Torsion subgroups of rational elliptic curves over the compositum of - PowerPoint PPT Presentation

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields Andrew V. Sutherland Massachusetts Institute of Technology October 2, 2015 joint work with Harris B. Daniels, ´ Alvaro Lozano-Robledo, and Filip Najman http://arxiv.org/abs/1509.00528

Torsion subgroups of elliptic curves over number fields Theorem (Mazur 1977) Let E be an elliptic curve over Q . � Z / M Z 1 � M � 10 , M = 12 ; E ( Q ) tors ≃ Z / 2 Z ⊕ Z / 2 M Z 1 � M � 4 . Theorem (Kenku,Momose 1988, Kamienny 1992) Let E be an elliptic curve over a quadratic number field K .  1 � M � 16 , M = 18 ; Z / M Z     Z / 2 Z ⊕ Z / 2 M Z 1 � M � 6 ; E ( K ) tors ≃ Z / 3 Z ⊕ Z / 3 M Z M = 1 , 2 ( K = Q ( ζ 3 ) only ) ;     ( K = Q ( i ) only ) . Z / 4 Z ⊕ Z / 4 Z

Torsion subgroups of elliptic curves over cubic fields Theorem (Jeon,Kim,Schweizer 2004) Let T be an abelian group for which E ( F ) tors ≃ T for infinitely many elliptic curves E over cubic number fields F with distinct j ( E ) . � 1 � M � 16 , M = 18 , 20 ; Z / M Z T ≃ Z / 2 Z ⊕ Z / 2 M Z 1 � M � 7 . Theorem (Najman 2012) Let E / Q be an elliptic curve and let K be a cubic number field. � Z / M Z 1 � M � 10 , M = 12 , 13 , 14 , 18 , 21 ; E ( K ) tors ≃ Z / 2 Z ⊕ Z / 2 M Z 1 � M � 4 , M = 7 . The case E ( K ) tors ≃ Z / 21 Z occurs only for 162b1 with K = Q ( ζ 9 ) + .

Elliptic curves over Q ( 2 ∞ ) Definition Let Q ( d ∞ ) be the compositum of all degree- d extensions K / Q in Q . Example: Q ( 2 ∞ ) is the maximal elementary 2-abelian extension of Q . Theorem (Frey,Jarden 1974) For E / Q the group E ( Q ( 2 ∞ )) is not finitely generated.

Elliptic curves over Q ( 2 ∞ ) Definition Let Q ( d ∞ ) be the compositum of all degree- d extensions K / Q in Q . Example: Q ( 2 ∞ ) is the maximal elementary 2-abelian extension of Q . Theorem (Frey,Jarden 1974) For E / Q the group E ( Q ( 2 ∞ )) is not finitely generated. Theorem (Laska,Lorenz 1985, Fujita 2004,2005) For E / Q the group E ( Q ( 2 ∞ )) tors is finite and  Z / M Z M = 1 , 3 , 5 , 7 , 9 , 15 ;      Z / 2 Z ⊕ Z / 2 M Z 1 � M � 6 , M = 8 ;   E ( Q ( 2 ∞ )) tors ≃ Z / 3 Z ⊕ Z / 3 Z    Z / 4 Z ⊕ Z / 4 M Z 1 � M � 4 ;     Z / 2 M Z ⊕ Z / 2 M Z 3 � M � 4 .

