Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary Torsion points on elliptic curves over number fields of small degree. Several variations of kamienny’s criterion Maarten Derickx Mathematisch Instituut Universiteit Leiden UW Number Theory Seminar 18-03-2011 Maarten Derickx Torsion points on elliptic curves
Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary Outline Introduction 1 Variations of Kamienny’s Criterion 2 The Original Version My version Parent’s version Results of testing the criterion 3 Maarten Derickx Torsion points on elliptic curves
Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary What is known � � d ⊇ Q ∃ E / K : E ( K ) [ p ] � = 0 S ( d ) = p prime | ∃ K 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 Torsion points on elliptic curves
Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary Reduce to Multiplicative Reduction d Let Q ⊂ K be a field extension, E / K an elliptic curve, l a prime m ⊆ O K a max. ideal lying over l with res. field F q , P ∈ E ( K ) of ∼ E the fiber over F q of the Néron model. If p ∤ q then order p and ∼ ∼ E ( F q ) has order p . Consider the three cases: P ∈ ∼ 1 2 + 1 ) 2 ≤ ( l d / 2 + 1 ) 2 E ( F q ) ≤ ( q Good reduction: p ≤ # ∼ Additive reduction: 0 → G a , F q → E → Φ → 0 hence p | #Φ( F q ) ≤ 4 < ( l d / 2 + 1 ) 2 ∼ Multiplicative reduction: 0 → T → E → Φ → 0 with ∼ T = G m , F q or T = G m , F q . Hence p | q − 1, p | q + 1 or p | #Φ( F q ) Conclusion: ( l d / 2 + 1 ) 2 is a bound for the torsion order in the good and the additive case. Maarten Derickx Torsion points on elliptic curves
Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary What happens in the multiplicative case Let x ∈ X 0 ( p ) and σ 1 , . . . , σ d be all embeddings of K in C . Then x ( d ) := [( σ 1 ( x ) , . . . , σ d ( x ))] ∈ X 0 ( p ) ( d ) ( Q ) . If s ′ = ( E , � P � ) ∈ X 0 ( p )( K ) and E has multiplicative reduction at ∼ all primes over l and P has nonzero image in Φ then all specializations of s ′ to characteristic l are the cusp 0. Define s = ( E / � P � , E [ p ] / � P � ) then all specializations of s to characteristic p are ∞ . This proves: Proposition If p ∤ l k + 1 , p ∤ l k − 1 for all k ≤ d then s ( d ) = ∞ ( d ) F l . F l In the rest of the talk we study s � = ∞ ∈ X 0 ( p ) such that s ( d ) = ∞ ( d ) F l . (and try to prove that no such s exist for certain p ). F l Maarten Derickx Torsion points on elliptic curves
Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary Mazur’s approach Derive a contradiction with formal immersions in the multiplicative case A morphism f : X → Y of noetherian schemes is a formal f : � O Y , f ( x ) → � immersion at x ∈ X if � O X , x is surjective. Or equivalently k ( x ) = k ( f ( x )) and f ∗ : Cot f ( x ) Y → Cot x X is surjective. Lemma (Mazur) Let A be the Néron model over Z ( l ) of an abelian variety over Q . Suppose there is a morphism f : X 0 ( p ) ( d ) → A normalized by f ( ∞ ( d ) ) = 0 . If s � = ∞ ∈ X 0 ( p ) , s ( d ) = ∞ ( d ) and F l F l f ( s ( d ) ) = 0 ( H ) then f is not a formal immersion at ∞ ( d ) F l Maarten Derickx Torsion points on elliptic curves
Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary If A ( Q ) has rank 0, use the following lemma to satisfy H Lemma If l > 2 prime and A a Z ( l ) group scheme with identity e. If also P ∈ A is a Z ( l ) valued torsion s.t. P F l = e F l then P = e. This is enough since ∞ ( d ) = s ( d ) implies F l F l e F l = f ( ∞ ( d ) ) F l = f ( s ( d ) ) F l ∈ A F l . Maarten Derickx Torsion points on elliptic curves
Introduction Variations of Kamienny’s Criterion Results of testing the criterion Summary Winding quotient The "largest" rank 0 quotient of J 0 ( p ) Definition (winding element) The winding element e ∈ H 1 ( X 0 ( p )( C ) , Q ) is the one � i ∞ ω ∈ H 0 ( X 0 ( p ) , Ω) ∨ corresponding to ω �→ 0 Definition (winding quotient) Let A e ⊆ T be the annihilator of e then J e ( p ) = J 0 ( p ) / A e J 0 ( p ) is called the winding quotient. This definition can also be made over X 1 ( p ) , in both cases J e ( Q ) has rank zero as a result of Kato’s theorem. Maarten Derickx Torsion points on elliptic curves
