Progress on Mazur’s Program B David Zureick-Brown Emory University Slides available at http://www.mathcs.emory.edu/~dzb/slides/ Southern California Number Theory Day October 21, 2017 David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 1 / 48
Background - Image of Galois G Q := Aut( Q / Q ) E [ n ]( Q ) ∼ = ( Z / n Z ) 2 ρ E , n : G Q → Aut E [ n ] ∼ = GL 2 ( Z / n Z ) GL 2 ( Z /ℓ n Z ) ρ E ,ℓ ∞ : G Q → GL 2 ( Z ℓ ) = lim ← − n ρ E : G Q → GL 2 ( � Z ) = lim GL 2 ( Z / n Z ) ← − n David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 2 / 48
Background - Galois Representations ρ E , n : G Q ։ H ( n ) ֒ → GL 2 ( Z / n Z ) Q ker ρ E , n G Q Q Q ( E [ n ]) H ( n ) Q Problem (Mazur’s “program B”) Classify all possibilities for H ( n ) . David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 3 / 48
Example - torsion on an ellitpic curve If E has a K -rational torsion point P ∈ E ( K )[ n ] (of exact order n ) then: 1 ∗ H ( n ) ⊂ 0 ∗ since for σ ∈ G K and Q ∈ E ( K )[ n ] such that E ( K )[ n ] ∼ = � P , Q � , σ ( P ) = P σ ( Q ) = a σ P + b σ Q David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 4 / 48
Example - Isogenies If E has a K -rational, cyclic isogeny φ : E → E ′ with ker φ = � P � then: ∗ ∗ H ( n ) ⊂ 0 ∗ since for σ ∈ G K and Q ∈ E ( K )[ n ] such that E ( K )[ n ] ∼ = � P , Q � , σ ( P ) = a σ P σ ( Q ) = b σ P + c σ Q David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 5 / 48
Example - other maximal subgroups Normalizer of a split Cartan: � � ∗ 0 0 1 N sp = , 0 ∗ − 1 0 H ( n ) ⊂ N sp and H ( n ) �⊂ C sp iff there exists an unordered pair { φ 1 , φ 2 } of cyclic isogenies, neither of which is defined over K but which are both defined over some quadratic extension of K and which are Galois conjugate. David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 6 / 48
Background - Galois Representations ρ E , n : G Q ։ H ( n ) ֒ → GL 2 ( Z / n Z ) Q ker ρ E , n G Q Q Q ( E [ n ]) H ( n ) Q Problem (Mazur’s “program B”) Classify all possibilities for H ( n ) . David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 7 / 48
Modular curves Definition X ( N )( K ) := { ( E / K , P , Q ) : E [ N ] = � P , Q �} ∪ { cusps } X ( N )( K ) ∋ ( E / K , P , Q ) ⇔ ρ E , N ( G K ) = { I } Definition Γ( N ) ⊂ H ⊂ GL 2 ( � Z ) (finite index) X H := X ( N ) / H X H ( K ) ∋ ( E / K , ι ) ⇔ H ( N ) ⊂ H mod N Stacky disclaimer This is only true up to twist; there are some subtleties if 1 j ( E ) ∈ { 0 , 12 3 } (plus some minor group theoretic conditions), or 2 if − I ∈ H . David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 8 / 48
Rational Points on modular curves Mazur’s program B Compute X H ( Q ) for all H . Remark = P 1 or elliptic with rank X H ( Q ) > 0. Sometimes X H ∼ Some X H have sporadic points. Can compute g ( X H ) group theoretically (via Riemann–Hurwitz). Fact � � GL 2 ( � g ( X H ) , γ ( X H ) → ∞ as Z ) : H → ∞ . David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 9 / 48
Gratuitous picture – subgroups of GL 2 ( Z 2 ) David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 10 / 48
� � � � Sample subgroup (Serre) ker φ 2 ⊂ H (8) ⊂ GL 2 ( Z / 8 Z ) dim F 2 ker φ 2 = 3 φ 2 = I + 2 M 2 ( Z / 2 Z ) ⊂ H (4) GL 2 ( Z / 4 Z ) dim F 2 ker φ 1 = 4 φ 1 = H (2) GL 2 ( Z / 2 Z ) χ : GL 2 ( Z / 8 Z ) → GL 2 ( Z / 2 Z ) × ( Z / 8 Z ) ∗ → Z / 2 Z × ( Z / 8 Z ) ∗ ∼ = F 3 2 . χ = sgn × det H (8) := χ − 1 ( G ), G ⊂ F 3 2 . David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 11 / 48
� � Sample subgroup (Dokchitser 2 ) � I + 2 E 1 , 1 , I + 2 E 2 , 2 � ⊂ H (4) ⊂ GL 2 ( Z / 4 Z ) dim F 2 ker φ 1 = 2 H (2) = GL 2 ( Z / 2 Z ) � � 0 1 0 1 ∼ H (2) = , = F 3 ⋊ D 8 . 3 0 1 1 im ρ E , 4 ⊂ H (4) ⇔ j ( E ) = − 4 t 3 ( t + 8). j X H ∼ = P 1 − → X (1). David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 12 / 48
