Progress on Mazur’s program B David Zureick-Brown Emory University Slides available at http://www.mathcs.emory.edu/~dzb/slides/ Rational points on irrational varieties June 25, 2019 David Zureick-Brown (Emory University) Program B June 25, 2019 1 / 44
Galois Representations Q ⊂ K G K := Aut( K / K ) ∼ = ( Z / n Z ) 2 E [ n ]( K ) ρ E , n : G K → Aut E [ n ] ∼ = GL 2 ( Z / n Z ) GL 2 ( Z /ℓ n Z ) ρ E ,ℓ ∞ : G K → GL 2 ( Z ℓ ) = lim ← − n ρ E : G K → GL 2 ( � Z ) = lim GL 2 ( Z / n Z ) ← − n David Zureick-Brown (Emory University) Program B June 25, 2019 2 / 44
Serre’s Open Image Theorem Theorem (Serre, 1972) Let E be an elliptic curve over K without CM. The image ρ E ( G K ) ⊂ GL 2 ( � Z ) of ρ E is open. Note: � GL 2 ( � Z ) ∼ = GL 2 ( Z p ) p David Zureick-Brown (Emory University) Program B June 25, 2019 3 / 44
Image of Galois 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) Program B June 25, 2019 4 / 44
Mazur’s Program B As presented at Modular functions in one variable V in Bonn Mazur - Rational points on modular curves (1977) David Zureick-Brown (Emory University) Program B June 25, 2019 5 / 44
Example - torsion on an elliptic 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) Program B June 25, 2019 6 / 44
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) Program B June 25, 2019 7 / 44
Example - other maximal subgroups Normalizer of a split Cartan: N sp = < > ∗ 0 0 1 , 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, whose kernels intersect trivially, 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) Program B June 25, 2019 8 / 44
Example - other maximal subgroups Normalizer of a non-split Cartan: � � F ∗ C ns = im p 2 → GL 2 ( F p ) ⊂ N ns H ( n ) ⊂ N ns and H ( n ) �⊂ C ns iff E admits a “necklace” (Rebolledo, Wuthrich) David Zureick-Brown (Emory University) Program B June 25, 2019 9 / 44
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) Program B June 25, 2019 10 / 44
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). Can compute # X H ( F q ) via moduli and enumeration [Sutherland]. Fact � � GL 2 ( � g ( X H ) , γ ( X H ) → ∞ as Z ) : H → ∞ . David Zureick-Brown (Emory University) Program B June 25, 2019 11 / 44
� � � � 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) Program B June 25, 2019 12 / 44
� � � � � � � � 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) Program B June 25, 2019 13 / 44
� � � � 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) ⇔ j ( E ) = 2 10 3 3 t 3 (1 − 4 t 3 ) ⇒ K ( E [2]) ⊂ K ( E [3]). j X H ∼ = P 1 − → X (1). David Zureick-Brown (Emory University) Program B June 25, 2019 14 / 44
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, H ( ℓ ) is not contained in a subgroup conjugate to 1 ∗ . 0 ∗ David Zureick-Brown (Emory University) Program B June 25, 2019 15 / 44
Classification of Images - Mazur; Bilu, Parent, Rebolledo Theorem (Mazur) Let E be an elliptic curve over Q without CM. Then for ℓ > 37 , H ( ℓ ) is not contained in a subgroup conjugate to ∗ ∗ . 0 ∗ Theorem (Bilu, Parent, Rebolledo) Let E be an elliptic curve over Q without CM. Then for ℓ > 13 , H ( ℓ ) is not contained in a subgroup conjugate to < > ∗ 0 0 1 , . 0 ∗ − 1 0 David Zureick-Brown (Emory University) Program B June 25, 2019 16 / 44
Main conjecture Conjecture (Serre) Let E be an elliptic curve over Q without CM. Then for ℓ > 37, ρ E ,ℓ is surjective. In other words, conjecturally, H ( ℓ ) = GL 2 ( Z /ℓ Z ) for ℓ > 37. David Zureick-Brown (Emory University) Program B June 25, 2019 17 / 44
“Vertical” image conjecture Conjecture There exists a constant N such that for every E / Q without CM � � GL 2 ( � Z ) : ρ E ( G Q ) ≤ N . Remark This follows from the “ ℓ > 37” conjecture. Problem Assume the “ ℓ > 37 ” conjecture and compute N. David Zureick-Brown (Emory University) Program B June 25, 2019 18 / 44
Main Theorem Rouse, ZB (2-adic) The index of ρ E , 2 ∞ ( G Q ) divides 64 or 96; all such indices occur. 1 All indices dividing 96 occur infinitely often; 64 occurs only twice. 2 The 2-adic image is determined by the mod 32 image. 3 1208 different images can occur for non-CM elliptic curves. 4 There are 8 “sporadic” subgroups. David Zureick-Brown (Emory University) Program B June 25, 2019 19 / 44
Subgroups of GL 2 ( Z 2 ) David Zureick-Brown (Emory University) Program B June 25, 2019 20 / 44
Cremona Database, 2-adic images Index , # of isogeny classes 1 , 727995 2 , 7281 3 , 175042 4 , 1769 6 , 57500 8 , 577 12 , 29900 16 , 235 24 , 5482 32 , 20 48 , 1544 64 , 0 (two examples) 96 , 241 (first example - X 0 (15)) CM , 1613 David Zureick-Brown (Emory University) Program B June 25, 2019 21 / 44
Cremona Database Index , # of isogeny classes 64 , 0 j = − 3 · 2 18 · 5 3 · 13 3 · 41 3 · 107 3 · 17 − 16 j = − 2 21 · 3 3 · 5 3 · 7 · 13 3 · 23 3 · 41 3 · 179 3 · 409 3 · 79 − 16 Rational points on X + ns (16) (Heegner, Baran) David Zureick-Brown (Emory University) Program B June 25, 2019 22 / 44
Applications 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 ) for particular d. Theorem (Daniels, Lozano-Robledo, Najman, Sutherland) Classification of E ( Q (3 ∞ )) tors Theorem (Gonzalez–Jimenez, Lozano–Robledo) Classify E / Q with ρ E , N ( G Q ) abelian. David Zureick-Brown (Emory University) Program B June 25, 2019 23 / 44
More applications Theorem (Sporadic points) Najman’s example X 1 (21) (3) ( Q ) ; “easy production” of other examples. Theorem (Jack Thorne) Elliptic curves over Q ∞ are modular. (One step is to show X 0 (15)( Q ∞ ) = X 0 (15)( Q ) = Z / 2 Z × Z / 4 Z .) David Zureick-Brown (Emory University) Program B June 25, 2019 24 / 44
Recent theorems Zywina (mod ℓ ) Classifies ρ E ,ℓ ( G Q ) (modulo some conjectures). Zywina (indices occuring infinitely often; modulo conjectures) The index of ρ E , N ( G Q ) divides 220 , 336 , 360 , 504 , 864 , 1152 , 1200 , 1296 or 1536 . Sutherland–Zywina Parametrizations in all prime power levels, g = 0 and g = 1 , r > 0 cases. Brau–N. Jones, N. Jones–McMurdy (in progress) Equations for X H for entanglement groups H . David Zureick-Brown (Emory University) Program B June 25, 2019 25 / 44
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