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 ρ E , n : G K → Aut E [ n ] ∼ = GL 2 ( Z / n Z ) GL 2 ( Z /ℓ n Z ) ρ E ,ℓ ∞ : G K → GL 2 ( Z ℓ ) = lim ← − n ρ E : G K → GL 2 ( � Z ) = lim GL 2 ( Z / n Z ) ← − n David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 2 / 23

Background - Galois Representations → Aut E [ n ] ∼ ρ E , n : G K ։ G n ֒ = GL 2 ( Z / n Z ) G n ∼ = Gal( K ( E [ n ]) / K ) David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 3 / 23

Example - torsion If E has a K -rational torsion point P ∈ E ( K )[ n ] (of exact order n ), then the image is constrained:    1 ∗   G n ⊂  0 ∗ since for σ ∈ G K and Q ∈ E ( K )[ n ] such that E ( K )[ n ] ∼ = � P , Q � , σ ( P ) = P σ ( Q ) = + a σ P b σ Q David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 4 / 23

Example - Isogenies If E has a K -rational, cyclic isogeny φ : E → E ′ with ker φ = � P � , then the image is constrained    ∗ ∗   G n ⊂  0 ∗ since for σ ∈ G K and Q ∈ E ( K )[ n ] such that E ( K )[ n ] ∼ = � P , Q � , σ ( P ) = a σ P σ ( Q ) = + b σ P c σ Q David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 5 / 23

Example - other maximal subgroups Normalizer of a split Cartan: �    �  ∗ 0 0 1     N sp =  ,   0 ∗ − 1 0 G n ⊂ N sp iff there exists an unordered pair { φ 1 , φ 2 } 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. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 6 / 23

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 ∗    . 0 ∗ David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 7 / 23

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    ∗ ∗    . 0 ∗ 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 �    �  ∗ 0 0 1      , .   0 ∗ − 1 0 David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 8 / 23

Main conjecture Conjecture Let E be an elliptic curve over Q without CM. Then for ℓ > 37, ρ E ,ℓ is surjective. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 9 / 23

Serre’s Open Image Theorem Theorem (Serre, 1972) Let E be an elliptic curve over K without CM. The image of ρ E ρ E ( G K ) ⊂ GL 2 ( � Z ) is open. Note: � GL 2 ( � Z ) ∼ = GL 2 ( Z p ) p David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 10 / 23

Sample Consequences of Serre’s Theorem Surjectivity For large ℓ , ρ E ,ℓ is surjective. Lang-Trotter Density of supersingular primes is 0. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 11 / 23

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 ( ζ n ) � � � ���������� � � � � � Q ( √ ∆ E ) Q David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 12 / 23

A surjective example Theorem (Greicius 2008) Let α be a real root of x 3 + x + 1 and let E be the elliptic curve y 2 + 2 xy + α y = x 3 − x 2 . Then ρ E is surjective. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 13 / 23

Random curves 1 (Duke) For a random E / Q , ρ E ,ℓ is surjective for all ℓ . 2 (Jones) For a random E / Q , [GL 2 ( � Z ) : ρ E ( G Q )] = 2. 3 (Cojocaru and Hall) Variants over Q . 4 (Zywina) For a random E / K , ρ E ( G K ) 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 ) : y 2 = x 3 + ax + b . Theorem (Zywina, 2008) Let K be a number field such that 1 K ∩ Q cyc = Q , 2 K � = Q . Let B K ( x ) = { ( a , b ) ∈ O 2 K : ∆ a , b � = 0 , | | ( a , b ) | | ≤ x } . Then |{ ( a , b ) ∈ B K ( x ) : ρ E ( a , b ) ( G K ) = GL 2 ( � Z ) }| lim = 1 . | B K ( x ) | x →∞ David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 15 / 23

� � � � � � � Other families E : y 2 = x ( x − a )( x − b ) � � E ( a , b ) E η E A 4 � � � � � � � � � � η = Spec K ( a , b ) A 2 ∋ ( a , b ) 1 Define H η = { M ∈ GL 2 ( � Z ) | M ≡ I mod 2 } 2 ρ E ( a , b ) ( G K ) ⊂ H η 3 ρ E η ( G K ( a , b ) ) = H η 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 P N K and let E → U be a family of elliptic curves. Let η be the generic point of U and let H η = ρ E η ( G K ( η ) ) . Then a random fiber has maximal image of Galois; i.e., |{ u ∈ B K ( N ) : ρ E u ( G K ) = H η }| lim = 1 | B K ( N ) | N →∞ where B K ( N ) = { u ∈ U ( K ) : ∆ E u � = 0 , | | u | | ≤ N } . David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 17 / 23

Main Theorem Definition We say that a principally polarized abelian variety A over a field K has big monodromy if the image of ρ A is open in GSp 2 g ( � Z ). Theorem (ZB-Zywina) Let U be a non-empty open subset of P N 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 A η / K ( η ) has big monodromy. Let H η be the image of ρ A η . Then a random fiber has maximal monodromy. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 18 / 23

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 | | , ρ E u ( G K ) is irreducible . 3 (Geometric) Tate curve – ρ E u ,ℓ ( G K ) contains a transvection if v p ( j ( E u )) < 0 and ℓ ∤ v p ( j ( E u )) . 4 (Group Theory) ‘irreducible + transvection’ ⇒ surjective. 5 (Arithmetic) For fixed ℓ , for most u , there exists p such that v p ( j ( E u )) < 0 and ℓ ∤ v p ( j ( E u )) . David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 19 / 23

Higher dimensional curve ball 1 Analytic, Transcendental, Geometric, and Group Theory steps all work for g > 1. 2 The condition v p ( j ( E u )) < 0 and ℓ ∤ v p ( j ( E u )) is a statement about variation of the component group of the N´ eron model of E u . 3 The analogue of this statement for a general family of abelian varieties fails, and new ideas are needed. David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 20 / 23

Uniform Semistable Approximation 1 Let ℓ > 4 g . 2 ρ E u ,ℓ ( G K ) =: G ℓ ⊂ GSp 2 g ( F ℓ ). 3 (Nori, ’87) Approximate G ℓ by G ( F ℓ ) 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 ρ E u , l ( G K ) =: G ℓ ⊂ GSp 2 g ( F ℓ ) Definition 1 Define G + ℓ := � unipotent elements of G ℓ � 2 For M ∈ G + ℓ , define φ M : G a , F ℓ → GSp 2 g , F ℓ . 3 G + ℓ := � im φ M : M ∈ G + ℓ � . 4 H ℓ := G m , F ℓ · G + ℓ David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 22 / 23

Main Theorem Theorem (ZB-Zywina) 1 H ℓ is reductive for ℓ > c (log | | ) γ . | u | 2 H ℓ := G ℓ ∩ H ℓ ( F ℓ ) has uniformly bounded index in G ℓ and H ℓ ( F ℓ ) . 3 For fixed ℓ , for most u, H ℓ = GSp 2 g , F ℓ David Zureick-Brown () Abelian Varieties with Big Monodromy January 9, 2013 23 / 23