Moduli spaces of genus 2 curves with split Jacobians through K3 surfaces

88 Pages·2015·1.19 MB·English

by  Abhinav Kumar

Moduli spaces of genus 2 curves with split
Jacobians through K3 surfaces
Abhinav Kumar
StonyBrook
October 22, 2015
AbhinavKumar Genus2curveswithsplitJacobians 1/33
Introduction
Let C be a genus 2 curve over a field k, and J = Jac(C) its
Jacobian.
Recall that we say J is simple if it does not contain a proper
abelian subvariety. Otherwise we say that J is reducible or
decomposable or split.
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Introduction
Let C be a genus 2 curve over a field k, and J = Jac(C) its
Jacobian.
Recall that we say J is simple if it does not contain a proper
abelian subvariety. Otherwise we say that J is reducible or
decomposable or split.
The only possibility for a genus 2 curve is that J is isogenous to a
′
product E E of elliptic curves. Equivalently, there is a degree n
×
map C E to some elliptic curve, for some natural number n.
→
AbhinavKumar Genus2curveswithsplitJacobians 2/33
Introduction
Let C be a genus 2 curve over a field k, and J = Jac(C) its
Jacobian.
Recall that we say J is simple if it does not contain a proper
abelian subvariety. Otherwise we say that J is reducible or
decomposable or split.
The only possibility for a genus 2 curve is that J is isogenous to a
′
product E E of elliptic curves. Equivalently, there is a degree n
×
map C E to some elliptic curve, for some natural number n.
→
Goal: give explicit examples of genus 2 curves with split Jacobians,
and explicit descriptions of moduli spaces of such (C,E).
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Examples
We give a few examples of split (over Q) Jacobians.
Example
Classical example: X (37) is a genus 2 curve with equation
0
y2 = x6+8x5 20x4+28x3 24x2+12x 4
− − −
Its Jacobian J (37) is (2,2)-isogenous to a product of elliptic
0
curves with j-invariants 37/542 and 372/543.
−
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Examples II
Example
Let C be the curve
y2 = (x3+420x 5600)(x3+42x2+1120)
−
Then J(C) is (3,3)-isogenous to a product of elliptic curves with
j-invariants 27 72 and 25 7 173.
− · − · ·
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Examples II
Example
Let C be the curve
y2 = (x3+420x 5600)(x3+42x2+1120)
−
Then J(C) is (3,3)-isogenous to a product of elliptic curves with
j-invariants 27 72 and 25 7 173.
− · − · ·
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Examples II
Example
Let C be the curve
y2 = (x3+420x 5600)(x3+42x2+1120)
−
Then J(C) is (3,3)-isogenous to a product of elliptic curves with
j-invariants 27 72 and 25 7 173. A morphism of degree 3 to
− · − · ·
the elliptic curve
y2 = x3+4900x2+7031500x [phone]
1 1 1 1
is given by
882000(x 14) 49000(x3 21x2 140)y
x = − − , y = − − .
1 x3+420x 5600 1 (x3+420x 5600)2
− −
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Examples III
Example
The genus two curve defined by
y2 = (x3 10x2 816x 204)(x3+4x2 893x 13861)
− − − − −
has Jacobian (7,7)-isogenous to a product of elliptic curves with
j-invariants 21932/55 and 2832721393/52.
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Examples III
Example
The genus two curve defined by
y2 = (x3 10x2 816x 204)(x3+4x2 893x 13861)
− − − − −
has Jacobian (7,7)-isogenous to a product of elliptic curves with
j-invariants 21932/55 and 2832721393/52.
Example
The genus two curve defined by
y2 = (x3+496x2+52302x 2673552)(x3+584x 271740x+24634652)
− −
has Jacobian (11,11)-isogenous to a product of elliptic curves with
j-invariants 212293 and 21231317093153913.
365243 355343210911
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