Elliptic curves over Q ( 3 ∞ ) Theorem (Daniels,Lozano-Robledo,Najman,S 2015) For E / Q the group E ( Q ( 3 ∞ )) tors is finite and  Z / 2 Z ⊕ Z / 2 M Z M = 1 , 2 , 4 , 5 , 7 , 8 , 13 ;     M = 1 , 2 , 4 , 7 ; Z / 4 Z ⊕ Z / 4 M Z E ( Q ( 3 ∞ )) tors ≃ Z / 6 Z ⊕ Z / 6 M Z M = 1 , 2 , 3 , 5 , 7 ;     Z / 2 M Z ⊕ Z / 2 M Z M = 4 , 6 , 7 , 9 . Of these, all but 4 arise for infinitely many j ( E ) . We give complete lists/parametrizations of the j ( E ) that arise in each case. E / Q E ( Q ( 3 ∞ )) tors E / Q E ( Q ( 3 ∞ )) tors 11a2 Z / 2 Z ⊕ Z / 2 Z 338a1 Z / 4 Z ⊕ Z / 28 Z 17a3 Z / 2 Z ⊕ Z / 4 Z 20a1 Z / 6 Z ⊕ Z / 6 Z 15a5 Z / 2 Z ⊕ Z / 8 Z 30a1 Z / 6 Z ⊕ Z / 12 Z 11a1 Z / 2 Z ⊕ Z / 10 Z 14a3 Z / 6 Z ⊕ Z / 18 Z Z / 2 Z ⊕ Z / 14 Z Z / 6 Z ⊕ Z / 30 Z 26b1 50a3 Z / 2 Z ⊕ Z / 16 Z Z / 6 Z ⊕ Z / 42 Z 210e1 162b1 Z / 2 Z ⊕ Z / 26 Z Z / 8 Z ⊕ Z / 8 Z 147b1 15a1 Z / 4 Z ⊕ Z / 4 Z Z / 12 Z ⊕ Z / 12 Z 17a1 30a2 Z / 4 Z ⊕ Z / 8 Z Z / 14 Z ⊕ Z / 14 Z 15a2 2450a1 Z / 4 Z ⊕ Z / 16 Z Z / 18 Z ⊕ Z / 18 Z 210e2 14a1

