Equations for Hilbert modular surfaces Abhinav Kumar MIT April 24, 2013 Introduction Outline of talk ◮ Elliptic curves, moduli spaces, abelian varieties 2/31 Introduction Outline of talk ◮ Elliptic curves, moduli spaces, abelian varieties ◮ Hilbert modular surfaces 2/31 Introduction Outline of talk ◮ Elliptic curves, moduli spaces, abelian varieties ◮ Hilbert modular surfaces ◮ An example: Y−(17). 2/31 Introduction Outline of talk ◮ Elliptic curves, moduli spaces, abelian varieties ◮ Hilbert modular surfaces ◮ An example: Y−(17). ◮ Applications 2/31 Introduction Outline of talk ◮ Elliptic curves, moduli spaces, abelian varieties ◮ Hilbert modular surfaces ◮ An example: Y−(17). ◮ Applications ◮ Method/proof 2/31 Elliptic curves An elliptic curve over C is the set of solutions to an equation y2 = x3+Ax +B with A,B C with ∆ = 4A3 27B2 = 0. ∈ − − 6 3/31 Elliptic curves An elliptic curve over C is the set of solutions to an equation y2 = x3+Ax +B with A,B C with ∆ = 4A3 27B2 = 0. ∈ − − 6 Geometrically, it’s a complex torus C/Λ where Λ = Zω +Zω is a 1 2 lattice in C. The periods ω and ω can be computed as elliptic integrals. 1 2 3/31 Uniformization and moduli space The map C/Λ E is given by z (x,y) = (℘(z),℘′(z)), where → 7→ ℘ is the Weierstrass function (depends on Λ). 4/31 Uniformization and moduli space The map C/Λ E is given by z (x,y) = (℘(z),℘′(z)), where → 7→ ℘ is the Weierstrass function (depends on Λ). So the set of elliptic curves/C is naturally the set of lattices in C, up to scaling and isometries. Parametrized by PSL (Z) H: write Λ = Z+Zτ. 2 \ 4/31