� � � � � � � � A typical subgroup ker φ 4 ⊂ H (32) ⊂ GL 2 ( Z / 32 Z ) dim F 2 ker φ 4 = 4 φ 4 ker φ 3 ⊂ H (16) ⊂ GL 2 ( Z / 16 Z ) dim F 2 ker φ 3 = 3 φ 3 ker φ 2 ⊂ H (8) ⊂ GL 2 ( Z / 8 Z ) dim F 2 ker φ 2 = 2 φ 2 ker φ 1 ⊂ H (4) ⊂ GL 2 ( Z / 4 Z ) dim F 2 ker φ 1 = 3 φ 1 = H (2) GL 2 ( Z / 2 Z ) David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 13 / 48
� � � � Non-abelian entanglements There exists a surjection θ : GL 2 ( Z / 3 Z ) → GL 2 ( Z / 2 Z ). H (6) := Γ θ ⊂ GL 2 ( Z / 6 Z ) � � GL 2 ( Z / 2 Z ) GL 2 ( Z / 3 Z ) im ρ E , 6 ⊂ H (6) ⇔ K ( E [2]) ⊂ K ( E [3]) David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 14 / 48
Classification of Images - Mazur’s Theorem Theorem Let E be an elliptic curve over Q . Then for ℓ > 11 , E ( Q )[ ℓ ] = { 0 } . In other words, for ℓ > 11 the mod ℓ image is not contained in a subgroup conjugate to 1 ∗ . 0 ∗ David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 15 / 48
Classification of Images - Mazur; Bilu, Parent Theorem (Mazur) Let E be an elliptic curve over Q without CM. Then for ℓ > 37 the mod ℓ image is not contained in a subgroup conjugate to ∗ ∗ . 0 ∗ Theorem (Bilu, Parent) Let E be an elliptic curve over Q without CM. Then for ℓ > 13 the mod ℓ image is not contained in a subgroup conjugate to � � ∗ 0 0 1 , . 0 ∗ − 1 0 David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 16 / 48
Main conjecture Conjecture Let E be an elliptic curve over Q without CM. Then for ℓ > 37, ρ E ,ℓ is surjective. David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 17 / 48
Serre’s Open Image Theorem Theorem (Serre, 1972) Let E be an elliptic curve over K without CM. The image of ρ E ρ E ( G K ) ⊂ GL 2 ( � Z ) is open. Note: � GL 2 ( � Z ) ∼ = GL 2 ( Z p ) p David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 18 / 48
“Vertical” image conjecture Conjecture There exists a constant N such that for every E / Q without CM � � ρ E ( G Q ) : GL 2 ( � Z ) ≤ N . Remark This follows from the “ ℓ > 37” conjecture. Problem Assume the “ ℓ > 37 ” conjecture and compute N. David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 19 / 48
Main Theorems Rouse, ZB (2-adic) The index of ρ E , 2 ∞ ( G Q ) divides 64 or 96; all such indicies occur. Zywina (mod ℓ ) Classifies ρ E ,ℓ ( G Q ) (modulo some conjectures). Zywina (all possible indicies) The index of ρ E , N ( G Q ) divides 220 , 336 , 360 , 504 , 864 , 1152 , 1200 , 1296 or 1536 . Morrow (composite level) Classifies ρ E , 2 · ℓ ( G Q ). Camacho–Li–Morrow–Petok–ZB (composite level) Classifies ρ E ,ℓ n 2 ( G Q ) (partially). 1 · ℓ m David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 20 / 48
Main Theorems continued Zywina–Sutherland Parametrizations in all prime power level, g = 0 and g = 1 , r > 0 cases. Gonzalez–Jimenez, Lozano–Robledo Classify E / Q with ρ E , n ( G Q ) abelian. Brau–Jones, Jones–McMurdy (in progress) Equations for X H for entanglement groups H . Rouse–ZB for other primes (in progress) Partial progress; e.g. for N = 3 n . Derickx–Etropolski–Morrow–van Hoejk–ZB (in progress) Classify possibilities for cubic torsion. David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 21 / 48
Some applications and complements Theorem (R. Jones, Rouse, ZB) 1 Arithmetic dynamics : let P ∈ E ( Q ) . 2 How often is the order of � P ∈ E ( F p ) odd? 3 Answer depends on ρ E , 2 ∞ ( G Q ) . 4 Examples: 11 / 21 (generic), 121 / 168 (maximal), 1 / 28 (minimal) Theorem (Various authors) Computation of S Q ( d ) and S ( d ) for particular d. Theorem (Daniels, Lozano-Robledo, Najman, Sutherland) Classification of E ( Q (3 ∞ )) tors David Zureick-Brown (Emory University) Progress on Mazur’s Program B October 21, 2017 22 / 48
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