Progress on Mazur s program B

Transkript

1 Progress on Mazur s program B David Zureick-Brown Emory University Slides available at Rational points on irrational varieties June 25, 2019 David Zureick-Brown (Emory University) Program B June 25, / 44

2 Galois Representations Q G K K := Aut(K/K) E[n](K) = (Z/nZ) 2 ρ E,n : G K Aut E[n] = GL2 (Z/nZ) ρ E,l : G K GL 2 (Z l ) = lim n GL 2 (Z/l n Z) ρ E : G K GL 2 (Ẑ) = lim n GL 2 (Z/nZ) David Zureick-Brown (Emory University) Program B June 25, / 44

3 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 (Ẑ) of ρ E is open. Note: GL 2 (Ẑ) = p GL 2 (Z p ) David Zureick-Brown (Emory University) Program B June 25, / 44

4 Image of Galois ρ E,n : G Q H(n) GL 2 (Z/nZ) Q G Q Q ker ρ E,n Q Q(E[n]) H(n) Problem (Mazur s program B ) Classify all possibilities for H(n). David Zureick-Brown (Emory University) Program B June 25, / 44

5 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, / 44

6 Example - torsion on an elliptic curve If E has a K-rational torsion point P E(K)[n] (of exact order n) then: H(n) 1 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, / 44

7 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, / 44

8 Example - other maximal subgroups Normalizer of a split Cartan: < > N sp = 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, / 44

9 Example - other maximal subgroups Normalizer of a non-split Cartan: ( ) C ns = im F p GL 2 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, / 44

10 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 (Ẑ) (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, / 44

11 Rational Points on modular curves Mazur s program B Compute X H (Q) for all H. Remark Sometimes X H = P 1 or elliptic with rank X H (Q) > 0. 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 g(x H ), γ(x H ) as ] [GL 2 (Ẑ) : H. David Zureick-Brown (Emory University) Program B June 25, / 44

12 Sample subgroup (Serre) ker φ 2 H(8) GL 2 (Z/8Z) dim F2 ker φ 2 = 3 φ 2 I + 2M 2 (Z/2Z) H(4) = GL 2 (Z/4Z) dim F2 ker φ 1 = 4 φ 1 H(2) = GL 2 (Z/2Z) χ: GL 2 (Z/8Z) GL 2 (Z/2Z) (Z/8Z) Z/2Z (Z/8Z) = F 3 2. χ = sgn det H(8) := χ 1 (G), G F 3 2. David Zureick-Brown (Emory University) Program B June 25, / 44

13 A typical subgroup ker φ 4 H(32) GL 2 (Z/32Z) dim F2 ker φ 4 = 4 φ 4 ker φ 3 H(16) GL 2 (Z/16Z) dim F2 ker φ 3 = 3 φ 3 ker φ 2 H(8) GL 2 (Z/8Z) dim F2 ker φ 2 = 2 φ 2 ker φ 1 H(4) GL 2 (Z/4Z) dim F2 ker φ 1 = 3 φ 1 H(2) = GL 2 (Z/2Z) David Zureick-Brown (Emory University) Program B June 25, / 44

14 Non-abelian entanglements There exists a surjection θ : GL 2 (Z/3Z) GL 2 (Z/2Z). H(6) := Γ θ GL 2 (Z/6Z) GL 2 (Z/2Z) GL 2 (Z/3Z) im ρ E,6 H(6) j(e) = t 3 (1 4t 3 ) K(E[2]) K(E[3]). X H = P 1 j X (1). David Zureick-Brown (Emory University) Program B June 25, / 44

15 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, / 44

16 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 > David Zureick-Brown (Emory University) Program B June 25, / 44

17 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/lZ) for l > 37. David Zureick-Brown (Emory University) Program B June 25, / 44

18 Vertical image conjecture Conjecture There exists a constant N such that for every E/Q without CM [ ] GL 2 (Ẑ) : 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, / 44

19 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 different images can occur for non-cm elliptic curves. 4 There are 8 sporadic subgroups. David Zureick-Brown (Emory University) Program B June 25, / 44

20 Subgroups of GL 2 (Z 2 ) David Zureick-Brown (Emory University) Program B June 25, / 44

21 Cremona Database, 2-adic images Index, # of isogeny classes 1, , , , , , , , , , 20 48, , 0 (two examples) 96, 241 (first example - X 0 (15)) CM, 1613 David Zureick-Brown (Emory University) Program B June 25, / 44

