MA342J: Introduction to Modular Forms
Tutorial 5, April 5
Families of elliptic curves and modular forms
Consider the one-parametric family of cubic curves given by
Et : y 2 = x3 − 2x2 + (1 − t)x
(t ∈ C) .
a) Determine singular values of t for this family, i.e. those t0 for which Et0
is not an elliptic curve
b) Write j-invariant for this family j(Et ) as a rational function of t. Check
that the map t 7→ j(t) is generically 3 : 1.
c) Show that in this family one can find a representative of every isomorphism
class of elliptic curves.
d) Consider the modular function
t(z) = 64
∆(2z)
∆(z) + 64∆(2z)
j(t(z)) =
1
+ 744 + O(q)
q
on Γ0 (2). Check that
e) Show that the point (0, 0) ∈ Et is 2-torsion, which explains appearance of
the group Γ0 (2).
f) Explain that the integrals
Z
√
1− t
Ω1 (t) =
0
Z
dx
p
0
Ω2 (t) =
−∞
x3
− 2x2 + (1 − t)x
,
dx
p
3
x − 2x2 + (1 − t)x
are periods for this family. In fact, Ω2 (t(z)) is a square root of an Eisenstein series of weight 2 on Γ0 (2) and Ω1 (t(z)) is its multiple by a simple
rational function in z.
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