12 CORBETT REDDEN
Here, logJ (x) is an example of an elliptic integral, and it naturally arises in physical
problems such as modeling the motion of a pendulum. Expanding logJ as a power
series in x produces a formal group law with a nice geometric description. Inverting
the function logJ (x) gives
f(z) := expJ (z) = (logJ
)−1(z),
which is an elliptic function (i.e. periodic with respect to a lattice Λ C) satisfying
the differential equation (f
)2(z)
= R(z). Hence, it parameterizes the elliptic curve
C defined by the Jacobi quartic equation
y2
= R(x) = 1
2δx2
+
x4

CP2
via the map
C/Λ −→
CP2
z −→ [x(z),y(z), 1] = [f(z),f (z), 1].
The additive group structure on the torus C/Λ induces a natural group struc-
ture on the elliptic curve C. This group structure coincides with the one given in
Chapter 2, defined by P + Q + R = 0 for points P, Q, R on a straight line. Near
the point [0, 1, 1], the group structure is given in the parameter x by
FJ (x1,x2) := f(f
−1(x1)
+ f
−1(x2))
= expJ (logJ (x1) + logJ (x2)).
The formal group law FJ defined by the logarithm logJ can therefore be expressed
by
x1
0
dt
R(t)
+
x2
0
dt
R(t)
=
FJ (x1,x2)
0
dt
R(t)
.
Despite the integral logJ having no closed form solution, the formal group law was
solved for explicitly by Euler.
Theorem 3.1. (Euler)
FJ (x1,x2) =
x1 R(x2) + x2 R(x1)
1 x1x222
.
While we previously worked over the field C, the Jacobi quartic is defined over
an arbitrary ring, and the universal curve is defined by the same equation over the
ring Z[δ, ]. The formal group law FJ can be expanded as a power series in the ring
Z[
1
2
, δ, ]. Any genus whose logarithm is of the form (3) is called an elliptic genus,
and the universal elliptic genus ϕJ corresponds to Euler’s formal group law FJ over
Z[
1
2
, δ, ]. When considering the grading, |δ| = −4 and | | = −8, so ϕJ also defines
an oriented genus. In fact, one can calculate that
ϕJ
(CP2)
= δ, ϕJ
(HP2)
= .
Example 3.2. The geometric description of FJ assumed Δ = (δ2 )2 = 0
so that the curve C has no singularities. However, the degenerate case δ = = 1
gives the L-genus, which equals the signature of an oriented manifold:
log(x) =
x
0
dt
1 t2
=
tanh−1(x),
Q(x) =
x
tanh x
.
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