Families of Abelian Varieties with Big Monodromydzb/slides/ZBrownAmsColoradofamilies.pdf · David...
Transcript of 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
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 )
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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), 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
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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).
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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)
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(Example) standard family of elliptic curves
E : y2 = x3 + ax + b
U = A2K −∆
H =A ∈ GL2(Z) : det(A) ∈ χK (Gal(K/K ))
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(Example) elliptic curves with full two torsion
E : y2 = x(x − a)(x − b)
U = A2Q −∆
H =A ∈ GL2(Z) : A ≡ I (mod 2)
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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)
.
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 − 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)
.
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 − 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).
David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 9 / 22
(Example) Hyperelliptic
E : y2 = x2g+2 + a2g+1x2g+1 + . . .+ a0
over U ⊂ A2g+2
H =A ∈ GSp2g (Z) : A (mod 2) ∈ S2g+2
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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.
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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).
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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)
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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.
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 ` > 2)
2 Mg−2 ⊂Mg
3 the mod 2 monodromy thus contains subgroups isomorphic to1 S2g+2
2 Sp2(g−2)+2(Z/2Z)
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(Example) Hyperelliptic
E : y2 = x2g+2 + a2g+1x2g+1 + . . .+ a0
over U ⊂ A2g+2
H =A ∈ GSp2g (Z) : A (mod 2) ∈ S2g+2
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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
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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.
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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).
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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 .
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Pryms
C → D ker0(JC → JD), generally not a Jacobian
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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.
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