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

.