Families of Abelian Varieties with Big Monodromy David Zureick-Brown - PowerPoint PPT Presentation

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

Background - Galois Representations ∼ ρ A , n : G K → Aut A [ n ] = GL 2 g ( Z / n Z ) GL 2 g ( Z /ℓ n Z ) ρ A ,ℓ ∞ : G K → GL 2 g ( Z ℓ ) = lim ← − n ρ A : G K → GL 2 g ( � Z ) = lim GL 2 g ( Z / n Z ) ← − n David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 2 / 22

Background - Galois Representations ρ A , n : G K ։ G n ֒ → GSp 2 g ( Z / n Z ) G n ∼ = 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:    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 (Emory) Families with Big Monodromy April 14, 2013 4 / 22

Monodromy of a family 1 U ⊂ P N K (non-empty open) 2 η ∈ U (generic point) 3 A → U (family of principally polarized abelian varieties) 4 ρ A η : G K ( U ) → GSp 2 g ( � Z ) Definition The monodromy of A → U is the image H η of ρ A η . We say that A → U has big monodromy if H η is an open subgroup of GSp 2 g ( � Z ). 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 ρ A . We say that A → U has big monodromy if H is an open subgroup of GSp 2 g ( � Z ). 1 Spec Ω η → U (geometric generic point) − 2 π 1 , et ( U ) 1 A → U (family of principally polarized abelian varieties) 2 ρ A : π 1 , et ( U ) → GSp 2 g ( � Z ) David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 6 / 22

(Example) standard family of elliptic curves E : y 2 = x 3 + ax + b U = A 2 K − ∆ � � A ∈ GL 2 ( � H = Z ) : det( A ) ∈ χ K (Gal( K / K )) David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 7 / 22

(Example) elliptic curves with full two torsion E : y 2 = x ( x − a )( x − b ) U = A 2 Q − ∆ � � A ∈ GL 2 ( � H = Z ) : A ≡ I (mod 2) David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 8 / 22

Exotic example from Zywina’s HIT paper 36 1 E : y 2 + xy = x 3 − j − 1728 x − j − 1728 . David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 9 / 22

Exotic example from Zywina’s HIT paper 36 1 E : y 2 + xy = x 3 − j − 1728 over U ⊂ A 1 j − 1728 x − K j = ( T 16 + 256 T 8 + 4096) 3 T 32 ( T 8 + 16) [GL 2 ( � Z ) : H ] = 1536 . David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 9 / 22

Exotic example from Zywina’s HIT paper 36 1 E : y 2 + xy = x 3 − j − 1728 over U ⊂ A 1 j − 1728 x − K j = ( T 16 + 256 T 8 + 4096) 3 T 32 ( T 8 + 16) [GL 2 ( � Z ) : H ] = 1536 H is the subgroup of matricies preserving h ( z ) = η ( z ) 4 /η (4 z ). David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 9 / 22

(Example) Hyperelliptic E : y 2 = x 2 g +2 + a 2 g +1 x 2 g +1 + . . . + a 0 over U ⊂ A 2 g +2 � � A ∈ GSp 2 g ( � H = Z ) : A (mod 2) ∈ S 2 g +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 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 . Let B K ( N ) = { u ∈ U ( K ) : h ( u ) ≤ N } . Then a random fiber has maximal monodromy , i.e. (if K � = Q ) |{ u ∈ B K ( N ) : ρ A u ( G K ) = H η }| lim = 1 . | B K ( N ) | N →∞ 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 ( A tors ) / Q ) ∼ = GSp 2 g ( � Z ) , i.e, for every n, Gal( Q ( A [ n ]) / Q ) ∼ = GSp 2 g ( Z / n Z ) . 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 T g of trigonal curves has monodromy GSp 2 g ( � Z ) , and 2 there is a family of trigonal curves over a nonempty rational base Q with monodromy GSp 2 g ( � U ⊂ P N Z ) 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 GSp 2 g ( � Z )? 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 T g of trigonal curves has monodromy GSp 2 g ( � Z ) . Proof. 1 M g , d − 1 ⊂ M g , d (suffices for ℓ > 2) 2 M g − 2 ⊂ M g 3 the mod 2 monodromy thus contains subgroups isomorphic to S 2 g +2 1 Sp 2( g − 2)+2 ( Z / 2 Z ) 2 David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 15 / 22

(Example) Hyperelliptic E : y 2 = x 2 g +2 + a 2 g +1 x 2 g +1 + . . . + a 0 over U ⊂ A 2 g +2 � � A ∈ GSp 2 g ( � H = Z ) : A (mod 2) ∈ S 2 g +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 y 2 = ( 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 : y 2 = x 2 g +2 + a 2 g +1 x 2 g +1 + . . . + a 0 � � A ∈ GSp 2 g ( � has monodromy Z ) : A (mod 2) ∈ S 2 g +2 . Proof. 1 U = space of distinct unordered 2 g + 2-tuples of points on P 1 2 U ։ H g , 2 3 H g , 2 ∼ = [ U / Aut P 1 ] 4 fibers are irreducible, thus π 1 , et ( U ) ։ π 1 , et ( H g , 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 Q with monodromy GSp 2 g ( � rational base U ⊂ P N Z ) Proof. 1 Main issue : f 3 ( x ) y 3 + f 2 ( x ) y 2 + f 1 ( x ) y + f 0 ( x ) = 0 2 The stack T g is unirational, need to make this explicit 3 (Bolognesi, Vistoli) T g ∼ = [ 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 → P g − 1 C ֒ → F n ֒ F n ∼ = P ( O ⊕ O ( − n )) = P 1 × P 1 F 0 ∼ F 1 ∼ = Bl P P 2 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 0 g ⊂ T g . David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 20 / 22

Pryms C → D � ker 0 ( J C → J D ), 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 X 1 (2). The associated family of Prym’s has big monodromy. David Zureick-Brown (Emory) Families with Big Monodromy April 14, 2013 22 / 22