Gonalities of Modular Curves Maarten Derickx 1 Mark van Hoeij 2 1 - PowerPoint PPT Presentation

Preliminaries (what are modular curves) Modular Units Gonalities Gonalities of Modular Curves Maarten Derickx 1 Mark van Hoeij 2 1 Algant (Leiden, Bordeaux and Milano) 2 Florida State University 17th Workshop on Elliptic Curve Cryptography 16 - 18 Sept. 2013 Leuven Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Gonalities Outline Preliminaries (what are modular curves) 1 Algebraic description of the modular curve Y 1 ( N ) Modular Units 2 Gonalities 3 Intro Computing gonalities Motivation Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities Main idea behind modular curves Let N ∈ N then the set:   Pairs ( E , P ) of elliptic   curve, point of order  / ∼ N  has a natural structure of a curve. One can study all pairs ( E , P ) at the same time by studying the curve C . ( E 1 , P 1 ) ∼ ( E 2 , P 2 ) if there exists an isomorphism φ : E 1 → E 2 such that φ ( P 1 ) = P 2 . Example: Multiplication by -1 gives ( E , P ) ∼ ( E , − P ) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities Definition (Tate normal form) Let b , c ∈ K then E ( b , c ) is the curve Y 2 + cXY + bY = X 3 + bX 2 Remark The discriminant of E ( b , c ) is: ∆( b , c ) := − b 3 ( 16 b 2 + ( 8 c 2 − 36 c + 27 ) b + ( c − 1 ) c 3 ) Proposition Let E / K an elliptic curve and P ∈ E ( K ) of order N ≥ 4 . Then there are unique b , c ∈ K and an unique isomorphism φ : E → E ( b , c ) such that φ ( P ) = ( 0 , 0 ) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities E ( b , c ) : Y 2 + cXY + bY = X 3 + bX 2 Proposition Let E / K an elliptic curve and P ∈ E ( K ) of order ≥ 4 . Then there are unique b , c ∈ K and an unique isomorphism φ : E → E ( b , c ) such that φ ( P ) = ( 0 , 0 ) Proof. E : Y 2 + a 1 XY + a 3 Y = X 3 + a 2 X 2 + a 4 X + a 6 , P = ( x , y ) Translate P to ( 0 , 0 ) . E : Y 2 + a ′ 3 Y = X 3 + a ′ 2 X 2 + a ′ 1 XY + a ′ 4 X , P = ( 0 , 0 ) Make the tangent line at ( 0 , 0 ) horizontal E : Y 2 + a ′′ 3 Y = X 3 + a ′′ 1 XY + a ′′ 2 X 2 , P = ( 0 , 0 ) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities E ( b , c ) : Y 2 + cXY + bY = X 3 + bX 2 Proposition Let E / K an elliptic curve and P ∈ E ( K ) of order ≥ 4 . Then there are unique b , c ∈ K and an unique isomorphism φ : E → E ( b , c ) such that φ ( P ) = ( 0 , 0 ) Proof. E : Y 2 + a ′′ 3 Y = X 3 + a ′′ 1 XY + a ′′ 2 X 2 , P = ( 0 , 0 ) Y �→ u 3 Y , X �→ u 2 X with u = a ′′ 2 / a ′′ 3 3 XY + a ′′ 3 3 Y = X 3 + a ′′ 3 E : Y 2 + a ′′ 1 a ′′ 3 X 2 , P = ( 0 , 0 ) 2 2 2 a ′′ 2 a ′′ 2 a ′′ 3 , b = a ′′ 3 E = E ( b , c ) , c = a ′′ 1 a ′′ 2 2 3 , P = ( 0 , 0 ) a ′′ a ′′ 2 Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities E ( b , c ) : Y 2 + cXY + bY = X 3 + bX 2 Proposition Let E / K an elliptic curve and P ∈ E ( K ) of order ≥ 4 . Then there are unique b , c ∈ K and an unique isomorphism φ : E → E ( b , c ) such that φ ( P ) = ( 0 , 0 ) � b , c ∈ A 2 ( K ) s.t   Pairs ( E , P ) of elliptic �   1 : 1 curve, point of order  / ∼ ← − → ∆( b , c ) � = 0 ≥ 4  Definition Let N ∈ N ≥ 4 and char K ∤ N then the modular curve Y 1 ( N ) K ⊂ A 2 K is the curve corresponding to the ( E , P ) where P has exactly order N . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities Definition (Division polynomials for E ( b , c ) at P = ( 0 : 0 : 1 ) ) Define Ψ n , Φ n , Ω n ∈ Z [ b , c ] by: Ψ 1 = 1 , Ψ 2 = b , Ψ 3 = b 3 , Ψ 4 = b 5 ( c − 1 ) Ψ m + n Ψ n − m Ψ 2 r = Ψ n + r Ψ n − r Ψ 2 m − Ψ m + r Ψ m − r Ψ 2 n n = m + 1 , r = 1 ⇒ Ψ 2 m + 1 = Ψ m + 2 Ψ 3 m − Ψ m − 1 Ψ 3 m + 1 n = m + 2 , r = 1 ⇒ b Ψ 2 m + 2 = Ψ m − 1 (Ψ m + 3 Ψ 2 m − Ψ m + 1 Ψ 2 m + 2 ) Φ n = − Ψ n − 1 Ψ n + 1 Ω n = Ψ 2 n 2 Ψ n − Ψ n ( c Φ n + b Ψ 2 n ) Proposition Let N ∈ Z and view E ( b , c ) as an elliptic curve over K ( b , c ) (or 1 Z [ b , c , ∆( b , c ) ]) then N ( 0 : 0 : 1 ) = (Φ N Ψ N : Ω N : Ψ 3 N ) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities N ( 0 : 0 : 1 ) = (Φ N Ψ N : Ω N : Ψ 3 N ) Proposition If ( b , c ) ∈ A 2 ( K ) , ∆( b , c ) � = 0 then N ( 0 : 0 : 1 ) = ( 0 : 1 : 0 ) ⇔ Ψ N ( b , c ) = 0 Define F N by removing form Ψ N all factors coming from some Ψ d with d | N , and all common factors with ∆( b , c ) . Corollary If char K ∤ N then Y 1 ( N ) K ⊂ A 2 K is given by F N = 0 , ∆( b , c ) � = 0 . Definition X 1 ( N ) K is the projective closure of Y 1 ( N ) , i.e. the unique smooth projective curve whose function field is K ( Y 1 ( N )) . The cusps are X 1 ( N ) K \ Y 1 ( N ) K . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Algebraic description of the modular curve Y 1 ( N ) Gonalities Example N = 5 ∆( b , c ) = − b 3 ( 16 b 2 + ( 8 c 2 − 36 c + 27 ) b + ( c − 1 ) c 3 ) Ψ 5 = ( − b + c − 1 ) b 8 F 5 = − b + c − 1 Y 1 ( N ) given by c = b + 1, ∆( b , c ) � = 0 ∆( b , b + 1 ) = − b 5 ( b 2 + 11 b − 1 ) X 1 ( N ) ∼ = P 1 , cusps are the points given by b = 0 , b = ∞ and b 2 + 11 b − 1 = 0, so not all cusps are always defined over K . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Gonalities Definition f ∈ K ( X 1 ( N )) is called a modular unit if all its poles and zero’s are cusps. Two modular units f , g are called equivalent if f / g ∈ K ∗ . Example (N=5) The cusps of X 1 ( 5 ) where b = 0, b = ∞ and b 2 + 11 b − 1 = 0. Over Q , b and b 2 + 11 b − 1 form a multiplicative basis for all modular units up to equivalence, over C one needs √ b + ( 5 5 + 11 ) / 2 as extra generator. Example If N ∤ M then Ψ M ∈ K ( X 1 ( N )) = K ( b )[ c ] / F N is a modular unit. Because if Ψ M ( b , c ) = 0 for ( b , c ) ∈ Y 1 ( N )( ¯ K ) then ( 0 : 0 : 1 ) ∈ E ( b , c ) ( ¯ K ) has order N and order dividing M . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Modular Units Gonalities Definition f ∈ K ( X 1 ( N )) is called a modular unit if all its poles and zero’s are cusps. Two modular units f , g are called equivalent if f / g ∈ K ∗ . Conjecture (Hoeij, D.) b , ∆ , Ψ 4 , Ψ 5 , . . . , Ψ ⌊ N / 2 ⌋ + 1 form a multiplicative basis for the modular units over Q up to equivalence. We used a computer to verify the conjecture for N ≤ 100. Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Intro Modular Units Computing gonalities Gonalities Motivation Definition of gonality Definition Let K be a field and C / K be a smooth projective curve then the K -gonality of C is: gon K ( C ) := min f ∈ K ( C ) \ K [ K ( C ) : K ( f )] = min f ∈ K ( C ) \ K deg f Example (N=5) K ( X 1 ( 5 )) = K ( c )[ b ] / ( − b + c − 1 ) = K ( c ) so gon K ( X 1 ( 5 )) = 1 Example For an elliptic curve E / K one has gon K ( E ) = 2. Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Intro Modular Units Computing gonalities Gonalities Motivation General bounds Theorem (Abramovich) Let N be a prime then: 1600 ( N 2 − 1 ) . 7 gon C ( X 1 ( N )) ≥ For general N: 6 1600 N 2 . 7 gon C ( X 1 ( N )) ≥ π 2 Theorem (Poonen) � 6 p − 1 If char K = p > 0 then gon K ( X 1 ( N )) ≥ 24 ( p 2 + 1 ) N π 2 Proposition gon K ( X 1 ( N )) ≤ N 2 24 Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves

Preliminaries (what are modular curves) Intro Modular Units Computing gonalities Gonalities Motivation Lowerbound for the Q -gonality by computing the F ℓ gonality Proposition Let C / Q be a smooth projective curve and ℓ be a prime of good reduction of C then: gon Q ( C ) ≥ gon F ℓ ( C F ℓ ) To use this we need to know how compute the F ℓ gonality of C . d C F ℓ ⊆ div + C F ℓ be the set of effective divisors of Let div + degree d . Then #( div + d C F ℓ ) < ∞ . The following algorithm computes the F ℓ -gonality: Step 1 set d = 1 Step 2 While for all D ∈ div + d C F ℓ : dim H 0 ( C , D ) = 1 set d = d + 1 Step 3 Output d. This is already becomes to slow for computing gon F 2 ( X 1 ( 29 )) . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves