Distributions in arithmetic geometry Edgar Costa (Dartmouth College) - PowerPoint PPT Presentation

Distributions in arithmetic geometry Edgar Costa (Dartmouth College) June 21st, 2018 University of Washington Presented at Communicating Mathematics Effectively 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 Theorem (Hasse, 1930s) In other words, p p 1 E p p 2 2 What can we say about the error term, p , as p ? | p + 1 − # E p | ≤ 2 √ p .

Hasse’s bound Theorem (Hasse, 1930s) In other words, | p + 1 − # E p | ≤ 2 √ p . λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] What can we say about the error term, λ p , as p → ∞ ?

Two types of elliptic curves 1 E p 0 p 1 2 0 p p 0 p E E CM non-CM 2 λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] There are two limiting distributions for λ p !

Two types of elliptic curves p E p 0 p 1 2 0 p p 1 0 2 non-CM CM E E λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] There are two limiting distributions for λ p ! - 2 - 1 1 2 - 2 - 1 0 1 2

Two types of elliptic curves p E p 0 p 1 2 0 p p 1 0 2 non-CM CM λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] There are two limiting distributions for λ p ! End E al = Z End E al ̸ = Z - 2 - 1 1 2 - 2 - 1 0 1 2

Two types of elliptic curves CM E p 0 p 2 non-CM λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] There are two limiting distributions for λ p ! End E al = Z End E al ̸ = Z - 2 - 1 1 2 - 2 - 1 0 1 2 Prob( λ p = 0 ) ∼ 1 / √ p Prob( λ p = 0 ) ∼ 1 / 2

Two types of elliptic curves non-CM CM λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] There are two limiting distributions for λ p ! End E al = Z End E al ̸ = Z - 2 - 1 1 2 - 2 - 1 0 1 2 Prob( λ p = 0 ) ∼ 1 / √ p Prob( λ p = 0 ) ∼ 1 / 2 ⇒ rk End E p al > 2 λ p = 0 ⇐

Two types of elliptic curves non-CM CM q λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] There are two limiting distributions for λ p ! End E al = Z End E al ̸ = Z - 2 - 1 1 2 - 2 - 1 0 1 2 Prob( λ p = 0 ) ∼ 1 / √ p Prob( λ p = 0 ) ∼ 1 / 2 ⇒ rk End E p al > 2 = min q rk End E al λ p = 0 ⇐

X p we study K3 surfaces 4 E p E p This is analogous to studying X p p In this case, instead of studying f K3 surfaces are a possible generalization of elliptic curves x f 0 f x y z w X 3 For example, smooth quartic surfaces in E p

X p we study K3 surfaces K3 surfaces are a possible generalization of elliptic curves In this case, instead of studying p X p This is analogous to studying E p E p E p For example, smooth quartic surfaces in P 3 X : f ( x , y , z , w ) = 0 , f ∈ Z [ x ] , deg f = 4

K3 surfaces K3 surfaces are a possible generalization of elliptic curves This is analogous to studying E p E p E p For example, smooth quartic surfaces in P 3 X : f ( x , y , z , w ) = 0 , f ∈ Z [ x ] , deg f = 4 In this case, instead of studying # X p , we study → rk NS X p al . p �−

K3 surfaces K3 surfaces are a possible generalization of elliptic curves For example, smooth quartic surfaces in P 3 X : f ( x , y , z , w ) = 0 , f ∈ Z [ x ] , deg f = 4 In this case, instead of studying # X p , we study → rk NS X p al . p �− This is analogous to studying rk End E p al = rk NS( E p al × E p al )

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 •

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 • � ρ ( X al ) NS X al ∈ { 1 , 2 , . . . , 20 } � � ??? � NS X p al � ρ ( X p al ) ∈ { 2 , 4 , . . . 22 }

Néron–Severi group � Theorem (Charles) X p � � � � X NS • = Néron–Severi group of • ≃ { curves on •} / ∼ ρ ( • ) = rk NS • � 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 p B 1 p Why? Generic K3 surfaces, ρ ( X al ) = 1 √ γ ( X , B ) ∼ c X , B → ∞

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

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

Could it be related to some integer being a square modulo p ? No obvious trend… Data for ρ ( X al ) = 2

Could it be related to some integer being a square modulo p ? No obvious trend… Data for ρ ( X al ) = 2

Could it be related to some integer being a square modulo p ? No obvious trend… Data for ρ ( X al ) = 2

jump X In general, d X is not a square. If d X is not a square: has infinitely many rational curves. d X • X 1 2 X B B • Corollary p inert in 2 p such that: , then there is a d X X p q X If Theorem (C, C–Elsenhans–Jahnel) Numerical experiments ⇝ Theoretical Results In most cases we can explain the 1 / 2!

If d X is not a square: has infinitely many rational curves. Corollary Theorem (C, C–Elsenhans–Jahnel) • X 1 2 X B B • Numerical experiments ⇝ Theoretical Results In most cases we can explain the 1 / 2! If ρ ( X al ) = min q ρ ( X p al ) , then there is a d X ∈ Z such that: { } √ p > 2 : p inert in Q ( d X ) ⊂ Π jump ( X ) . In general, d X is not a square.

Theorem (C, C–Elsenhans–Jahnel) Corollary Numerical experiments ⇝ Theoretical Results In most cases we can explain the 1 / 2! If ρ ( X al ) = min q ρ ( X p al ) , then there is a d X ∈ Z such that: { } √ p > 2 : p inert in Q ( d X ) ⊂ Π jump ( X ) . In general, d X is not a square. If d X is not a square: • lim inf B →∞ γ ( X , B ) ≥ 1 / 2 • X al has infinitely many rational curves.

γ � � � γ � � � γ � � � ���� � ����� ���� � ����� ���� � ����� if d X is not a square modulo p jump X otherwise p 1 1 p Why?!? What if we ignore B B c B d X X Experimental data for ρ ( X al ) = 2 (again) p > 2 : p inert in Q ( √ d X ) { } ⊂ Π jump ( X ) ?

γ � � � γ � � � γ � � � ���� � ����� ���� � ����� ���� � ����� if d X is not a square modulo p jump X Why?!? What if we ignore otherwise p 1 1 p B c d X Experimental data for ρ ( X al ) = 2 (again) p > 2 : p inert in Q ( √ d X ) { } ⊂ Π jump ( X ) ? ( ) ( √ √ γ ) , B ∼ , B → ∞ X Q

Why?!? d X otherwise 1 1 What if we ignore B c Experimental data for ρ ( X al ) = 2 (again) p > 2 : p inert in Q ( √ d X ) { } ⊂ Π jump ( X ) ? ( ) ( √ √ γ ) , B ∼ , B → ∞ X Q 1 1 1 γ ( � � � ) γ ( � � � ) γ ( � � � ) 0.50 0.50 0.50 ���� � - ����� ���� � - ����� ���� � - ����� 0.10 0.10 0.10 0.05 0.05 0.05 10 4 10 5 10 4 10 5 10 4 10 5 100 1000 1000 100 1000   if d X is not a square modulo p Prob( p ∈ Π jump ( X )) = ∼ √ p 