Abelian Varieties with Big Monodromy David Zureick-Brown (Emory - - PowerPoint PPT Presentation

Abelian Varieties with Big Monodromy

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

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

2013 Joint Math Meetings Special Session on Number Theory and Geometry San Diego, CA January 9, 2013

Background - Galois Representations ρE,n : GK → Aut E[n] ∼ = GL2(Z/nZ) ρE,ℓ∞ : GK → GL2(Zℓ) = lim ← −

n

GL2 (Z/ℓnZ) ρE : GK → GL2( Z) = lim ← −

n

GL2 (Z/nZ)

David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 2 / 23

Background - Galois Representations

ρE,n : GK ։ Gn ֒ → Aut E[n] ∼ = GL2(Z/nZ) Gn ∼ = Gal(K (E[n]) /K)

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Example - torsion

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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Example - Isogenies

If E has a K-rational, cyclic isogeny φ: E → E ′ with ker φ = P, then the image is constrained Gn ⊂    ∗ ∗ ∗    since for σ ∈ GK and Q ∈ E(K)[n] such that E(K)[n] ∼ = P, Q, σ(P) = aσP σ(Q) = bσP + cσQ

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Example - other maximal subgroups

Normalizer of a split Cartan: Nsp =    ∗ ∗    ,    1 −1   

• Gn ⊂ Nsp iff there exists an unordered pair {φ1, φ2} of cyclic isogenies,

neither of which is defined over K, but which are both defined over some quadratic extension of K and which are Galois conjugate.

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Classification of Images - Mazur’s Theorem

Theorem

Let E be an elliptic curve over Q. Then for ℓ > 11, E(Q)[ℓ] = {∞}. In other words, for ℓ > 11 the mod ℓ image is not contained in a subgroup conjugate to    1 ∗ ∗    .

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Classification of Images - Mazur; Bilu, Parent

Theorem (Mazur)

Let E be an elliptic curve over Q without CM. Then for ℓ > 37 the mod ℓ image is not contained in a subgroup conjugate to    ∗ ∗ ∗    .

Theorem (Bilu, Parent)

Let E be an elliptic curve over Q without CM. Then for ℓ > 13 the mod ℓ image is not contained in a subgroup conjugate to    ∗ ∗    ,    1 −1   

• .

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Main conjecture

Conjecture

Let E be an elliptic curve over Q without CM. Then for ℓ > 37, ρE,ℓ is surjective.

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Serre’s Open Image Theorem

Theorem (Serre, 1972)

Let E be an elliptic curve over K without CM. The image of ρE ρE(GK) ⊂ GL2( Z) is open.

Note:

GL2( Z) ∼ =

• p

GL2(Zp)

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Sample Consequences of Serre’s Theorem

Surjectivity

For large ℓ, ρE,ℓ is surjective.

Lang-Trotter

Density of supersingular primes is 0.

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Index 2

Fact

Serre also observed that for an elliptic curve over Q, the index is always divisible by 2. Q(E[2])

• Q(E[n])

Q(ζn)

• Q(√∆E)

Q

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A surjective example

Theorem (Greicius 2008)

Let α be a real root of x3 + x + 1 and let E be the elliptic curve y2 + 2xy + αy = x3 − x2. Then ρE is surjective.

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Random curves

1 (Duke) For a random E/Q, ρE,ℓ is surjective for all ℓ. 2 (Jones) For a random E/Q, [GL2(

Z) : ρE(GQ)] = 2.

3 (Cojocaru and Hall) Variants over Q. 4 (Zywina) For a random E/K, ρE(GK) is maximal. 5 (Wallace) Variant for genus 2. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 14 / 23

Zywina’s Theorem - I

Let E(a,b) : y2 = x3 + ax + b.

Theorem (Zywina, 2008)

Let K be a number field such that

1 K ∩ Qcyc = Q, 2 K = Q.

Let BK(x) = {(a, b) ∈ O2

K : ∆a,b = 0, |

|(a, b)| | ≤ x}. Then lim

x→∞

|{(a, b) ∈ BK(x) : ρE(a,b)(GK) = GL2( Z)}| |BK(x)| = 1.

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Other families

E : y2 = x(x − a)(x − b) Eη

• E
• A4
• E(a,b)
• η = Spec K(a, b)

A2 ∋ (a, b)

1 Define Hη = {M ∈ GL2(

Z) | M ≡ I mod 2}

2 ρE(a,b)(GK) ⊂ Hη 3 ρEη(GK(a,b)) = Hη David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 16 / 23

Zywina’s Theorem - II

Theorem (Zywina, 2010)

Let K be a number field, let U be a non-empty open subset of PN

K and let

E → U be a family of elliptic curves. Let η be the generic point of U and let Hη = ρEη(GK(η)). Then a random fiber has maximal image of Galois; i.e., lim

N→∞

|{u ∈ BK(N) : ρEu(GK) = Hη}| |BK(N)| = 1 where BK(N) = {u ∈ U(K) : ∆Eu = 0, | |u| | ≤ N}.

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Main Theorem

Definition

We say that a principally polarized abelian variety A over a field K has big monodromy if the image of ρA is open in GSp2g( Z).

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. Then a random fiber has maximal monodromy.

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Outline of proof - 1 dimensional case

1 (Analytic) Zywina’s refinement of Hilbert’s irreducibility theorem. 2 (Transcendental) Masser-W¨

ustholz: for ℓ ≫ log | |u| | , ρEu(GK) is irreducible.

3 (Geometric) Tate curve – ρEu,ℓ(GK) contains a transvection if

vp(j(Eu)) < 0 and ℓ ∤ vp(j(Eu)).

4 (Group Theory) ‘irreducible + transvection’ ⇒ surjective. 5 (Arithmetic) For fixed ℓ, for most u, there exists p such that

vp(j(Eu)) < 0 and ℓ ∤ vp(j(Eu)).

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Higher dimensional curve ball

1 Analytic, Transcendental, Geometric, and Group Theory steps all

work for g > 1.

2 The condition

vp(j(Eu)) < 0 and ℓ ∤ vp(j(Eu)) is a statement about variation of the component group of the N´ eron model of Eu.

3 The analogue of this statement for a general family of abelian

varieties fails, and new ideas are needed.

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Uniform Semistable Approximation

1 Let ℓ > 4g. 2 ρEu,ℓ(GK) =: Gℓ ⊂ GSp2g(Fℓ). 3 (Nori, ’87) Approximate Gℓ by G(Fℓ) for some reductive group G. 4 Idea – use classification of reductive groups and independence of ℓ. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 21 / 23

Uniform Semistable Approximation

ρEu,l(GK) =: Gℓ ⊂ GSp2g(Fℓ)

Definition

1 Define G +

ℓ := unipotent elements of Gℓ

2 For M ∈ G +

ℓ , define φM : Ga,Fℓ → GSp2g,Fℓ.

3 G+

ℓ := im φM : M ∈ G + ℓ .

4 Hℓ := Gm,Fℓ · G+

ℓ

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Main Theorem

Theorem (ZB-Zywina)

1 Hℓ is reductive for ℓ > c (log |

|u| |)γ.

2 Hℓ := Gℓ ∩ Hℓ(Fℓ) has uniformly bounded index in Gℓ and Hℓ(Fℓ). 3 For fixed ℓ, for most u, Hℓ = GSp2g,Fℓ David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 23 / 23