The rational group structure of modular Jacobians with applications to torsion points on elliptic curves over number fields Maarten Derickx 1 Algant (Leiden, Bordeaux and Milano) LMFDB Workshop 05-06-2014 Talk will only start after you opened: bit.ly/rat-points-mod-jac Maarten Derickx Group structure of modular Jacobians 05-06-2014 1 / 22
Outline Introduction 1 Determining J H ( Q ) 2 When has J H ( Q ) rank 0 Determining J H ( N )( Q ) tors 3 Application to torsion points on elliptic curves (with Mark van Hoeij) Maarten Derickx Group structure of modular Jacobians 05-06-2014 2 / 22
Definitions and notation N ∈ N ≥ 5 , H ⊆ Z / N Z ∗ K a field E / K and E ′ / K elliptic curves (EC). E ( K )[ N ] are the points of order N E ( K )[ N ] ′ are the points of order exactly N . Y 1 ( N )( K ) := { ( E , p ) | E / K EC , p ∈ E ( K )[ N ] ′ } / ∼ . n ∈ Z / N Z ∗ acts on Y 1 ( N ) by sending ( E , p ) to ( E , np ) Y H := Y 1 ( N ) / H , Y 0 ( N ) = Y H with H = Z / N Z ∗ . Let p ∈ E ( K )[ N ] ′ and p ′ ∈ E ′ ( K )[ N ] ′ then ( E , p ) ∼ H ( E ′ , p ′ ) if → E ′ and n ∈ H such that φ ( p ) = np ′ . there exists φ : E ˜ 1 : 1 Y H ( ¯ ( E , p ) | E / ¯ K EC , p ∈ E ( ¯ � K )[ N ] ′ � K ) ← → / ∼ H X H , X 0 ( N ) , X 1 ( N ) are the compactifications of Y H , Y 0 ( N ) , Y 1 ( N ) J H , J 0 ( N ) , J 1 ( N ) are the Jacobians of X H , X 0 ( N ) , X 1 ( N ) . Maarten Derickx Group structure of modular Jacobians 05-06-2014 3 / 22
Why J H is awesome used to prove part of BSD Theorem (Wiles, Breuil, Conrad, Diamond, Taylor) Every EC / Q occurs as an isogeny factor of J 0 ( N ) Conjecture (Weak Birch and Swinnerton-Dyer (Weak BSD)) Let A / K be an abelian variety over a number field then the order of vanishing of L ( A , s ) at s = 1 equals the rank of A ( K ) Part of Weak BSD has been proven for modular abelian varieties Q : Theorem ( J 0 ( N ) : Kolyvagin, Logachev. J H ( N ) : Kato) Let A / Q be an abelian variety isogenous to a sub abelian variety of J H ( N ) such that L ( A , 1 ) � = 0 then A ( Q ) has rank 0. Maarten Derickx Group structure of modular Jacobians 05-06-2014 4 / 22
Why J H is awesome Studying questions about rational points on modular curves. The structure of J H ( Q ) plays a crucial role in the proof of the following theorems: Theorem (Mazur) Let E → E ′ / Q by an isogeny of prime degree p, then p ≤ 19 or p = 37 , 43 , 67 , 163 Theorem (Mazur) Let E / Q be an EC then either E ( Q ) tors ∼ = Z / N Z for 1 ≤ N ≤ 10 , N = 12 or, E ( Q ) tors ∼ = Z / 2 N Z × Z / 2 Z for 1 ≤ N ≤ 4 Theorem (Merel) Let E / K by an EC over a number field, then # E ( K ) < M d for some constant M d only depending on d := [ K : Q ] Maarten Derickx Group structure of modular Jacobians 05-06-2014 5 / 22
Why J H is awesome Studying questions about rational points on modular curves. Let ( E , p ) be a pair such that it’s H equivalence class is defined over Q , then ( E , p ) gives a rational point on X H . Let µ ∞ : X H → J H p �→ p − ∞ Let π : J H → A be a map of abelian varieties s.t. # A ( Q ) < ∞ . π ◦ µ ∞ maps X H ( Q ) to the finite set A ( Q ) this gives a lot of restrictions on X H ( Q ) . Maarten Derickx Group structure of modular Jacobians 05-06-2014 6 / 22
