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This chapter is about how to use Magma to compute with the Hecke
module D(N, p) of divisors on the supersingular points on X0(N) in
characteristic p.
Let p be a prime. A divisor on the supersingular points of X0(1)
in characteristic p is a finite formal linear combination of
supersingular j-invariants j∈(F)p2. More generally,
suppose N is an integer that is not divisible by p. The module
D(N, p) of divisors on the supersingular points of X0(N) in
characteristic p is the free abelian group generated by
isomorphism classes of pairs (E, C) where E is a supersingular
elliptic curve over (F)p2
and C is a cyclic subgroup of E of order N.
(We call such a pair (E, C) an elliptic curve enhanced with level N
structure.)
The abelian group D(N, p) of divisors is equipped in a natural
way with an action of Hecke operators Tn.
The module of supersingular points is a special case of the Brandt
module construction (see Chapter BRANDT MODULES) because of
the following equivalence between objects:
 - pairs (E, C) as above,
- isomorphism classes of left ideals of an Eichler order of
level N
in the quaternion algebra over Q ramified at, and
p and ∞,
- supersingular points on X0(N)/Fp over /line(F)p.
The supersingular points of X0(N)/(F)p correspond to
singularities of the special fiber of a minimal model of X0(Np)
at p. This special fiber, X0(Np)/Fp, is isomorphic to two
copies of X0(N)/Fp joined at the supersingular points as simple
double points.
The structure of the multiplicative part of the Jacobian
J0(Np) at p is captured by the behavior of the supersingular
points of X0(N)/Fp (see Grothendieck [Gro72] and
Deligne-Rapoport [DR73]). The system of Hecke
operators on the supersingular divisor group gives a representation
of the p-new subspace of modular forms of level Np.
We therefore refer to the level of the supersingular module
as Np, and use the term auxiliary level to refer to N.
There are many reasons why one might be interested in computing with
D(N, p). Foremost, D(N, p) is isomorphic as a module over the Hecke
algebra to a subspace of the modular forms for Γ0(N) of weight
2 and level Np (more precisely, D(N, p) tensor (C) is
isomorphic to the p-new subspace of M2(Γ0(Np))). If
N=1, 2, 3, 5, 7, 13, then Magma computes D(N, p) using the
highly-efficient method of graphs, which for small q quickly produce
very sparse matrices that represent Hecke operators Tq. There are
also formulas of Gross, Kudla, Merel, and others that involve the
explicit representation of an eigenform in terms of a basis of
supersingular j-invariants, and Stein has a formula, which involves
D(N, p), for orders of component groups of modular abelian varieties
(see the ComponentGroupOrder command in the modular symbols
package).
Magma computes the module of supersingular divisors using either the
method of graphs of Mestre--Oesterlé when N=1, 2, 3, 5, 7, 13 or
Brandt modules in general. When it is applicable, the method of graphs
is much faster (in Magma) than Brandt modules, but the Brandt modules
method works in general.
The modular curve approach computes correspondences on the modular
curves X0(N) by means of pre-computed models for the system of
covering maps X0(N ell) -> X0(N).
Such correspondences give rise to the Hecke operators Tell
as the adjacency matrices of the graphs of ell-isogenies of the
basis of D(N, p).
For the alternative approach through Brandt modules, the reader should
consult Chapter BRANDT MODULES and the articles of Pizer [Piz80]
and Kohel [Koh01].
This package still has some unnecessary limitations.
When using Brandt modules to compute the module of supersingular
divisor in Magma, no facility is currently provided for describing the
divisors in terms of supersingular elliptic curves. Also, it is
currently only possible to work with the module of supersingular
divisors over the integers.
Modules of supersingular points belong to the category ModSS, and
the elements of these modules belong to ModSSElt.
To set the verbosity level use the command
SetVerbose("SupersingularModule",n),
where n is 0 (silent), 1 (verbose), or 2 (very verbose).
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