Motivation Modular Curves Gonalities Gonalities of Modular Curves Maarten Derickx 1 Mark van Hoeij 2 1 Algant (Leiden, Bordeaux and Milano) 2 Florida State University Intercity Number Theory Seminar 01-03-2013 Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities Outline Gonalities 1 Lower bounds Upper bounds Summary Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities What is known S ( d ) := { p prime | ∃ K / Q : [ K : Q ] ≤ d , ∃ E / K : E ( K ) [ p ] � = 0 } Primes ( n ) := { p prime | p ≤ n } S ( d ) is finite (Merel) S ( d ) ⊆ Primes (( 3 d / 2 + 1 ) 2 ) (Oesterlé) S ( 1 ) = Primes ( 7 ) (Mazur) S ( 2 ) = Primes ( 13 ) (Kamienny, Kenku, Momose) S ( 3 ) = Primes ( 13 ) (Parent) S ( 4 ) = Primes ( 17 ) (Kamienny, Stein, Stoll) to be published. Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities New results S ( d ) := { p prime | ∃ K / Q : [ K : Q ] ≤ d , ∃ E / K : E ( K ) [ p ] � = 0 } Primes ( n ) := { p prime | p ≤ n } S ( 5 ) = Primes ( 19 ) (Kamienny, Stein, Stoll and D.) S ( 6 ) ⊆ Primes ( 23 ) ∪ { 37 , 73 } (Kamienny, Stein, Stoll and D.) 73 is the only prime p for which we do not know whether p ∈ S ( 6 ) . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities j -invariant Over C the j -invariant gives a 1-1 correspondence: j : { E / C } / ∼ ← → C Now C ∼ = H / SL 2 ( Z ) where SL 2 ( Z ) acts on H by: � a � b τ = a τ + b c d c τ + d Analytic description: E = C / ( τ Z + Z ) , q = e 2 π i τ j ( E ) = q − 1 + 744 + 196884 q + 21493760 q 2 + . . . Algebraic description: E = Z ( y 2 − x 3 − ax − b ) j ( E ) = 1728 · 4 a 3 4 a 3 + 27 b 2 Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities Analytic description of the modular curve Y 1 ( N ) �� a � � a � � 1 � � b b ∗ Γ 1 ( N ) := ∈ SL 2 ( Z ) | ≡ mod N c d c d 0 1 Y 1 ( N )( C ) := H / Γ 1 ( N ) There is again a 1-1 correspondence: 1 : 1 ψ : { ( E , P ) | E / C , P ∈ E of order N } / ∼ ← → Y 1 ( N )( C ) Analytic description ( E , P ) = ( C / ( τ Z + Z ) , 1 / N mod τ Z + Z ) ψ ( E , P ) = τ mod SL 2 ( Z ) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities Algebraic description of the modular curve Y 1 ( N ) Proposition Let K be a field, E / K and P ∈ E ( K ) of order N ≥ 4 . Then there are unique b , c ∈ K such that = Z ( Y 2 + cXY + bY − X 3 − bX 2 ) and P = ( 0 , 0 ) E ∼ R := Z [ b , c , 1 ∆ ] with ∆ := − b 3 ( 16 b 2 + ( 8 c 2 − 36 c + 27 ) b + ( c − 1 ) c 3 ) E / R elliptic curve given by Y 2 + cXY + bY = X 3 + bX 2 P := ( 0 : 0 : 1 ) Let Φ N , Ψ N , Ω N ∈ R be s.t. (Φ N Ψ N : Ω N : Ψ 3 N ) = NP The equation Ψ N = 0 means P has order dividing N . Define F N by removing form Ψ N all factors coming from some Ψ d with d | N . Y 1 ( N ) Z [ 1 / N ] := Spec ( R [ 1 / N ] / F N ) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities Algebraic description of the modular curve Y 1 ( N ) R := Z [ b , c , 1 ∆ ] E / R elliptic curve given by Y 2 + cXY + bY = X 3 + bX 2 P := ( 0 : 0 : 1 ) Let Φ N , Ψ N , Ω N ∈ R be s.t. (Φ N Ψ N : Ω N : Ψ 3 N ) = NP Define F N by removing form Ψ N all factors coming from some Ψ d with d | N . Y 1 ( N ) Z [ 1 / N ] := Spec ( R [ 1 / N ] / F N ) Let N ≥ 4 and let K be a field with char ( K ) ∤ N then 1 : 1 ψ : { ( E , P ) | E / K , P ∈ E ( K ) of order N } / ∼ ← → Y 1 ( N )( K ) Let ( E , P ) = ( Z ( y 2 − cxy − by − x 3 − bx 2 ) , ( 0 , 0 )) then ψ ( E , P ) = ( b , c ) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities Relation between Y 1 ( N ) and S ( d ) The 1-1 correspondence 1 : 1 ψ : { ( E , P ) | E / K , P ∈ E ( K ) of order N } / ∼ ← → Y 1 ( N )( K ) gives S ( d ) := { p prime | ∃ K / Q : [ K : Q ] ≤ d , ∃ E / K : E ( K ) [ p ] � = 0 } = = { p prime | ∃ K / Q : [ K : Q ] ≤ d , Y 1 ( p )( K ) � = ∅} So we want to know whether Y 1 ( p ) has any points of degree ≤ d over Q . