# Families of Abelian Varieties with Big dzb/slides/ZBro ¢ David Zureick-Brown

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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

http://www.mathcs.emory.edu/~dzb/slides/

Background - Galois Representations

ρA,n : GK → AutA[n] ∼= GL2g(Z/nZ)

ρA,`∞ : GK → GL2g(Z`) = lim←− n

GL2g (Z/`nZ)

ρA : GK → GL2g(Ẑ) = 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 )

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 3 / 22

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 ∗ 0 ∗

since for σ ∈ GK and Q ∈ E (K )[n] such that E (K )[n] ∼= 〈P,Q〉,

σ(P) = P

σ(Q) = aσP + bσQ

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 4 / 22

Monodromy of a family

1 U ⊂ PNK (non-empty open) 2 η ∈ U (generic point) 3 A → U (family of principally polarized abelian varieties) 4 ρAη : GK(U) → GSp2g (Ẑ)

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 (Ẑ).

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 (Ẑ).

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 (Ẑ)

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 6 / 22

(Example) standard family of elliptic curves

E : y2 = x3 + ax + b

U = A2K −∆

H = { A ∈ GL2(Ẑ) : 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 = A2Q −∆

H = { A ∈ GL2(Ẑ) : 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 − 1728

x − 1 j − 1728

over U ⊂ A1K

j = (T 16 + 256T 8 + 4096)3

T 32(T 8 + 16)

[GL2(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, 2013 9 / 22

Exotic example from Zywina’s HIT paper

E : y2 + xy = x3 − 36 j − 1728

x − 1 j − 1728

over U ⊂ A1K

j = (T 16 + 256T 8 + 4096)3

T 32(T 8 + 16)

[GL2(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, 2013 9 / 22

Exotic example from Zywina’s HIT paper

E : y2 + xy = x3 − 36 j − 1728

x − 1 j − 1728

over U ⊂ A1K

j = (T 16 + 256T 8 + 4096)3

T 32(T 8 + 16)

[GL2(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, 2013 9 / 22

(Example) Hyperelliptic

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

over U ⊂ A2g+2

H = { A ∈ GSp2g (Ẑ) : 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 PNK 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 6= Q)

lim N→∞

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

= 1.

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 11 / 22

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 (Ẑ),

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 (Ẑ), and 2 there is a family of trigonal curves over a nonempty rational base

U ⊂ PNQ with monodromy GSp2g (Ẑ)

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 (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, 2013 14 / 22

Sketch of trigonal proof

Theorem

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

Proof.

1 Mg ,d−1 ⊂Mg ,d (suffices for ` > 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+1x 2g+1 + . . .+ a0

over U ⊂ A2g+2

H = { A ∈ GSp2g (Ẑ) : 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+1x 2g+1 + . . .+ a0

has monodromy { A ∈ GSp2g (Ẑ) : 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/AutP1] 4 fibers are irreducible, thus

π1,et(U)� π1,et(Hg ,2)

is surjective.

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 18 / 22

Sketch of trigonal proof

Theorem (ZB, Zywina)

For every g > 2 there is a family of trigonal curves over a nonempty rational base U ⊂ PNQ with monodromy GSp2g (Ẑ)

Proof. 1 Main issue:

f3(x)y 3 + f2(x)y

2 + 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 of bi-degree (3, d)

5 [U/G ] ∼= T 0g ⊂ Tg .

David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 20 / 22

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.

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

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