Manin conjectures for K3 surfaces

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Manin Conjectures for K3 surfaces

Ronald van Luijk CRM, Montreal MSRI, Berkeley PIMS, UBC/SFU

May 17, 2006 Banff

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A K3 surface

with 2 singular curves of genus 0 and

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Question 1. Is there a K3 surface, defined over a number field, such that its set of rational points is neither empty, nor dense?

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Examples of K3 surfaces are smooth quartic surfaces in P3. Question 2 (Swinnerton-Dyer).

Does _{X ⊂} P3 given by

x4 + 2y4 = z4 + 4w4 have more than 2 rational points?

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Examples of K3 surfaces are smooth quartic surfaces in P3. Question 2 (Swinnerton-Dyer).

Does _{X ⊂} P3 given by

x4 + 2y4 = z4 + 4w4 have more than 2 rational points?

Answer (Elsenhans, Jahnel, 2004):

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We will look at the growth of the number of rational points of bounded height. Consider a surface X/K, choose a height H, and set

N_U_{(B) = #{x ∈ U(K) : H(x) ≤ B}.}

Conjecture 1 (Batyrev, Manin). Let X be a smooth, geometrically integral, projective variety over a number field _{K, and let D be a} hyper-plane section. Assume that the canonical sheaf _K_X satisfies _−K_X = aD for some _{a > 0. Then there exists a finite field extension L, a constant} C, and an open subset U ⊂ X, such that with b = rk Pic(XL) we have

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We will look at the growth of the number of rational points of bounded height. Consider a surface X/K, choose a height H, and set

N_U_{(B) = #{x ∈ U(K) : H(x) ≤ B}.}

Conjecture 1 (Batyrev, Manin). Let X be a smooth, geometrically integral, projective variety over a number field _{K, and let D be a} hyper-plane section. Assume that the canonical sheaf _K_X satisfies _−K_X = aD for some _{a > 0. Then there exists a finite field extension L, a constant} C, and an open subset U ⊂ X, such that with b = rk Pic(XL) we have

N_U_L_{(B) ≈ CB}a(log B)b−1.

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Define the height-zeta function Z(U, s) = X x∈U(K) H(x)−s. From 1 2πi Z c+i∞ c−i∞ x s ds s =      1 if x > 1 1 2 if x = 1 0 if x < 1 (c > 0) we get N (U, x) = 1 2πi Z c+i∞ c−i∞ Z(U, s) x s ds s (c >> 0)

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Assuming Z(U, s) is analytic on <(s) > a − ², except for a pole of order b at a, we can write

N (U, x) = 1 2πi Z c+i∞ c−i∞ Z(U, s) x s ds s (c >> 0) = res_s=a hZ(U, s)s−1 exp(s log x)i + 1

2πi

Z a−²+i∞

a−²−i∞ Z(U, s) x

s ds

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2πi

Z a−²+i∞

a−²−i∞ Z(U, s) x

s ds

s .

The main term is

xap(log x)

for some polynomial p of degree (

b − 1 if a 6= 0, b if a = 0.

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2πi

Z a−²+i∞

a−²−i∞ Z(U, s) x

s ds

s .

The main term is

xap(log x)

for some polynomial p of degree (

b − 1 if a 6= 0, b if a = 0.

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For X a K3 surface: N (U, B) ∼ C(log B)rk Pic X ?

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Not if X admits an elliptic fibration (in particular, if rk Pic X ≥ 5). More problems:

• infinitely many curves of genus 0 or 1,

• infinitely many automorphisms (cf. A. Baragar), • Swinnerton-Dyer’s surface has very slow growth, • which height to use?

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Not if X admits an elliptic fibration (in particular, if rk Pic X ≥ 5). More problems:

• infinitely many curves of genus 0 or 1,

• infinitely many automorphisms (cf. A. Baragar), • Swinnerton-Dyer’s surface has very slow growth, • which height to use?

We will run experiments on quartic surfaces, comparing exponents, and the main coefficient C to a relatively naive heuristic coefficient ˜C (part of Peyre’s constant for del Pezzo).

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10 20 30 40 # pt’s ˜ C ∼ 6.24 line : _{C(log B) − 9}˜ ρ(X_Q) = 1 # Aut X_Q _{< ∞} no elliptic fibration f = x3_{w − 3x}2y2 _{− x}2_{yw − x}2zw + x2w2

−xy2z − xy2w − 4xyz2 − xyzw + 2xyw2 − 2xz3 +xz2w + 2xzw2 + y3w + y2_{zw − y}2w2 _{− 5yz}3 +yz2w + yzw2 _{− yw}3 _{− z}4 + z2w2 + zw3 + 2w4

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2 4 6 8 10 # pt’s ˜ C ∼ 0.57 line : C(log B)˜ ρ(X) = 1 ρ(X_Q) = 20 # Aut X_Q _{= ∞}

no elliptic fibration over Q

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10 20 30 40 # pt’s ˜ C ∼ 7.8 line : 3₄_{C(log B) − 10}˜ ρ(X_Q) = 1 # Aut X_Q _{< ∞} no elliptic fibration f = x3_{w − 3x}2y2 + 4x2_{yz − x}2z2 −x2z2 + x2_{zw − xy}2_{z − xyz}2 +xw3 + y3w + y2z2 + z3w 17

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Consider X given by

f + 6m(w4 + 2x4 _{− (y + w)}4 _{− 4z}4_{) for m ∈ {±1, ±2, ±3}} with f as on the previous page.

