Families of Abelian Varieties with Big Monodromydzb/slides/ZBrownAmsColoradofamilies.pdf · David...

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 meetingSpecial 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(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 )


Example - torsion on an ellitpic curve

If E has a K -rational torsion point P ∈ E (K )[n] (of exact order n), thenthe 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


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 (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).


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 → Uhas big monodromy if H is an open subgroup of GSp2g (Z).

1 Spec Ωη−→ U (geometric generic point)

2 π1,et(U)

1 A → U (family of principally polarized abelian varieties)

2 ρA : π1,et(U)→ GSp2g (Z)


(Example) standard family of elliptic curves

E : y2 = x3 + ax + b

U = A2K −∆

H =A ∈ GL2(Z) : det(A) ∈ χK (Gal(K/K ))


(Example) elliptic curves with full two torsion

E : y2 = x(x − a)(x − b)

U = A2Q −∆

H =A ∈ GL2(Z) : A ≡ I (mod 2)


Exotic example from Zywina’s HIT paper

E : y2 + xy = x3 − 36

j − 1728x − 1

j − 1728

over U ⊂ A1K

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)

.



E : y2 + xy = x3 − 36

j − 1728x − 1

j − 1728over 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)

.



E : y2 + xy = x3 − 36

j − 1728x − 1

j − 1728over 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).


(Example) Hyperelliptic

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

over U ⊂ A2g+2

H =A ∈ GSp2g (Z) : A (mod 2) ∈ S2g+2


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 andsuppose moreover that Aη/K (η) has big monodromy. Let Hη be theimage of ρAη .

LetBK (N) = u ∈ U(K ) : h(u) ≤ N.

Then a random fiber has maximal monodromy, i.e. (if K 6= Q)

limN→∞

|u ∈ BK (N) : ρAu(GK ) = Hη||BK (N)|

= 1.


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).


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 baseU ⊂ PN

Q with monodromy GSp2g (Z)


Monodromy of families of Pryms

Question

For every g , does there exists a family A → U of PP abelian varieties ofdimension g , U rational, which are not generically isogenous toJacobians, with monodromy GSp2g (Z)?

1 One can (probably) take A → U to be a family of Prym varietiesassociated to tetragonal curves, or

2 (Tsimerman) one can take A → U to be a family of Prym varietiesassociated to bielliptic curves.


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 ` > 2)

2 Mg−2 ⊂Mg

3 the mod 2 monodromy thus contains subgroups isomorphic to1 S2g+2

2 Sp2(g−2)+2(Z/2Z)


(Example) Hyperelliptic

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

over U ⊂ A2g+2

H =A ∈ GSp2g (Z) : A (mod 2) ∈ S2g+2


Hyperelliptic example continued

Theorem1 (Yu) unpublished

2 (Achter, Pries) the stack of hyperelliptic curves has maximalmonodromy

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


Hyperelliptic example proof

Corollary

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

has monodromyA ∈ 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/AutP1]

4 fibers are irreducible, thus

π1,et(U) π1,et(Hg ,2)

is surjective.


Sketch of trigonal proof

Theorem (ZB, Zywina)

For every g > 2 there is a family of trigonal curves over a nonemptyrational 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 aconnected algebraic group.

4 Maroni-invariant (normal form for trigonal curves).


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 polynomialsof bi-degree (3, d)

5 [U/G ] ∼= T 0g ⊂ Tg .


Pryms

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


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


Families of Abelian Varieties with Big Monodromydzb/slides/ZBrownAmsColoradofamilies.pdf · David...

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