Explicit moduli spaces of abelian varieties with automorphisms PDF
Preview Explicit moduli spaces of abelian varieties with automorphisms
Explicit moduli spacesof abelian varietieswith automor- phisms Explicit moduli spaces of abelian Bertvan varieties with automorphisms Geemen (jointwork withMatthias Schu¨tt) Bert van Geemen (joint work with Matthias Schu¨tt) Trento,September2010 Outline Explicit moduli spacesof abelian varietieswith automor- phisms 1 Introduction Bertvan Geemen (jointwork withMatthias Schu¨tt) 2 The moduli spaces of ppav’s Outline 3 The Shimura varieties The Shimura curve The Shimura surface Introduction Explicit Example of a Shimura variety: moduli spacesof a moduli space of ppav’s with an automorphism, abelian varietieswith i.e. of triples (X,L,φ): automor- ∼ phisms X a complex torus (X = V/Γ), Bertvan L ample line bundle on X, which gives a principal Geemen (jointwork polarization (equiv: h0(L) = 1), withMatthias Schu¨tt) φ is an automorphism of (X,L): Introduction φ : X −∼=→ X, φ(0) = 0, φ∗L ∼ L. Themoduli spacesof ppav’s A : Moduli space of ppav’s with level structure ∗ TheShimura g,∗ vTahreiSehtiimeusracurve (for example, ∗ = level n: α : A[n] −∼=→ (Z/nZ)2g) TheShimura surface which is a Galois cover with group G of A : g A −→ A = A /G. g,∗ g g,∗ Shimura variety as fixed point set Explicit moduli spacesof abelian varietieswith Given (X,L,φ) and a point [(X,L,α)] ∈ A then automor- g,∗ phisms define φ∗[(X,L,α)] = [(X,L,α◦φ)], Bertvan Geemen (jointwork you get φ∗ ∈ G, (more precisely: α◦φ∗◦α−1 ∈ G) withMatthias Schu¨tt) [(X,L,α)] = [(X,L,α◦φ)] (isomorphic objects), so Introduction [(X,L,α)] is a fixed point for φ∗ ∈ G in A g,∗ Themoduli spacesof ppav’s Hence: moduli space of triples (X,L,φ), TheShimura with level structure ∗, varieties TheShimuracurve is the fixed point locus (Ag,∗)φ∗, a Shimura variety. TheShimura surface G-equivariant map Explicit moduli spacesof abelian varietieswith Given a G-equivariant embedding automor- phisms Bertvan Θ : Ag,∗ −→ PN, Θ◦g = Mg ◦Θ, Geemen (jointwork withMatthias for g ∈ G, M ∈ Aut(PN), Schu¨tt) g the image of the moduli space of triples (X,L,φ) Introduction with level structure ∗ is Themoduli spacesof ppav’s Θ((A )φ∗) = Θ(A ) ∩ P TheShimura g,∗ g,∗ λ varieties TheShimuracurve where P is an eigenspace of M . TheShimura λ g surface Outline Explicit moduli spacesof abelian To do: varietieswith automor- phisms specify the triples (A,L,φ), Bertvan Geemen specify level structure ∗, (jointwork withMatthias find G-equivariant map Θ : A −→ PN, Schu¨tt) g,∗ determine M ∈ Aut(PN) and its eigenspaces P , Introduction φ∗ λ Themoduli find equations for Θ(A ), spacesof g,∗ ppav’s study the intersection Θ(A )∩P . TheShimura g,∗ λ varieties TheShimuracurve Applications to Arithmetic and Geometry of Shimura TheShimura surface varieties The Abelian varieties Explicit moduli spacesof (B0,L0) := Jac(C), C : y2 = x5+1 abelian varietieswith automor- C is a genus 2 curve, B0 is a ppav with automorphism phisms Bertvan φ : B −→ B , φ = φ∗, φ (x,y) = (ζx,y) Geemen 0 0 C C (jointwork withMatthias where ζ is a primitve 5-th root of unity ((B ,L ,φ) is unique). Schu¨tt) 0 0 (B ,L ,φ) is rigid. Consider the 4 dim ppav with 0 0 Introduction automorphism Themoduli spacesof ppav’s (A ,L,φ ) := (B ×B ,L (cid:2)L ,φ×φk). 0 k 0 0 0 0 TheShimura varieties TheShimuracurve Deformation space has dimension: TheShimura surface (cid:26) 1 k = 2,3, dim (Deformations (A ,L,φ )) = 0 k 2 k = 4. The level structure ∗=(2,4) Explicit Symmetric theta structure of level two, (2,4). moduli spacesof abelian varietieswith A −→ A −→ A −→ A . automor- g,4 g,(2,4) g,2 g phisms (cid:124) (cid:123)(cid:122) (cid:125) group(Z/2Z)2g Bertvan Geemen (cid:124) (cid:123)(cid:122) (cid:125) (jointwork groupG withMatthias Schu¨tt) There is a non-split exact sequence: Introduction Themoduli 0 −→ (Z/2Z)2g −→ G −→ Sp(2g,F ) −→ 0. spacesof 2 ppav’s TheShimura Sp(2g,F ) is generated by transvections: for v ∈ F2g varieties 2 2 TheShimuracurve TheShimura surface t : F2g −→ F2g, w (cid:55)−→ w +E(w,v)v, v 2 2 E : F2g ×F2g → F = Z/2Z is the symplectic form. 2 2 2 The G-equivariant map Θ : A −→ PN g,(2,4) Explicit moduli The theta constants provide a natural G-equivariant map spacesof abelian varietieswith Θ : A −→ PN, N +1 = 2g. automor- g,(2,4) phisms Bertvan Over C, the map Θ is induced by the map Geemen (jointwork withMatthias Schu¨tt) H −→ PN, τ (cid:55)−→ (... : Θ[σ](τ) : ...) g σ∈(Z/2Z)g Introduction with theta constants Themoduli spacesof ppav’s Θ[σ](τ) = (cid:88) e2πit(m+σ/2)τ(m+σ/2). TheShimura varieties m∈Zg TheShimuracurve TheShimura surface Θ(A ) is birationally isomorphic with A . g,(2,4) g,(2,4) For g = 2: Θ(A ) = P3 − {30 lines}. 2,(2,4) Determine M ∈ Aut(PN) g Explicit moduli Can easily find M ∈ Aut(PN) for g ∈ (Z/2Z)2g ⊂ G. spacesof g abelian (”Heisenberg group action”) varietieswith automor- phisms For any transvection t ∈ Sp(2g,F ) can find M ∈ Aut(PN) Bertvan v 2 tv (jGoienetmweonrk (Mtv is a linear combination of I and Mv). withMatthias Schu¨tt) Hence can find M for any g ∈ G. g Introduction In case g = 2, Sp(2g,F ) ∼= S (symmetric group). Themoduli 2 6 spacesof Transvections correspond to transpositions. ppav’s TheShimura Can easily find element of order five h ∈ G and varieties TheShimuracurve corresponding Mh ∈ Aut(P3). TheShimura surface M has four fixed points in P3−{30 lines} = Θ(A ), h 2,(2,4) By unicity, each fixed point is a [(B ,L ,α)] and h = φ∗. 0 0
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