Rigorous computation of the endomorphism ring of a Jacobian Edgar - PowerPoint PPT Presentation

Rigorous computation of the endomorphism ring of a Jacobian Edgar Costa (MIT) Simons Collab. on Arithmetic Geometry, Number Theory, and Computation November 13th, 2019 University of New South Wales Slides available at edgarcosta.org under Research 1/25

• 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 . Polynomials 2, quadratic reciprocity gives us that • What about for higher degrees? f . depending only on p 0 f p p • For f f x x • factorization of p in f x irreducible • factorization of f x 2/25 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 . Polynomials • What about for higher degrees? f . depending only on p 0 f p p f 2, quadratic reciprocity gives us that • For 2/25 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 . Polynomials 2, quadratic reciprocity gives us that • What about for higher degrees? f . depending only on p 0 f p p f • For 2/25 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 2/25 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 2/25 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 2/25 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 2 3 x 3 if k 0 1 3 if k 3 g 3 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) 3/25 Prob( N f ( p ) = i ) = Prob( g ∈ Gal( f ) : g fixes i roots ) ,

Example: Cubic polynomials 3 3 g S 3 f Theorem (Frobenius) 3 3/25 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 2 Theorem (Frobenius) 3 3 3/25 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 = ⇒ 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 . 

Elliptic curves An elliptic curve is a smooth curve defined by or There is a natural group structure ! If P , Q , and R are colinear, then P Q R 0 4/25 y 2 = x 3 + ax + b Over R it might look like Over C this is a torus

Elliptic curves An elliptic curve is a smooth curve defined by or There is a natural group structure ! If P , Q , and R are colinear, then 4/25 y 2 = x 3 + ax + b Over R it might look like Over C this is a torus P + Q + R = 0

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 . 5/25 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 . 5/25 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 . 5/25 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 5/25 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 .

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 ? 6/25 | p + 1 − # E p | ≤ 2 √ p .

Hasse’s bound Theorem (Hasse, 1930s) In other words, 6/25 | 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 p 1 2 0 p p 1 0 E E special ordinary 7/25 λ p := p + 1 − # E p √ p ∈ [ − 2 , 2 ] There are two limiting distributions for λ p

Two types of elliptic curves p 1 2 0 p p 1 0 7/25 ordinary E E special 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 1 2 0 p p 1 0 7/25 ordinary special 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 ordinary special 7/25 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 ∼ 1 / √ p Prob( λ p = 0 ) ? Prob( λ p = 0 ) = 1 / 2

Two types of elliptic curves ordinary special d 2 1 d for some d 0 non-CM CM 8/25 Over C an elliptic curve E is a torus E C ≃ C / Λ , where Λ = ω 1 Z + ω 2 Z = and we have End E al = End Λ

Two types of elliptic curves special CM non-CM 8/25 ordinary Over C an elliptic curve E is a torus E C ≃ C / Λ , where Λ = ω 1 Z + ω 2 Z = and we have End E al = End Λ √ End Λ = Z Z ⊊ End(Λ) ⊂ Q ( − d ) √ ω 2 /ω 1 ∈ Q ( − d ) for some d > 0 - 2 - 1 1 2 - 2 - 1 0 1 2

How to distinguish between the two types? • non-CM p 4 p • CM d a 2 p 4 p . a 2 E p 4 p a 2 q 4 q for p q w/prob 1. a 2 0 non-CM It is enough to count points! CM a p 9/25 • p 1 E p √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2

How to distinguish between the two types? a 2 • CM d a 2 p 4 p . • non-CM p non-CM 4 p a 2 q 4 q for p q w/prob 1. a 2 9/25 CM It is enough to count points! √ End Q E al = Q End Q E al = Q ( − d ) - 2 - 1 1 2 - 2 - 1 0 1 2 (√ ⇒ End Q E al ⊂ Q ) • p + 1 − # E p =: a p ̸ = 0 = p − 4 p

How to distinguish between the two types? a 2 w/prob 1. q for p 4 q q a 2 4 p p a 2 • non-CM . a 2 non-CM 9/25 It is enough to count points! 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 ⊂ Q ) • p + 1 − # E p =: a p ̸ = 0 = p − 4 p √ (√ ) • CM ⇒ Q ( − d ) ≃ Q p − 4 p

How to distinguish between the two types? It is enough to count points! w/prob 1. a 2 a 2 . a 2 non-CM a 2 9/25 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 ⊂ Q ) • p + 1 − # E p =: a p ̸ = 0 = p − 4 p √ (√ ) • CM ⇒ Q ( − d ) ≃ Q p − 4 p (√ ) (√ ) • non-CM ⇒ Q p − 4 p ̸≃ Q q − 4 q for p ̸ = q