general elliptic curves have an automorphism group of order two, and there are two
special curves with automorphism groups of orders four and six. That picture needs
to be modified when dealing with small primes. At the prime p = 3 (respectively
p = 2), there is only one special ‘orbifold point’, and the automorphism group of
the corresponding elliptic curve has order 12 (respectively 24). The automorphism
group of that curve is given by Z/4 Z/3 at the prime 3, and by Z/3 Q8 (also
known as the binary tetrahedral group) at the prime 2.
In characteristic p, there is an important dichotomy between ordinary and
supersingular elliptic curves. An elliptic curve C is ordinary if its group of p-
torsion points has p connected components, and supersingular if the group of p-
torsion points is connected. This dichotomy is also reflected in the structure of the
multiplication-by-p map, which is purely inseparable in the supersingular case, and
the composite of an inseparable map with a degree p covering in the case of an
ordinary elliptic curve. The supersingular elliptic curves form a zero-dimensional
substack of (Mell )Fp —the stack of elliptic curves in characteristic p—whose car-
dinality grows roughly linearly in p. If one counts supersingular curves with a
multiplicity equal to the inverse of the order of their automorphism group, then
there are exactly (p 1)/24 of them.
The stratification of (Mell )Fp into ordinary and supersingular elliptic curves is
intimately connected to the stratification of the moduli stack of formal groups by
the height of the formal group. A formal group has height n if the first non-zero
coefficient of the multiplication-by-p map is that of
. The ordinary elliptic curves
are the ones whose associated formal group has height 1, and the supersingular
elliptic curves are the ones whose associated formal group has height 2. Higher
heights cannot occur among elliptic curves.
Chapter 4: The Landweber exact functor theorem. The next main re-
sult is that there this a presheaf Ell of homology theories on the (affines of the)
flat site of the moduli stack of elliptic curves—the category whose objects are flat
maps Spec(R) Mell . That presheaf is defined as follows. Given an elliptic curve
C over a ring R, classified by a flat map Spec(R) Mell , the associated formal
group C corresponds to a map MP0 R, where MP∗ =
MU∗+2n is the
periodic version of complex cobordism. EllC is then defined by
:= MP∗(X)
We claim that for every elliptic curve C whose classifying map Spec(R) Mell
is flat, the functor
is a homology theory, i.e., satisfies the exactness axiom.
An example of an elliptic curve whose classifying map is flat, and which therefore
admits an associated elliptic homology theory, is the universal smooth Weierstrass
The proof of this claim is a combination of several ingredients. The main
one is the Landweber exact functor theorem, which provides an algebraic criterion
(Landweber exactness, which is weaker than flatness) on a ring map MP0 R,
that ensures the functor X MP∗(X) ⊗MP0 R satisfies exactness. The other
ingredients, due to Hopkins and Miller, relate the geometry of Mell and MFG to
the Landweber exactness criterion. These results are the following: (1) A formal
group law MP0 R over R is Landweber exact if and only if the corresponding
map Spec(R) MFG is flat; together with Landweber exactness, this gives a
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