Frobenius Distributions Edgar Costa (MIT) Simons Collab. on - PowerPoint PPT Presentation

Frobenius Distributions Edgar Costa (MIT) Simons Collab. on Arithmetic Geometry, Number Theory, and Computation April 23rd, 2019 University of Washington Slides available at edgarcosta.org under Research

• factorization of f p x e.g.: f p x irreducible • What can we say about f p x for arbitrary p ? N f p studying the statistical properties N f p . • factorization of f x p • What about for higher degrees? f . depending only on p 0 f p 2, quadratic reciprocity gives us that f • For f x x • factorization of p in f x irreducible Polynomials f ( x ) = a n x n + · · · + a 0 ∈ Z [ x ] Write f p ( x ) := f ( x ) mod p • Given f p ( x ) what can we say about f ( x ) ?

• What can we say about f p x for arbitrary p ? N f p studying the statistical properties N f p . 2, quadratic reciprocity gives us that • What about for higher degrees? f . depending only on p 0 f p p Polynomials f • For f ( x ) = a n x n + · · · + a 0 ∈ Z [ x ] Write f p ( x ) := f ( x ) mod p • Given f p ( x ) what can we say about f ( x ) ? • factorization of f p ( x ) ⇝ • factorization of f ( x ) e.g.: f p ( x ) irreducible ⇒ f ( x ) irreducible • factorization of p in Q [ x ] / f ( x )

N f p studying the statistical properties N f p . 2, quadratic reciprocity gives us that • What about for higher degrees? f . depending only on p 0 f p p Polynomials • For f f ( x ) = a n x n + · · · + a 0 ∈ Z [ x ] Write f p ( x ) := f ( x ) mod p • Given f p ( x ) what can we say about f ( x ) ? • factorization of f p ( x ) ⇝ • factorization of f ( x ) e.g.: f p ( x ) irreducible ⇒ f ( x ) irreducible • factorization of p in Q [ x ] / f ( x ) • What can we say about f p ( x ) for arbitrary p ?

studying the statistical properties N f p . • What about for higher degrees? Polynomials f ( x ) = a n x n + · · · + a 0 ∈ Z [ x ] Write f p ( x ) := f ( x ) mod p • Given f p ( x ) what can we say about f ( x ) ? • factorization of f p ( x ) ⇝ • factorization of f ( x ) e.g.: f p ( x ) irreducible ⇒ f ( x ) irreducible • factorization of p in Q [ x ] / f ( x ) • What can we say about f p ( x ) for arbitrary p ? • For deg f = 2, quadratic reciprocity gives us that N f ( p ) := # { α ∈ F p : f p ( α ) = 0 } depending only on p mod ∆( f ) .

studying the statistical properties N f p . • What about for higher degrees? Polynomials f ( x ) = a n x n + · · · + a 0 ∈ Z [ x ] Write f p ( x ) := f ( x ) mod p • Given f p ( x ) what can we say about f ( x ) ? • factorization of f p ( x ) ⇝ • factorization of f ( x ) e.g.: f p ( x ) irreducible ⇒ f ( x ) irreducible • factorization of p in Q [ x ] / f ( x ) • What can we say about f p ( x ) for arbitrary p ? • For deg f = 2, quadratic reciprocity gives us that N f ( p ) := # { α ∈ F p : f p ( α ) = 0 } depending only on p mod ∆( f ) .

• What about for higher degrees? Polynomials f ( x ) = a n x n + · · · + a 0 ∈ Z [ x ] Write f p ( x ) := f ( x ) mod p • Given f p ( x ) what can we say about f ( x ) ? • factorization of f p ( x ) ⇝ • factorization of f ( x ) e.g.: f p ( x ) irreducible ⇒ f ( x ) irreducible • factorization of p in Q [ x ] / f ( x ) • What can we say about f p ( x ) for arbitrary p ? • For deg f = 2, quadratic reciprocity gives us that N f ( p ) := # { α ∈ F p : f p ( α ) = 0 } depending only on p mod ∆( f ) . ⇝ studying the statistical properties N f ( p ) .

3 2 3 2 e 2 i 3 3 2 e 4 i 3 N f p N g p 2 x 1 x 1 x 2 x 3 Example: Cubic polynomials k x 3 2 3 if k 0 1 3 if k 3 g x 2 f g x 1 3 f x x 3 2 x x x k if k S 3 0 1 2 if k 1 1 6 if k 3 Theorem (Frobenius) 3 Prob( N f ( p ) = i ) = Prob( g ∈ Gal( f ) : g fixes i roots ) ,

Example: Cubic polynomials 3 g S 3 f Theorem (Frobenius) 3 3 2 3 Prob( N f ( p ) = i ) = Prob( g ∈ Gal( f ) : g fixes i roots ) , √ √ √ ( ) ( 2 e 2 π i / 3 ) ( 2 e 4 π i / 3 ) f ( x ) = x 3 − 2 = x − x − x −  1 / 3 if k = 0    ( ) Prob N f ( p ) = k = 1 / 2 if k = 1   1 / 6 if k = 3 .  g ( x ) = x 3 − x 2 − 2 x + 1 = ( x − α 1 ) ( x − α 2 ) ( x − α 3 )  2 / 3 if k = 0  Prob ( N g ( p ) = k ) = 1 / 3 if k = 3 . 

Example: Cubic polynomials 3 3 3 2 Theorem (Frobenius) Prob( N f ( p ) = i ) = Prob( g ∈ Gal( f ) : g fixes i roots ) , √ √ √ ( ) ( 2 e 2 π i / 3 ) ( 2 e 4 π i / 3 ) f ( x ) = x 3 − 2 = x − x − x −  1 / 3 if k = 0    ( ) Prob N f ( p ) = k = ⇒ Gal( f ) = S 3 1 / 2 if k = 1   1 / 6 if k = 3 .  g ( x ) = x 3 − x 2 − 2 x + 1 = ( x − α 1 ) ( x − α 2 ) ( x − α 3 )  2 / 3 if k = 0  Prob ( N g ( p ) = k ) = ⇒ Gal( g ) = Z / 3 Z 1 / 3 if k = 3 . 

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

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 , for p a prime of good reduction • 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 , for p a prime of good reduction • 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 , for p a prime of good reduction • 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 .

p H 1 E Hasse’s bound p as p What can we say about the error term a p Question p p 2 2 p E p 1 p a p T Theorem (Hasse) 1 pT 2 a p T 1 L T where L 1 E 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 where 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 , # E p = L ( 1 ) , L ( T ) = 1 − a p T + pT 2 = det( 1 − T Frob p | H 1 ( E )) 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 where Question a p ∈ [ − 2 √ p , 2 √ p ] # E p = p + 1 − a p , # E p = L ( 1 ) , L ( T ) = 1 − a p T + pT 2 = det( 1 − T Frob p | H 1 ( E )) 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

Two types of elliptic curves a p p 0 a p 1 2 0 a p p 1 0 E p 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 a p p 0 E p 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 √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2 ∼ 1 / √ p Prob( a p = 0 ) ? Prob( a p = 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 ∼ 1 / √ p Prob( a p = 0 ) ? Prob( a p = 0 ) = 1 / 2 ⇒ Q (Frob p ) ⊊ End Q E p al a p = 0 ⇐

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 ∼ 1 / √ p Prob( a p = 0 ) ? Prob( a p = 0 ) = 1 / 2 ⇒ Q (Frob p ) ⊊ End Q E p al a p = 0 ⇐ ⇒ dim End Q E p al > 2 ⇐