Variation of NronSeveri ranks of reductions of K3 surfaces Edgar - PowerPoint PPT Presentation

Variation of Néron–Severi ranks of reductions of K3 surfaces Edgar Costa (Massachusetts Institute of Technology) May 28th, 2019 The University of Tenessee Knoxville Presented at 2019 John H. Barrett Memorial Lectures Slides available at edgarcosta.org under Research

E p for many p , what can we say about E ? Elliptic curves • Given studying the statistical properties E p . E : y 2 = x 3 + ax + b , a , b ∈ Z Write E p := E mod p • What can we say about # E p for an arbitrary p ?

studying the statistical properties Elliptic curves E p . E : y 2 = x 3 + ax + b , a , b ∈ Z Write E p := E mod p • What can we say about # E p for an arbitrary p ? • Given # E p for many p , what can we say about E ?

Elliptic curves E : y 2 = x 3 + ax + b , a , b ∈ Z Write E p := E mod p • What can we say about # E p for an arbitrary p ? • Given # E p for many p , what can we say about E ? ⇝ studying the statistical properties # E p .

Hasse’s bound p p as p What can we say about the error term a p Question p p 2 2 E p Theorem (Hasse) 1 p a p Alternatively, we could also have written the formula above as ? a p ∈ [ − 2 √ p , 2 √ p ] # E p = p + 1 − a p ,

Hasse’s bound Theorem (Hasse) Alternatively, we could also have written the formula above as Question What can we say about the error term a p p as p ? a p ∈ [ − 2 √ p , 2 √ p ] # E p = p + 1 − a p , a p := p + 1 − # E p = Tr Frob p ∈ [ − 2 √ p , 2 √ p ]

Hasse’s bound Theorem (Hasse) Alternatively, we could also have written the formula above as Question a p ∈ [ − 2 √ p , 2 √ p ] # E p = p + 1 − a p , a p := p + 1 − # E p = Tr Frob p ∈ [ − 2 √ p , 2 √ p ] What can we say about the error term a p / √ p as p → ∞ ?

Two types of elliptic curves non-CM CM E E d Over an elliptic curve E is a torus E where 1 2 and we have E a p := p + 1 − # E p = Tr Frob p ∈ [ − 2 √ p , 2 √ p ] There are two limiting distributions for a p / √ p

Two types of elliptic curves Over and we have 2 1 where E an elliptic curve E is a torus E E d E CM non-CM a p := p + 1 − # E p = Tr Frob p ∈ [ − 2 √ p , 2 √ p ] There are two limiting distributions for a p / √ p - 2 - 1 1 2 - 2 - 1 0 1 2

Two types of elliptic curves Over and we have 2 1 where E an elliptic curve E is a torus E non-CM CM a p := p + 1 − # E p = Tr Frob p ∈ [ − 2 √ p , 2 √ p ] There are two limiting distributions for a p / √ p √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2

Two types of elliptic curves non-CM CM a p := p + 1 − # E p = Tr Frob p ∈ [ − 2 √ p , 2 √ p ] There are two limiting distributions for a p / √ p √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2 Over C an elliptic curve E is a torus E C ≃ C / Λ , where Λ = Z ω 1 + Z ω 2 = and we have End E al = End Λ

E p . How to distinguish between the two types? non-CM E q E p • If E is non-CM, then E 0, then • If E has CM and a p with prob. 1. CM √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2 • End Q E al ֒ → End Q E al p ← ֓ Q (Frob p ) ⇒ End Q E p al is a quadratic field • a p ̸ = 0 ⇐

How to distinguish between the two types? non-CM E q E p • If E is non-CM, then with prob. 1. CM √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2 • End Q E al ֒ → End Q E al p ← ֓ Q (Frob p ) ⇒ End Q E p al is a quadratic field • a p ̸ = 0 ⇐ • If E has CM and a p ̸ = 0, then End Q E al ≃ End Q E al p .

How to distinguish between the two types? non-CM CM √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2 • End Q E al ֒ → End Q E al p ← ֓ Q (Frob p ) ⇒ End Q E p al is a quadratic field • a p ̸ = 0 ⇐ • If E has CM and a p ̸ = 0, then End Q E al ≃ End Q E al p . • If E is non-CM, then End Q E al p ∩ End Q E al q ≃ Q with prob. 1.

E p is a Quaternion algebra Examples 3 E • 3 E p 3 1 • p 0 a p 2 • p 7 (27.a2) x 3 y y 2 E 3 E : y 2 + y = x 3 − x 2 − 10 x − 20 (11.a2) 2 ≃ Q ( √− 1 ) • End Q E al 3 ≃ Q ( √− 11 ) • End Q E al • ⇒ End Q E al = Q

Examples E : y 2 + y = x 3 − x 2 − 10 x − 20 (11.a2) 2 ≃ Q ( √− 1 ) • End Q E al 3 ≃ Q ( √− 11 ) • End Q E al • ⇒ End Q E al = Q E : y 2 + y = x 3 − 7 (27.a2) • p = 2 mod 3 ⇒ a p = 0 ⇒ End Q E al p is a Quaternion algebra √ • p = 1 mod 3 ⇒ End Q E al p ≃ Q ( − 3 ) √ • ⇝ End Q E al = Q ( − 3 )

X p or p we study K3 surfaces Can we play similar game as before? 2 4 X p p In this case, instead of studying 22 They may arise in many ways: K3 surfaces are a possible generalization of elliptic curves • smooth quartic surfaces in P 3 X : f ( x , y , z , w ) = 0 , deg f = 4 • double cover of P 2 branched over a sextic curve X : w 2 = f ( x , y , z ) , deg f = 6

