Families of Abelian Varieties with Big Monodromy David Zureick-Brown - - PowerPoint PPT Presentation

Families of Abelian Varieties with Big Monodromy

David Zureick-Brown (Emory University) David Zywina (IAS)

Slides available at http://www.mathcs.emory.edu/~dzb/slides/

2013 Colorado AMS meeting Special Session on Arithmetic statistics and big monodromy Boulder, CO April 14, 2013

Background - Galois Representations ρA,n : GK → Aut A[n] ∼ = GL2g(Z/nZ) ρA,ℓ∞ : GK → GL2g(Zℓ) = lim ← −

n

GL2g (Z/ℓnZ) ρA: GK → GL2g( Z) = lim ← −

n

GL2g (Z/nZ)

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 2 / 22

Background - Galois Representations

ρA,n : GK ։ Gn ֒ → GSp2g(Z/nZ) Gn ∼ = Gal(K (A[n]) /K)

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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: Gn ⊂    1 ∗ ∗    since for σ ∈ GK and Q ∈ E(K)[n] such that E(K)[n] ∼ = P, Q, σ(P) = P σ(Q) = aσP + bσQ

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Monodromy of a family

1 U ⊂ PN

K (non-empty open)

2 η ∈ U (generic point) 3 A → U (family of principally polarized abelian varieties) 4 ρAη : GK(U) → GSp2g(

Z)

Definition

The monodromy of A → U is the image Hη of ρAη. We say that A → U has big monodromy if Hη is an open subgroup of GSp2g( Z).

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 5 / 22

Monodromy of a family over a stack

1 U is now a stack.

Definition

The monodromy of A → U is the image H of ρA . We say that A → U has big monodromy if H is an open subgroup of GSp2g( Z).

1 Spec Ω

η

− → U (geometric generic point)

2 π1,et(U) 1 A → U (family of principally polarized abelian varieties) 2 ρA : π1,et(U) → GSp2g(

Z)

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(Example) standard family of elliptic curves

E : y2 = x3 + ax + b U = A2

K − ∆

H =

• A ∈ GL2(

Z) : det(A) ∈ χK(Gal(K/K))

• David Zureick-Brown (Emory)

Families with Big Monodromy April 14, 2013 7 / 22

(Example) elliptic curves with full two torsion

E : y2 = x(x − a)(x − b) U = A2

Q − ∆

H =

• A ∈ GL2(

Z) : A ≡ I (mod 2)

• David Zureick-Brown (Emory)

Families with Big Monodromy April 14, 2013 8 / 22

Exotic example from Zywina’s HIT paper

E : y2 + xy = x3 − 36 j − 1728x − 1 j − 1728 .

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 9 / 22

Exotic example from Zywina’s HIT paper

E : y2 + xy = x3 − 36 j − 1728x − 1 j − 1728 over U ⊂ A1

K

j = (T 16 + 256T 8 + 4096)3 T 32(T 8 + 16) [GL2( Z) : H] = 1536 .

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 9 / 22

Exotic example from Zywina’s HIT paper

E : y2 + xy = x3 − 36 j − 1728x − 1 j − 1728 over U ⊂ A1

K

j = (T 16 + 256T 8 + 4096)3 T 32(T 8 + 16) [GL2( Z) : 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, 2013 9 / 22

(Example) Hyperelliptic

E : y2 = x2g+2 + a2g+1x2g+1 + . . . + a0

• ver U ⊂ A2g+2

H =

• A ∈ GSp2g(

Z) : A (mod 2) ∈ S2g+2

• David Zureick-Brown (Emory)

Families with Big Monodromy April 14, 2013 10 / 22

Main Theorem

Theorem (ZB-Zywina)

Let U be a non-empty open subset of PN

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 Aη/K(η) has big monodromy. Let Hη be the image of ρAη. Let BK(N) = {u ∈ U(K) : h(u) ≤ N}. Then a random fiber has maximal monodromy, i.e. (if K = Q) lim

N→∞

|{u ∈ BK(N) : ρAu(GK) = Hη}| |BK(N)| = 1.

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Corollary - Variant of Inverse Galois Problem

Corollary

For every g > 2, there exists an abelian variety A/Q such that Gal(Q(Ators)/Q) ∼ = GSp2g( Z), i.e, for every n, Gal(Q(A[n])/Q) ∼ = GSp2g(Z/nZ).

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 12 / 22

Monodromy of trigonal curves

Theorem (ZB, Zywina)

For every g > 2

1 the stack Tg of trigonal curves has monodromy GSp2g(

Z), and

2 there is a family of trigonal curves over a nonempty rational base

U ⊂ PN

Q with monodromy GSp2g(

Z)

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 13 / 22

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 GSp2g( Z)?

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, 2013 14 / 22

Sketch of trigonal proof

Theorem

For every g the stack Tg of trigonal curves has monodromy GSp2g( Z).

Proof.

1 Mg,d−1 ⊂ Mg,d (suffices for ℓ > 2) 2 Mg−2 ⊂ Mg 3 the mod 2 monodromy thus contains subgroups isomorphic to 1

S2g+2

2

Sp2(g−2)+2(Z/2Z)

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 15 / 22

(Example) Hyperelliptic

E : y2 = x2g+2 + a2g+1x2g+1 + . . . + a0

• ver U ⊂ A2g+2

H =

• A ∈ GSp2g(

Z) : A (mod 2) ∈ S2g+2

• David Zureick-Brown (Emory)

Families with Big Monodromy April 14, 2013 16 / 22

Hyperelliptic example continued

Theorem

1 (Yu) unpublished 2 (Achter, Pries) the stack of hyperelliptic curves has maximal

monodromy

3 (Hall) any 1-paramater family y2 = (t − x)f (t) over K(x) has full

monodromy

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 17 / 22

Hyperelliptic example proof

Corollary

E : y2 = x2g+2 + a2g+1x2g+1 + . . . + a0 has monodromy

• A ∈ GSp2g(

Z) : A (mod 2) ∈ S2g+2

• .

Proof.

1 U = space of distinct unordered 2g + 2-tuples of points on P1 2 U ։ Hg,2 3 Hg,2 ∼

= [U/ Aut P1]

4 fibers are irreducible, thus

π1,et(U) ։ π1,et(Hg,2) is surjective.

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Sketch of trigonal proof

Theorem (ZB, Zywina)

For every g > 2 there is a family of trigonal curves over a nonempty rational base U ⊂ PN

Q with monodromy GSp2g(

Z)

Proof.

1 Main issue:

f3(x)y3 + f2(x)y2 + f1(x)y + f0(x) = 0

2 The stack Tg is unirational, need to make this explicit 3 (Bolognesi, Vistoli) Tg ∼

= [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, 2013 19 / 22

Sketch of trigonal proof - Maroni Invariant

Maroni-invariant

1 The image of the canonical map lands in a scroll

C ֒ → Fn ֒ → Pg−1 Fn ∼ = P(O ⊕ O(−n)) F0 ∼ = P1 × P1 F1 ∼ = BlPP2

2 n has the same parity as g 3 generically n = 0 or 1 4 e.g., if g even we can take U = space of bihomogenous polynomials

• f bi-degree (3, d)

5 [U/G] ∼

= T 0

g ⊂ Tg.

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Pryms

C → D ker0(JC → JD), generally not a Jacobian

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 21 / 22

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 X1(2). The associated family of Prym’s has big monodromy.

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