T j ( t ) Z / 2 Z ⊕ Z / 2 Z t ( t 2 + 16 t + 16 ) 3 Z / 2 Z ⊕ Z / 4 Z t ( t + 16 ) ( t 4 − 16 t 2 + 16 ) 3 Z / 2 Z ⊕ Z / 8 Z t 2 ( t 2 − 16 ) ( t 4 − 12 t 3 + 14 t 2 + 12 t + 1 ) 3 Z / 2 Z ⊕ Z / 10 Z t 5 ( t 2 − 11 t − 1 ) ( t 2 + 13 t + 49 )( t 2 + 5 t + 1 ) 3 Z / 2 Z ⊕ Z / 14 Z t ( t 16 − 8 t 14 + 12 t 12 + 8 t 10 − 10 t 8 + 8 t 6 + 12 t 4 − 8 t 2 + 1 ) 3 Z / 2 Z ⊕ Z / 16 Z t 16 ( t 4 − 6 t 2 + 1 )( t 2 + 1 ) 2 ( t 2 − 1 ) 4 ( t 4 − t 3 + 5 t 2 + t + 1 )( t 8 − 5 t 7 + 7 t 6 − 5 t 5 + 5 t 3 + 7 t 2 + 5 t + 1 ) 3 Z / 2 Z ⊕ Z / 26 Z t 13 ( t 2 − 3 t − 1 ) ( t 2 + 192 ) 3 − 16 ( t 4 − 14 t 2 + 1 ) 3 − 4 ( t 2 + 2 t − 2 ) 3 ( t 2 + 10 t − 2 ) Z / 4 Z ⊕ Z / 4 Z ( t 2 − 64 ) 2 , , t 2 ( t 2 + 1 ) 4 t 4 16 ( t 4 + 4 t 3 + 20 t 2 + 32 t + 16 ) 3 − 4 ( t 8 − 60 t 6 + 134 t 4 − 60 t 2 + 1 ) 3 Z / 4 Z ⊕ Z / 8 Z , t 4 ( t + 1 ) 2 ( t + 2 ) 4 t 2 ( t 2 − 1 ) 2 ( t 2 + 1 ) 8 ( t 16 − 8 t 14 + 12 t 12 + 8 t 10 + 230 t 8 + 8 t 6 + 12 t 4 − 8 t 2 + 1 ) 3 Z / 4 Z ⊕ Z / 16 Z t 8 ( t 2 − 1 ) 8 ( t 2 + 1 ) 4 ( t 4 − 6 t 2 + 1 ) 2 � 351 � 4 , − 38575685889 Z / 4 Z ⊕ Z / 28 Z 16384 ( t + 27 )( t + 3 ) 3 Z / 6 Z ⊕ Z / 6 Z t ( t 2 − 3 ) 3 ( t 6 − 9 t 4 + 3 t 2 − 3 ) 3 Z / 6 Z ⊕ Z / 12 Z t 4 ( t 2 − 9 )( t 2 − 1 ) 3 ( t + 3 ) 3 ( t 3 + 9 t 2 + 27 t + 3 ) 3 , ( t + 3 )( t 2 − 3 t + 9 )( t 3 + 3 ) 3 Z / 6 Z ⊕ Z / 18 Z t ( t 2 + 9 t + 27 ) t 3 � − 121945 � , 46969655 Z / 6 Z ⊕ Z / 30 Z 32 32768 � 3375 � , − 140625 , − 1159088625 , − 189613868625 Z / 6 Z ⊕ Z / 42 Z 2 8 2097152 128 ( t 8 + 224 t 4 + 256 ) 3 Z / 8 Z ⊕ Z / 8 Z t 4 ( t 4 − 16 ) 4 � − 35937 ( t 2 + 3 ) 3 ( t 6 − 15 t 4 + 75 t 2 + 3 ) 3 � , 109503 Z / 12 Z ⊕ Z / 12 Z , t 2 ( t 2 − 9 ) 2 ( t 2 − 1 ) 6 4 64 � 2268945 � Z / 14 Z ⊕ Z / 14 Z 128 27 t 3 ( 8 − t 3 ) 3 432 t ( t 2 − 9 )( t 2 + 3 ) 3 ( t 3 − 9 t + 12 ) 3 ( t 3 + 9 t 2 + 27 t + 3 ) 3 ( 5 t 3 − 9 t 2 − 9 t − 3 ) 3 Z / 18 Z ⊕ Z / 18 Z , ( t 3 + 1 ) 3 ( t 3 − 3 t 2 − 9 t + 3 ) 9 ( t 3 + 3 t 2 − 9 t − 3 ) 3

Characterizing Q ( 3 ∞ ) Definition A finite group G is of generalized S 3 -type if it is isomorphic to a subgroup of S 3 × · · · × S 3 . Example: D 6 . Nonexamples: A 4 , C 4 , B ( 2 , 3 ) . Lemma G is of generalized S 3 -type if and only if G is a supersolvable group whose exponent divides 6 and whose Sylow subgroups are abelian. Corollary The class of generalized S 3 -type groups is closed under products, subgroups, and quotients.

Characterizing Q ( 3 ∞ ) Definition A finite group G is of generalized S 3 -type if it is isomorphic to a subgroup of S 3 × · · · × S 3 . Example: D 6 . Nonexamples: A 4 , C 4 , B ( 2 , 3 ) . Lemma G is of generalized S 3 -type if and only if G is a supersolvable group whose exponent divides 6 and whose Sylow subgroups are abelian. Corollary The class of generalized S 3 -type groups is closed under products, subgroups, and quotients. Proposition A number field lies in Q ( 3 ∞ ) if and only if its Galois group is of generalized S 3 -type.

Uniform boundedness for base extensions of E / Q Theorem Let F / Q be a Galois extension with finitely many roots of unity. There is a uniform bound B such that # E ( F ) tors � B for all E / Q .