X p or p we study K3 surfaces Can we play similar game as before? 2 4 X p p In this case, instead of studying 22 They may arise in many ways: K3 surfaces are a possible generalization of elliptic curves • smooth quartic surfaces in P 3 X : f ( x , y , z , w ) = 0 , deg f = 4 • double cover of P 2 branched over a sextic curve X : w 2 = f ( x , y , z ) , deg f = 6

K3 surfaces K3 surfaces are a possible generalization of elliptic curves They may arise in many ways: Can we play similar game as before? • smooth quartic surfaces in P 3 X : f ( x , y , z , w ) = 0 , deg f = 4 • double cover of P 2 branched over a sextic curve X : w 2 = f ( x , y , z ) , deg f = 6 In this case, instead of studying # X p or Tr Frob p we study → rk NS X p al ∈ { 2 , 4 , . . . , 22 } p �−

K3 Surfaces 0 p is ramified or inert in p 0 a p p In the later case, d if E has CM by 1 2 if E is non-CM (Lang–Trotter) p 1 a p • 0 a p 4 E p • Recall that: This is analogous to studying: d X / Q a K3 surface → rk NS X p al ∈ { 2 , 4 , . . . , 22 } p �− → rk End E p al ∈ { 2 , 4 } p �−

K3 Surfaces 1 p is ramified or inert in p 0 a p p In the later case, if E is non-CM (Lang–Trotter) d This is analogous to studying: Recall that: X / Q a K3 surface → rk NS X p al ∈ { 2 , 4 , . . . , 22 } p �− → rk End E p al ∈ { 2 , 4 } p �− • rk End E p al = 4 ⇐ ⇒ a p = 0  ? ∼ √ p  • Prob( a p = 0 ) = √ 1 / 2 if E has CM by Q ( − d ) 

K3 Surfaces This is analogous to studying: Recall that: if E is non-CM (Lang–Trotter) In the later case, 1 X / Q a K3 surface → rk NS X p al ∈ { 2 , 4 , . . . , 22 } p �− → rk End E p al ∈ { 2 , 4 } p �− • rk End E p al = 4 ⇐ ⇒ a p = 0  ? ∼ √ p  • Prob( a p = 0 ) = √ 1 / 2 if E has CM by Q ( − d )  √ { p : a p = 0 } = { p : p is ramified or inert in Q ( − d ) }

Néron–Severi group X p X q q X p For infinitely many p we have Theorem (Charles) 22 2 4 X p X p 20 1 2 X X X . • NS • = Néron–Severi group of • ≃ { curves on •} / ∼ • ρ ( • ) = rk NS • • X p := X mod p

Néron–Severi group � X q q X p For infinitely many p we have Theorem (Charles) X p � . � � � X • NS • = Néron–Severi group of • ≃ { curves on •} / ∼ • ρ ( • ) = rk NS • • X p := X mod p � ρ ( X al ) NS X al ∈ { 1 , 2 , . . . , 20 } � � ??? � NS X p al � ρ ( X p al ) ∈ { 2 , 4 , . . . 22 }

Néron–Severi group � X � � Theorem (Charles) X p � � • NS • = Néron–Severi group of • ≃ { curves on •} / ∼ • ρ ( • ) = rk NS • • X p := X mod p � ρ ( X al ) NS X al ∈ { 1 , 2 , . . . , 20 } � � ??? � NS X p al � ρ ( X p al ) ∈ { 2 , 4 , . . . 22 } For infinitely many p we have ρ ( X p al ) = min q ρ ( X q al ) .

jump X The Problem X as B B p p B p X B • What can we say about the following: Theorem (Charles) Let’s do some numerical experiments! � � � � � X p � ρ ( X al ) NS X al ∈ { 1 , 2 , . . . , 20 } � � ??? � NS X p al � ρ ( X p al ) ∈ { 2 , 4 , . . . 22 } For infinitely many p we have ρ ( X p al ) = min q ρ ( X q al ) . p : ρ ( X p al ) > min q ρ ( X q al ) { } • Π jump ( X ) :=

The Problem � What can we say about the following: Theorem (Charles) X X p � Let’s do some numerical experiments! � � � � ρ ( X al ) NS X al ∈ { 1 , 2 , . . . , 20 } � � ??? � NS X p al � ρ ( X p al ) ∈ { 2 , 4 , . . . 22 } For infinitely many p we have ρ ( X p al ) = min q ρ ( X q al ) . p : ρ ( X p al ) > min q ρ ( X q al ) { } • Π jump ( X ) := • γ ( X , B ) := # { p ≤ B : p ∈ Π jump ( X ) } as B → ∞ # { p ≤ B }

The Problem � What can we say about the following: Theorem (Charles) X X p � Let’s do some numerical experiments! � � � � ρ ( X al ) NS X al ∈ { 1 , 2 , . . . , 20 } � � ??? � NS X p al � ρ ( X p al ) ∈ { 2 , 4 , . . . 22 } For infinitely many p we have ρ ( X p al ) = min q ρ ( X q al ) . p : ρ ( X p al ) > min q ρ ( X q al ) { } • Π jump ( X ) := • γ ( X , B ) := # { p ≤ B : p ∈ Π jump ( X ) } as B → ∞ # { p ≤ B }

jump X B p 1 p Why? Two generic K3 surfaces, ρ ( X al ) = 1 γ ( X , B ) ? √ ∼ c X , B → ∞

Why? B Two generic K3 surfaces, ρ ( X al ) = 1 γ ( X , B ) ? √ ∼ c X , B → ∞ ∼ 1 / √ p ⇒ Prob( p ∈ Π jump ( X )) ? =