Families of Abelian Varieties with Big Monodromy

Transcription

1 Families of Abelian Varieties with Big Monodromy David Zureick-Brown (Emory University) David Zywina (IAS) Slides available at Colorado AMS meeting Special Session on Arithmetic statistics and big monodromy Boulder, CO April 14, 2013

2 Background - Galois Representations ρ A,n : G K Aut A[n] = GL2g (Z/nZ) ρ A,l : G K GL 2g (Z l ) = lim n GL 2g (Z/l n Z) ρ A : G K GL 2g (Ẑ) = lim n GL 2g (Z/nZ) David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

3 Background - Galois Representations ρ A,n : G K G n GSp 2g (Z/nZ) G n = Gal(K (A[n]) /K) David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

4 Example - torsion on an ellitpic curve If E has a K-rational torsion point P E(K)[n] (of exact order n), then the image is constrained: G 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) Families with Big Monodromy April 14, / 22

5 Monodromy of a family 1 U P N K (non-empty open) 2 η U (generic point) 3 A U (family of principally polarized abelian varieties) 4 ρ Aη : G K(U) GSp (Ẑ) 2g Definition The monodromy of A U is the image H η of ρ Aη. We say that A U has big monodromy if H η is an open subgroup of GSp 2g (Ẑ). David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

6 Monodromy of a family over a stack 1 U is now a stack. Definition The monodromy of A U is the image H of ρ A. We say that A U has big monodromy if H is an open subgroup of GSp 2g (Ẑ). 1 Spec Ω η U (geometric generic point) 2 π 1,et (U) 1 A U (family of principally polarized abelian varieties) 2 ρ A : π 1,et (U) GSp 2g (Ẑ) David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

7 (Example) standard family of elliptic curves E : y 2 = x 3 + ax + b U = A 2 K H = { A GL 2 (Ẑ) : det(a) χ K (Gal(K/K)) } David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

8 (Example) elliptic curves with full two torsion E : y 2 = x(x a)(x b) U = A 2 Q H = { A GL 2 (Ẑ) : A I (mod 2)} David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

9 Exotic example from Zywina s HIT paper E : y 2 + xy = x 3 36 j 1728 x 1 j David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

10 Exotic example from Zywina s HIT paper E : y 2 + xy = x 3 36 j 1728 x 1 j 1728 over U A1 K j = (T T ) 3 T 32 (T ) [GL 2 (Ẑ) : H] = David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

11 Exotic example from Zywina s HIT paper E : y 2 + xy = x 3 36 j 1728 x 1 j 1728 over U A1 K j = (T T ) 3 T 32 (T ) [GL 2 (Ẑ) : H] = 1536 H is the subgroup of matricies preserving h(z) = η(z) 4 /η(4z). David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

12 (Example) Hyperelliptic E : y 2 = x 2g+2 + a 2g+1 x 2g a 0 over U A 2g+2 H = { A GSp (Ẑ) : A (mod 2) S } 2g 2g+2 David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

13 Main Theorem Theorem (ZB-Zywina) Let U be a non-empty open subset of P N K and let A U be a family of principally polarized abelian varieties. Let η be the generic point of U and suppose moreover that A η /K(η) has big monodromy. Let H η be the image of ρ Aη. Let B K (N) = {u U(K) : h(u) N}. Then a random fiber has maximal monodromy, i.e. (if K Q) {u B K (N) : ρ Au (G K ) = H η } lim = 1. N B K (N) David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

14 Corollary - Variant of Inverse Galois Problem Corollary For every g > 2, there exists an abelian variety A/Q such that Gal(Q(A tors )/Q) = GSp 2g (Ẑ), i.e, for every n, Gal(Q(A[n])/Q) = GSp 2g (Z/nZ). David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

15 Monodromy of trigonal curves Theorem (ZB, Zywina) For every g > 2 1 the stack T g of trigonal curves has monodromy GSp 2g (Ẑ), and 2 there is a family of trigonal curves over a nonempty rational base U P N Q with monodromy GSp 2g (Ẑ) David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

16 Monodromy of families of Pryms Question For every g, does there exists a family A U of PP abelian varieties of dimension g, U rational, which are not generically isogenous to Jacobians, with monodromy GSp 2g (Ẑ)? 1 One can (probably) take A U to be a family of Prym varieties associated to tetragonal curves, or 2 (Tsimerman) one can take A U to be a family of Prym varieties associated to bielliptic curves. David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

17 Sketch of trigonal proof Theorem For every g the stack T g of trigonal curves has monodromy GSp 2g (Ẑ). Proof. 1 M g,d 1 M g,d (suffices for l > 2) 2 M g 2 M g 3 the mod 2 monodromy thus contains subgroups isomorphic to 1 S 2g+2 2 Sp 2(g 2)+2 (Z/2Z) David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

18 (Example) Hyperelliptic E : y 2 = x 2g+2 + a 2g+1 x 2g a 0 over U A 2g+2 H = { A GSp (Ẑ) : A (mod 2) S } 2g 2g+2 David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

19 Hyperelliptic example continued Theorem 1 (Yu) unpublished 2 (Achter, Pries) the stack of hyperelliptic curves has maximal monodromy 3 (Hall) any 1-paramater family y 2 = (t x)f (t) over K(x) has full monodromy David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

20 Hyperelliptic example proof Corollary E : y 2 = x 2g+2 + a 2g+1 x 2g a 0 has monodromy { A GSp 2g (Ẑ) : A (mod 2) S 2g+2}. Proof. 1 U = space of distinct unordered 2g + 2-tuples of points on P 1 2 U H g,2 3 H g,2 = [U/ Aut P 1 ] 4 fibers are irreducible, thus is surjective. π 1,et (U) π 1,et (H g,2 ) David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

21 Sketch of trigonal proof Theorem (ZB, Zywina) For every g > 2 there is a family of trigonal curves over a nonempty rational base U P N Q with monodromy GSp 2g (Ẑ) Proof. 1 Main issue: f 3 (x)y 3 + f 2 (x)y 2 + f 1 (x)y + f 0 (x) = 0 2 The stack T g is unirational, need to make this explicit 3 (Bolognesi, Vistoli) T g = [U/G] where U is rational and G is a connected algebraic group. 4 Maroni-invariant (normal form for trigonal curves). David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

22 Sketch of trigonal proof - Maroni Invariant Maroni-invariant 1 The image of the canonical map lands in a scroll C F n P g 1 2 n has the same parity as g 3 generically n = 0 or 1 F n = P(O O( n)) F 0 = P 1 P 1 F 1 = BlP P 2 4 e.g., if g even we can take U = space of bihomogenous polynomials of bi-degree (3, d) 5 [U/G] = T 0 g T g. David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

23 Pryms C D ker 0 (J C J D ), generally not a Jacobian David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22

24 Monodromy of families of Pryms, bielliptic target Example (Tsimerman) The space of (ramified) double covers of a fixed elliptic curve is rational, so the space of Pryms is also rational, with base isomorphic to a projective space over X 1 (2). The associated family of Prym s has big monodromy. David Zureick-Brown (Emory) Families with Big Monodromy April 14, / 22