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities X 1 ( N ) and cusps Let N ≥ 5. Then Y 1 ( N ) can be embedded in a projective Z [ 1 / N ] -scheme X 1 ( N ) . Let K = K and N prime. Then #( X 1 ( N )( K ) \ Y 1 ( N )( K )) = N − 1 . These N − 1 elements are called the cusps. Over Q we have #( X 1 ( N )( Q ) \ Y 1 ( N )( Q )) = ( N − 1 ) / 2 . i.e. only half of the cusps are defined over Q . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Modular Curves Gonalities A useful proposition of Michael Stoll Proposition Let C / Q be a smooth proj. geom. irred. curve with Jacobian J, d ≥ 1 and ℓ a prime of good reduction for C. Let P ∈ C ( Q ) and ι : C ( d ) → J the canonical map normalized by ι ( dP ) = 0 . Suppose that: there is no non-constant f ∈ Q ( C ) of degree ≤ d . 1 J ( Q ) is finite. 2 → J ( F ℓ ) . ℓ > 2 or J ( Q )[ 2 ] ֒ 3 C ( Q ) ։ C ( F ℓ ) 4 The intersection of ι ( C ( d ) ( F ℓ )) ⊆ J ( F ℓ ) with the image of 5 J ( Q ) under reduction mod ℓ is contained in the image of C d ( F ℓ ) . Then C ( Q ) is the set of points of degree ≤ d on C. Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Lower bounds Modular Curves Upper bounds Gonalities Summary Definition of gonality Definition Let K be a field and C / K be a smooth proj. geom. irred. 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 Theorem (Abramovich) Let N be a prime then: 1600 ( N 2 − 1 ) . 7 gon C ( X 1 ( N )) ≥ If Selberg’s eigenvalue conjecture holds then: 192 ( N 2 − 1 ) . 1 gon C ( X 1 ( N )) ≥ So gon Q ( X 1 ( 41 )) ≥ gon C ( X 1 ( 41 )) ≥ 7 / 1600 ( 41 2 − 1 ) > 7. But, even with the conjecture, this doesn’t give a good enough bound for showing gon Q ( X 1 ( 29 )) , gon Q ( X 1 ( 31 )) > 6 Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Lower bounds Modular Curves Upper bounds Gonalities Summary The F ℓ gonality is smaller than the Q -gonality Proposition Let C / Q be a smooth proj. geom. irred. 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 too slow to compute gon F 2 ( X 1 ( 29 )) and gon F 2 ( X 1 ( 31 )) Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Lower bounds Modular Curves Upper bounds Gonalities Summary Divisors dominating all functions of degree ≤ d C / F l a smooth proj. geom. irr. curve. View f ∈ F l ( C ) as a map f : C → P 1 F l . For g ∈ Aut C , h ∈ Aut P 1 F l : deg f = deg h ◦ f ◦ g Definition A set of divisors S ⊆ div C dominates all functions of degree ≤ d if for all dominant f : C → P 1 F l of degree ≤ d there are D ∈ S , g ∈ Aut C and h ∈ Aut P 1 F l such that div h ◦ f ◦ g ≥ − D Proposition If S ⊆ div C dominates all functions of degree ≤ d then gon F l C ≥ min ( d + 1 , inf deg f ) . D ∈ S , f ∈ H 0 ( C , D ) , degf � = 0 Example: div + d C dominates all functions of degree ≤ d . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Lower bounds Modular Curves Upper bounds Gonalities Summary A smaller set of divisors dominating functions of degree ≤ d Proposition Define n := ⌈ # C ( F l ) / ( l + 1 ) ⌉ and D := � p ∈ C ( F l ) p. Then s ′ + D | s ′ ∈ div + div + � � d − n C + D := d − n C dominates all functions of degree ≤ d. Proof. There is a g ∈ Aut P 1 F l such that g ◦ f has poles at at least n distinct points in C ( F l ) . If f has degree ≤ d then there is an element s ∈ div + d − n C such that div g ◦ f ≥ − s − D . Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
Motivation Lower bounds Modular Curves Upper bounds Gonalities Summary An even smaller set of divisors dominating functions of degree ≤ d Proposition If S ⊆ div C dominates all functions of degree ≤ d and S ′ ⊆ div C is such that for all s ∈ S there are s ′ ∈ S ′ and g ∈ Aut C such that g ( s ′ ) ≥ s. Then S ′ also dominates all functions of degree ≤ d. This means that only 1 representative of each Aut C orbit of S is needed. This will be useful in the cases C = X 1 ( p ) with p = 29 , 31. In these case we have an automorphism of C for each d ∈ ( Z / p Z ) ∗ / {± 1 } given by ( E , P ) �→ ( E , dP ) . This gives 14 and 15 automorphisms respectively. Maarten Derickx , Mark van Hoeij Gonalities of Modular Curves
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