Charles 2014]

    Order-5 example: 17≤ρ(Xal)<18

  • E totally real, E≠ℚ, m odd: η=d≥2

    min pρ(Xpal)=r+d; two-prime upper bound: r+d−1

Certified quadratic RM

F↪E, [F:ℚ]=2; ρ(Xpal)=ρ(Xqal)=18

disc Pic (Xpal)≢disc Pic (Xqal) mod (ℚ×)2

ρ(Xal)≤17, ρ(Xal) even ⇒ ρ(Xal)≤16

A real multiplication example

  • Elsenhans-Jahnel [2014, Thms. 5.12 and 6.6]
  • X: minimal resolution of

w2=(−y2/8+yz−z2)(7x2/8+5xz+7z2)(2x2+3xy+y2)

  • 6 lines; 15=(62) nodes; 15 exceptional (−2)-curves
  • H,Eij: 16 independent classes
  • ρ(Xal)=16
  • RM: E=ℚ(√(2)); dim E T=(22−16)/2=3
  • η=2; ρ(Xpal)≥18 at every good prime
  • Rank-18 pair, unequal square classes, certified RM: ρ(Xal)≤16