Prime Numbers

7.7 Exercises 375
7.4. As in Exercise 7.3 the nonsingularity condition for affine curves is that
the discriminant 4a 3 +27b 2 be nonzero in the field Fp. Show that for the
parameterization
Y 2 = X 3 + CX 2 + AX + B
and characteristic p > 3 the nonsingularity condition is different on a
discriminant ∆, namely
∆=4(A − C 2 /3) 3 + 27(B − AC/3+2C 3 /27) 2 =0.
Then show that in the computationally useful Montgomery parameterization
is nonsingular if and only if C 2 =4.
Y 2 = X 3 + CX 2 + X
7.5. For an elliptic curve over Fp, p>3, with cubic
we define the j-invariant of E as
Y 2 = X 3 + CX 2 + AX + B
j(E) = 4096 (C2 − 3A) 3
,
∆
where the discriminant ∆ is given in Exercise 7.4. Carry out the following
computational exercise. By choosing a conveniently small prime that allows
hand computation or easy machine work (you might assess curve orders via the
direct formula (7.8)), create a table of curve orders vs. j-invariants. Based on
such empirical evidence, state an apparent connection between curve orders
and j-invariant values. For an excellent overview of the beautiful theory of
j-invariants and curve isomorphisms see [Seroussi et al. 1999] and numerous
references therein, especially [Silverman 1986].
7.6. Here we investigate just a little of the beautiful classical theory of
elliptic integrals and functions, with a view to the connections of same
to the modern theory of elliptic curves. Good introductory references are
[Namba 1984], [Silverman 1986], [Kaliski 1988]. One essential connection is
the observation of Weierstrass that the elliptic integral
∞
ds
Z(x) =
x 4s3 − g2s − g3
can be considered as a solution to an implicit relation
℘g2,g3(Z) =x,
where ℘ is the Weierstrass function. Derive, then, the differential equations
℘(z1 + z2) = 1
4
′ ℘ (z1) − ℘ ′ 2 (z2)
− ℘(z1) − ℘(z2)
℘(z1) − ℘(z2)