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

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

  • 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