Outline Introduction 1 Determining J H ( Q ) 2 When has J H ( Q ) rank 0 Determining J H ( N )( Q ) tors 3 Application to torsion points on elliptic curves (with Mark van Hoeij) Maarten Derickx Group structure of modular Jacobians 05-06-2014 7 / 22
Theorem (Mazur) J 0 ( N ) has rank > 0 for N = 37 , 43 , 53 , 61 , 67 or N a prime ≥ 73 . Using magma (W. Stein) one can compute L ( J 1 ( N ) , 1 ) / Ω J 1 ( N ) L ( J 1 ( N ) , 1 ) / Ω J 1 ( N ) � = 0 for all other primes N . So the proven part of BSD implies rank J 1 ( N )( Q ) = rank J H ( Q ) = 0 in the other cases. Same method allows everybody with access to magma to proof: Proposition If N ∈ N , N � = 37 , 43 , 53 , 57 , 58 , 61 , 63 , . . . then rank J H ( Q ) = 0 . Remark: there are N such that J 0 ( N ) has rank 0 but J 1 ( N ) not. Maarten Derickx Group structure of modular Jacobians 05-06-2014 8 / 22
Outline Introduction 1 Determining J H ( Q ) 2 When has J H ( Q ) rank 0 Determining J H ( N )( Q ) tors 3 Application to torsion points on elliptic curves (with Mark van Hoeij) Maarten Derickx Group structure of modular Jacobians 05-06-2014 9 / 22
A lot is known for prime level. Theorem (Mazur) Let N be prime and 0 , ∞ the two cusps of X 0 ( N ) then J 0 ( N )( Q ) tors is cyclic of order numerator ( N − 1 12 ) and generated by 0 − ∞ . Definition Cl Q − cusp , 0 X 1 ( N )( Q ) ⊆ J 1 ( N )( Q ) tors is the subgroup generated by the differences of Q -rational cusps in X 1 ( N )(¯ Q ) . Conjecture (Conrad,Edixhoven,Stein (CES)) Let N be a prime then Cl Q − cusp , 0 X 1 ( N )( Q ) = J 1 ( N )( Q ) tors Theorem (Ohta) The index of Cl Q − cusp , 0 X 1 ( N )( Q ) in J 1 ( N )( Q ) tors is a power of 2 for N prime. Maarten Derickx Group structure of modular Jacobians 05-06-2014 10 / 22
Three different cuspidal class groups Definition Cl cusp X H ⊆ Pic X H is the group variety of sums of cusps in X H (¯ Q ) . Cl Gal ( Q ) − cusp X H ⊆ Cl cusp X H is the group variety of sums of Gal ( Q ) -orbits of cusps in X H (¯ Q ) . Cl Q − cusp X H ⊆ Cl Gal ( Q ) − cusp X H is the group variety of sums of Q -rational cusps in X H (¯ Q ) . in general Cl Gal ( Q ) − cusp X H � = Cl cusp X H computations suggest Cl Gal ( Q ) − cusp X H ( Q ) = Cl cusp X H ( Q ) If N prime then Cl Q − cusp X H = Cl Gal ( Q ) − cusp X H but for composite N one often has Cl Q − cusp X H ( Q ) � = Cl Gal ( Q ) − cusp X H ( Q ) Maarten Derickx Group structure of modular Jacobians 05-06-2014 11 / 22
The right generalization of the Conrad Edixhoven Stein conjecture Definition Cl cusp X H ⊆ Pic X H is the group variety of sums of cusps in X H (¯ Q ) . Cl Gal ( Q ) − cusp X H ⊆ Cl cusp X H is the group variety of sums of Gal ( Q ) -orbits of cusps in X H (¯ Q ) . Cl Q − cusp X H ⊆ Cl Gal ( Q ) − cusp X H is the group variety of sums of Q -rational cusps in X H (¯ Q ) . Theorem (Manin-Drinfeld) Cl cusp , 0 X H (¯ Q ) ⊆ J H (¯ Q ) tors Conjecture (Generalized CES) Cl cusp , 0 X H ( Q ) = J H ( Q ) tors Maarten Derickx Group structure of modular Jacobians 05-06-2014 12 / 22
