Equations for Hilbert modular surfaces

82 Pages·2013·0.19 MB·English

by  Abhinav Kumar

Equations for Hilbert modular surfaces
Abhinav Kumar
MIT
April 24, 2013
Introduction
Outline of talk
◮ Elliptic curves, moduli spaces, abelian varieties
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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
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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
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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).
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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
\
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