Uniform boundedness for base extensions of E / Q Theorem Let F / Q be a Galois extension with finitely many roots of unity. There is a uniform bound B such that # E ( F ) tors � B for all E / Q . Proof sketch. 1. E [ n ] �⊆ E ( F ) for all sufficiently large n (Weil pairing). 2. If E [ p k ] ⊆ E ( F ) with k maximal and p j | λ ( E ( F )[ p ∞ ]) , then E admits a Q -rational cyclic p j − k -isogeny (Galois stability). 3. E does not admit a Q -rational cyclic p n -isogeny for p n > 163 (Mazur+Kenku). Corollary E ( Q ( 3 ∞ )) tors is finite. Indeed, # E ( Q ( 3 ∞ )) must divide 2 10 3 7 5 2 7 3 13 .

Determining E ( Q ( 3 ∞ ))[ p ∞ ] for p ∈ { 2 , 3 , 5 , 7 , 13 } Lemma For j ( E ) � = 1728 the structure of E ( Q ( 3 ∞ )) tors is determined by j ( E ) . For j ( E ) = 1728 we have E ( Q ( 3 ∞ )) tors ≃ Z / 2 Z ⊕ Z / 2 Z or Z / 4 Z ⊕ Z / 4 Z . Now we start computing possible Galois images G in GL 2 ( Z / p n Z ) and corresponding modular curves X G , leaning heavily on results of Rouse-Zureick-Brown and S-Zywina. The most annoying case is 27-torsion. We get the genus 4 curve X : x 3 y 2 − x 3 y − y 3 + 6 y 2 − 3 y = 1 . Fortunately Aut ( X Q ( ζ 3 ) ) ≃ Z / 3 Z ⊕ Z / 3 Z , and the two cyclic quotients are hyperelliptic curves over Q ( ζ 3 ) with only 3 Q ( ζ 3 ) -rational points.

Determining E ( Q ( 3 ∞ ))[ p ∞ ] for p ∈ { 2 , 3 , 5 , 7 , 13 } Lemma For j ( E ) � = 1728 the structure of E ( Q ( 3 ∞ )) tors is determined by j ( E ) . For j ( E ) = 1728 we have E ( Q ( 3 ∞ )) tors ≃ Z / 2 Z ⊕ Z / 2 Z or Z / 4 Z ⊕ Z / 4 Z . Now we start computing possible Galois images G in GL 2 ( Z / p n Z ) and corresponding modular curves X G , leaning heavily on results of Rouse-Zureick-Brown and S-Zywina. The most annoying case is 27-torsion. We get the genus 4 curve X : x 3 y 2 − x 3 y − y 3 + 6 y 2 − 3 y = 1 . Fortunately Aut ( X Q ( ζ 3 ) ) ≃ Z / 3 Z ⊕ Z / 3 Z , and the two cyclic quotients are hyperelliptic curves over Q ( ζ 3 ) with only 3 Q ( ζ 3 ) -rational points. We eventually find E ( Q ( 3 ∞ )) tors must be isomorphic to a subgroup of Z / 8 Z ⊕ Z / 16 Z ⊕ Z / 9 Z ⊕ Z / 9 Z ⊕ Z / 5 Z ⊕ Z / 7 Z ⊕ Z / 7 Z ⊕ Z / 13 Z .

An algorithm to compute E ( Q ( 3 ∞ )) tors Naive approach is not practical, need to be clever. ◮ Compute each E ( Q ( 3 ∞ ))[ p ∞ ] separately. ◮ Q ( E [ p n ]) ⊆ Q ( 3 ∞ ) iff Q ( E [ p n ]) is of generalized S 3 -type. ◮ Q ( P ) ⊆ Q ( 3 ∞ ) iff Q ( P ) is of generalized S 3 -type. ◮ Use fields defined by division polynomials (+ quadratic ext). ◮ If the exponent does not divide 6 you can detect this locally. ◮ Use isogeny kernel polynomials to speed things up. ◮ Prove theorems to rule out annoying cases. theorem ⇒ algorithm ⇒ theorem ⇒ algorithm ⇒ theorem ⇒ · · ·