Introduction The Original Version Variations of Kamienny’s Criterion My version Results of testing the criterion Parent’s version Summary Kamienny’s Criterion The original case: X 0 ( p ) and l � = 2 , p Theorem (Kamienny) Let l � = 2 , p be a prime and f : X 0 ( p ) ( d ) → J e ( p ) be the canonical map normalized by f ( ∞ ( d ) ) = 0 then f is a formal immersion at ∞ ( d ) if and only if T 1 , . . . , T d are F l linearly F l independent in T / ( l T + A e ) . Corollary If p > ( l d / 2 + 1 ) 2 and T 1 , . . . , T d are F l linearly independent in T / ( l T + A e ) . Then p / ∈ S ( d ) . Maarten Derickx Torsion points on elliptic curves
Introduction The Original Version Variations of Kamienny’s Criterion My version Results of testing the criterion Parent’s version Summary What goes wrong at 2 Point orders don’t always stay the same under reduction Need again a lemma to satisfy ( 1 ) Lemma If l = 2 and A a Z ( l ) group scheme with identity e. If also P ∈ A is a Z ( l ) valued torsion s.t. P F l = e F l then P = e or P generates a µ 2 , Z ( l ) immersion. So we need to kill all the 2 torsion: Proposition If q � = p prime. Then T q − q − 1 kills all the Q -rational torsion of J 0 ( p ) of order co prime to pq. Maarten Derickx Torsion points on elliptic curves
Introduction The Original Version Variations of Kamienny’s Criterion My version Results of testing the criterion Parent’s version Summary What goes wrong at 2 Kamienny’s criterion doesn’t work. The criterion is proved by calculating when the composition X 0 ( p ) ( d ) Cot 0 J e ( p ) F l → Cot 0 J 0 ( p ) F l → Cot ∞ ( d ) F l F l is surjective and then translate this to the dual condition in Tan J e ( p ) F l ∼ = T / ( l T + A e ) . The problems at l = 2 arise in proving the isomorphism: Cot J e ( p ) Z ( l ) ∼ = Cot J 0 ( p ) Z ( l ) [ A e ] ⊆ Cot J 0 ( p ) Z ( l ) ∼ = S 2 (Γ 0 ( p ) , Z ( l ) ) Approach by Parent: Instead of looking at f : X 0 ( p ) ( d ) → J e ( p ) construct an f : X 0 ( p ) ( d ) → J 0 ( p ) which factors through J e ( p ) . Maarten Derickx Torsion points on elliptic curves
Introduction The Original Version Variations of Kamienny’s Criterion My version Results of testing the criterion Parent’s version Summary Kamienny’s criterion Parent’s version translated to X 0 ( p ) Theorem Let l � = p be a prime and f : X 0 ( p ) ( d ) → J 0 ( p ) be the canonical map normalized by f ( ∞ ( d ) ) = 0 and t ∈ T then t ◦ f is a formal immersion at ∞ ( d ) if and only if T 1 t , . . . , T d t are F l linearly F l independent in T / ( l T ) . Corollary Take l = 2 and q > 2 prime, if the independence holds for p > ( 2 d / 2 + 1 ) 2 and t = a q · t 1 with t 1 ∈ A ⊥ e then p / ∈ S ( d ) . Maarten Derickx Torsion points on elliptic curves
Introduction The Original Version Variations of Kamienny’s Criterion My version Results of testing the criterion Parent’s version Summary Proof of the corollary Proof. Need to show that for s ∈ X 0 ( p )( K ) with multiplicative reduction at 2 that t ◦ f ( s ( d ) ) = 0. Now t 1 ◦ f factors through J e ( p ) since e hence t 1 ◦ f ( s ( d ) ) is torsion. s ( d ) F 2 = ∞ ( d ) t 1 ∈ A ⊥ F 2 so t 1 ◦ f ( s ( d ) ) is 2 torsion hence killed by a q . Maarten Derickx Torsion points on elliptic curves
Introduction The Original Version Variations of Kamienny’s Criterion My version Results of testing the criterion Parent’s version Summary Some notation to formulate Kamienny for X 1 ( p ) This is why I explained everything for X 0 ( p ) first Let π : X 1 ( p ) → X 0 ( p ) the canonical map. And S := π ( − 1 ) ( ∞ ) then as in the X 0 ( p ) case s ′ ∈ X 1 ( p )( K ) which reduce multiplicative give rise to an s s.t. s F q = ∞ s , F q . Now take σ i ∈ S and n i ∈ N s.t. = � m s ( d ) i = 0 n i σ i , F l F l σ i pairwise distinct n m ≥ n m − 1 ≥ . . . ≥ n 0 ≥ 1 � n i = d . Also write σ 0 = � j � σ j (ok since � d � act transitively on S ) and σ = � m i = 0 n i σ i . Maarten Derickx Torsion points on elliptic curves
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