Then X has geometric Picard number 1.

m C˜ _{C · log(12000) #{x : H(x) ≤ 12000}}˜ 0 7.8 73 46 1 0.30 2.8 5 −1 0.58 5.4 9 2 0.39 3.7 2 −2 0.35 3.3 6 3 0.18 1.7 4

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5 10 15 20 25 30 # pt’s ˜ C ∼ 6.66 line : 1₂C(log B)˜ ρ(X) = 1 ρ(X_Q) = 2 (over Q(√3)) Aut X_Q _{= {1}} no elliptic fibration f = x3_{w − x}2y2 _{− 2x}2yz −x2z2 + x2zw + xy2z + xyz2 xw3 + y3_{w − y}2z2 + z3w 19

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20 40 60 80 100 # pt’s ˜ C ∼ 0.96 parabola : 1.1(log B)2 ρ(X) = 2 ρ(X_Q) = 2 Aut X_Q _{= {1}} no elliptic fibration f = 3x3_{w − x}2y2 _{− 2x}2yz

−x2z2 _{− 4xy}2z + 3xy2_{w − 4xyz}2 +4xyw2 + 3xzw2 + 4xw3 + y3w −3y2z2 + 4y2w2 + 2z3w + w4 20

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1 2 3 4 5 log(# pt’s) ˜ C ∼ 0.96

line : 2(log log B) + log 1.1 ρ(X) = 2

ρ(X_Q) = 2 Aut X_Q _{= {1}}

no elliptic fibration

f = 3x3_{w − x}2y2 _{− 2x}2yz

−x2z2 _{− 4xy}2z + 3xy2_{w − 4xyz}2 +4xyw2 + 3xzw2 + 4xw3 + y3w −3y2z2 + 4y2w2 + 2z3w + w4 21

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10 20 30 40 # pt’s ˜ C ∼ 0.53 parabola : 3₂_{C((log B) − 2)}˜ 2 + 2 ρ(X) = 2 ρ(X_Q) = 2 Aut X_Q _{= {1}} no elliptic fibration f = −6x3y − 6x3z − 3x3w − x2y2 + 16x2yz +35x2z2 + 2xy2z + 3xy2w + 8xyz2 _{− 2xyw}2 +6xz3 + 3xzw2 _{− 2xw}3 + y3w + 3y2z2

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1 2 3 4 y log(# pt’s) ˜ C ∼ 0.53

line : _{2(log log B) − 0.75} ρ(X) = 2

ρ(X_Q) = 2 Aut X_Q _{= {1}}

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50 100 150 200 250 # pt’s ˜ C ∼ 0.012 cubic : _{0.9((log B) − 3)}3 + 10 ρ(X) = 3 ρ(X_Q) = 3 # Aut X_Q _{= ∞ (A.Baragar)} no elliptic fibration f = −7x3z + 11x3w + 7x2y2 _{− 11x}2_{yz − 11x}2_{yw − x}2z2 +16x2_{zw − 9x}2w2 + 10xy2z + 5xy2_{w − 44xyz}2 + 10xyzw

−14xyw2 − 10xz3 + 18xz2_{w − 15xzw}2 + 5xw3 + 2y4 + 23y3_{z − 7y}3w + 16y2z2 −3y2zw + 12y2w2 _{− yz}3 _{− 15yz}2_{w − 21yzw}2 _{− 3yw}3 + 18z4 + 16z3w + 3z2w2

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1 2 3 4 5 y log(# pt’s) ˜ C ∼ 0.012

line : _{3(log log B) − 1.2} ρ(X) = 3

ρ(X_Q) = 3

# Aut X_Q _{= ∞}

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10 20 30 40 50 60 # pt’s ˜ C ∼ 0.00061 parabola : 0.074(log B)3 ρ(X) = 3 ρ(X_Q) = 3 # Aut X_Q _{= ∞ (A.Baragar)} no elliptic fibration f = · · · 26

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1 2 3 4 log(# pt’s) ˜ C ∼ 0.00061

line : _{3(log log B) − 2.6} ρ(X) = 3

ρ(X_Q) = 3

# Aut X_Q _{= ∞}