Proposition Let N ≤ 55 . If N � = 24 , 32 , 33 , 40 , 48 , 54 then Cl cusp , 0 X 1 ( Q ) = J 1 ( N )( Q ) tors . If N = 24 , 32 , 33 , 40 , 48 respectively 54 then [ Cl cusp , 0 X 1 ( Q ) : Cl csp , 0 X 1 ( N )] Q is a divisor of 2 , 2 , 2 , 4 , 16 respectively 3 . The proposition is proved using two different approaches for computing multiplicative upper bounds on J 1 ( N )( Q ) tors CES: count point on J 1 ( N )( F p ) for different values of p . Other approach based on finding hecke operators that kill J 1 ( N )( Q ) tors . Sometimes taking gcd of both multiplicative upper bounds gives a better result. Maarten Derickx Group structure of modular Jacobians 05-06-2014 13 / 22
Killing the torsion Proposition Let q ∤ 2 N be a prime then T q − q � q � − 1 kills every element in J H ( Q ) tors . Proof. Since q � = 2 we have J H ( Q ) tors ֒ → J H ( F q ) . So it suffices to prove the statement for J H ( F q ) . On J H ( F q ) on has 1 = Frob q and q = Ver q . So the statement follows from T q − Ver q � q � − Frob q = 0 (Eichler-Shimura). Maarten Derickx Group structure of modular Jacobians 05-06-2014 14 / 22
Proposition Let N ≤ 55 . If N � = 24 , 32 , 33 , 40 , 48 , 54 then Cl cusp , 0 X 1 ( Q ) = J 1 ( N )( Q ) tors . If N = 24 , 32 , 33 , 40 , 48 respectively 54 then [ J 1 ( N )( Q ) tors : Cl cusp , 0 X 1 ( Q )] is a divisor of 2 , 2 , 2 , 4 , 16 respectively 3 . Idea behind the proof. Use that T q − q � q � − 1 kills all elements in J 1 ( N )( Q ) . Compute M q := ker ( T q − q � q � − 1 : J 1 ( N )(¯ Q ) tors → J 1 ( N )(¯ Q ) tors ) for several small q 1 , . . . , q n ∤ 2 N . Compute M = ∩ i M q i and let M ′ ⊂ M be the ones invariant under complex conjugation. If M ′ ⊂ Cl cusp , 0 X 1 (¯ Q ) then Cl cusp , 0 X 1 ( Q ) = J 1 ( N )( Q ) tors . If M ′ � Cl csp , 0 X 1 ( N ) then one can still get an upper bound on the index. Maarten Derickx Group structure of modular Jacobians 05-06-2014 15 / 22
Outline Introduction 1 Determining J H ( Q ) 2 When has J H ( Q ) rank 0 Determining J H ( N )( Q ) tors 3 Application to torsion points on elliptic curves (with Mark van Hoeij) Maarten Derickx Group structure of modular Jacobians 05-06-2014 16 / 22
A finite problem Proposition Let N ≤ 55 , N � = 37 , 43 , 53 then the rank of J 1 ( N )( Q ) is 0 . Let N ≤ 55 , N � = 24 , 32 , 33 , 40 , 48 , 54 then Cl cusp , 0 X 1 ( Q ) = J 1 ( N )( Q ) tors . So for N ≤ 55, N � = 24 , 32 , 33 , 37 , 40 , 43 , 48 , 53 , 54 finding all places of degree d (more general finding all g r d ’s since places are g 0 d ’s) is a finite problem, "just" compute the inverse of X 1 ( N ) ( d ) ( Q ) → Pic d X 1 ( N )( Q ) . Algorithm solving this finite problem for D in Pic d X 1 ( N )( Q ) = Cl cusp , 0 X 1 ( Q ) do: write D = � n i C i with C i cusps an n i ∈ Z . compute H := H 0 ( X 1 ( N ) , O ( � n i C i )) if dim H = 0 then D is not linearly equivalent to a D ′ ≥ 0. else | D | = P ( H ) is a g r d with r = dim H − 1 Maarten Derickx Group structure of modular Jacobians 05-06-2014 17 / 22
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