Introduction

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.

Contents

Categories

Modules of supersingular points belong to the category ModSS, and the elements of these modules belong to ModSSElt.

Verbose Output

To set the verbosity level use the command SetVerbose("SupersingularModule",n), where n is 0 (silent), 1 (verbose), or 2 (very verbose).

V2.28, 13 July 2023