22 Cremona Database Index, # of isogeny classes 64, 0 j = j = Rational points on X ns(16) + (Heegner, Baran) David Zureick-Brown (Emory University) Program B June 25, / 44

23 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, / 44

24 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/2Z Z/4Z.) David Zureick-Brown (Emory University) Program B June 25, / 44

25 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 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, / 44

26 In progress Morrow; Camacho Li Morrow Petok ZB (composite level) Classifies ρ E,l n 1 l m(g 2 Q) (partially). Derickx Rouse Sutherland ZB for other prime powers (in progress) Partial progress; e.g. for N = 3 n. David Zureick-Brown (Emory University) Program B June 25, / 44

27 Proof template 1 Compute all arithmetically minimal H GL 2 (Z 2 ) 2 Compute equations for each X H 3 Find (with proof) all rational points on each X H. David Zureick-Brown (Emory University) Program B June 25, / 44

29 Finding Equations Basic idea 1 The canoncial map C P g 1 is given by P [ω 1 (P) : : ω g (P)]. 2 For a general curve, this is an embedding, and the relations are quadratic. 3 For a modular curve, M k (H) = H 0 (X H, Ω 1 ( ) k/2 ) given by f (z) f (z) dz k/2. David Zureick-Brown (Emory University) Program B June 25, / 44

30 Equations Example: X 1 (17) P 4 q 11q q 7 + O(q 8 ) q 2 7q 5 + 6q 7 + O(q 8 ) q 3 4q 5 + 2q 7 + O(q 8 ) q 4 2q 5 + O(q 8 ) q 6 3q 7 + O(q 8 ) xu + 2xv yz + yu 3yv + z 2 4zu + 2u 2 + v 2 = 0 xu + xv yz + yu 2yv + z 2 3zu + 2uv = 0 2xz 3xu + xv 2y 2 + 3yz + 7yu 4yv 5z 2 3zu + 4zv = 0 David Zureick-Brown (Emory University) Program B June 25, / 44

31 Equations general 1 H H of index 2, X H X H degree 2. 2 Given equations for X H, compute equations for X H. 3 Compute a new modular form on H, compute (quadratic) relations between this and modular forms on H. 4 Main technique if X H has new cusps, then write down Eisenstein series which vanish at one new cusp, not others. David Zureick-Brown (Emory University) Program B June 25, / 44

32 Rational points rundown, l = curves (excluding pointless conics) Genus Number Rank of Jacobian ?? ?? 2 7?? 3 9?? 4?? 5 10?? David Zureick-Brown (Emory University) Program B June 25, / 44

33 More 2-adic facts 1 There are 8 sporadic subgroups 1 Only one genus 2 curve has a sporadic point 2 Six genus 3 curves each have a single sporadic point 3 The genus 1, 5, and 7 curves have no sporadic points 2 Many accidental isomorphisms of X H = XH. 3 There is one H such that g(x H ) = 1 and X H X H (Q). David Zureick-Brown (Emory University) Program B June 25, / 44

40 Rational Points: summary of remaining work. 3 g = 12 5 g = 2, 4, 14 7 g = 9, 12, g = 41, X S4 (13) (genus 3) David Zureick-Brown (Emory University) Program B June 25, / 44

41 Rational Points: summary of remaining work more info. The Untouchables X + ns(27), X + ns(25), X + ns(49), X + ns(121) g = 12, 14, 69, 511 Also probably untouchable (r g) X 13, X 21, X 14 g = 9, 9, 41 level 7, 7, 11 Cautiously optimistic (r g) X 11, X 15, X 16, X S4 g = 2, 2, 4, 3 level 5, 5, 5, 13 Optimistic (r = 3 < g) g = 12, level 7 David Zureick-Brown (Emory University) Program B June 25, / 44

42 Explicit methods: highlight reel Local methods Chabauty Elliptic Chabauty Mordell Weil sieve étale descent Pryms Equationless descent via group theory. New techniques for computing Aut C. David Zureick-Brown (Emory University) Program B June 25, / 44

43 Thanks Thank you! David Zureick-Brown (Emory University) Program B June 25, / 44