A positive integer n such that the Hecke operators
T1, ..., Tn generate the Hecke algebra as a Z-module. When
the character is trivial, the default bound is (k/12).[(SL)2(Z):Γ0(N)]. That this suffices follows from
[Stu87], as is explained in [AS02].
When the character of the space of modular symbols M is nontrivial, the
default bound is twice the above bound; however, it is not known
that this bound is large enough in all cases in which the character is
nontrivial, so one may wish to increase the bound using SetHeckeBound.
Many computations require a bound n such that T1, ..., Tn
generate the Hecke algebra associated to the space of modular symbols M
as a Z-module. This command allows you
to set the bound that is used internally. Setting it too low can
result in functions quickly producing incorrect results.
The Hecke algebra associated to the space of modular symbols M. This is an algebra
TQ over Q, such that Generators(TQ) is a set that
generates the ring Z[T1, T2, T3, ... ], as a Z-module.
If the optional integer parameter Bound is set, then HeckeAlgebra
only computes the algebra generated by those Tn,
with n≤ Bound.
The discriminant of the Hecke algebra associated to the space of modular symbols M.
If the optional parameter Bound is set, then the discriminant
of the algebra generated by only those Tn, with n≤ Bound,
is computed instead.
Bound: RngIntElt Default: -1
The order generated by the Fourier coefficients of one of the
q-expansions of a newform corresponding to the space of modular symbols M, along with a map
from the ring containing the coefficients of qExpansion(A) to
the order. If the optional parameter Bound is set,
then the order generated only by those an, with n ≤
Bound, is computed.
The number field generated by the Fourier coefficients of one
of the q-expansions of a newform corresponding to the space of modular symbols M, along with
a map from the ring containing the coefficients of qExpansion(M)
to the number field. We require that M be defined over Q.
In this example, we compute the discriminant of the Hecke algebra of
prime level 389.
> M := ModularSymbols(389,2,+1);
> C := CuspidalSubspace(M);
> DiscriminantOfHeckeAlgebra(C);
62967005472006188288017473632139259549820493155023510831104000000
> Factorization($1);
[ <2, 53>, <3, 4>, <5, 6>, <31, 2>, <37, 1>, <389, 1>, <3881, 1>,
<215517113148241, 1>, <477439237737571441, 1> ]
The prime 389 is the only prime p<10000 such that p divides the
discriminant of the Hecke algebra associated to S
2(Γ
0(p)).
It is an open problem to decide whether or not there are any other
such primes. Are there infinitely many?
V2.28, 13 July 2023