Modular forms modulo p explained
In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms.
Reduction of modular forms modulo 2
Conditions to reduce modulo 2
Modular forms are analytic functions, so they admit a Fourier series. As modular forms also satisfy a certain kind of functional equation with respect to the group action of the modular group, this Fourier series may be expressed in terms of
.So if
is a modular form, then there are coefficients
such that
.To reduce modulo 2, consider the subspace of modular forms with coefficients of the
-series being all integers (since
complex numbers, in general, may not be reduced modulo 2).It is then possible to reduce all coefficients modulo 2, which will give a modular form modulo 2.
Basis for modular forms modulo 2
Modular forms are generated by
and
.
[1] It is then possible to normalize
and
to
and
, having integers coefficients in their
-series.This gives generators for modular forms, which may be reduced modulo 2.Note the Miller basis has some interesting properties:
[2] once reduced modulo 2,
and
are just
; that is, a trivial reduction.To get a non-trivial reduction, one must use the modular discriminant
.Thus, modular forms are seen as polynomials of
,
and
(over the complex
in general, but seen over integers
for reduction), once reduced modulo 2, they become just polynomials of
over
.
The modular discriminant modulo 2
The modular discriminant is defined by an infinite product,
\Delta(q)=q
(1-qn)24=
\tau(n)qn,
where
is the
Ramanujan tau function.Results from Kolberg
[3] and Jean-Pierre Serre
[4] demonstrate that, modulo 2, we have
i.e., the
-series of
modulo 2 consists of
to powers of odd squares.
Hecke operators modulo 2
The action of the Hecke operators is fundamental to understanding the structure of spaces of modular forms. It is therefore justified to try to reduce them modulo 2.
The Hecke operators for a modular form
are defined as follows:
[5] Tnf(z)=n2k-1\sumad-2kf\left(
\right)
with
.
Hecke operators may be defined on the
-series as follows:
[5] if
, then
with
\gamma(z)=\sumaa2k-1c\left(
\right).
Since modular forms were reduced using the
-series, it makes sense to use the
-series definition. The sum simplifies a lot for Hecke operators of primes (i.e. when
is prime): there are only two summands. This is very nice for reduction modulo 2, as the formula simplifies a lot.With more than two summands, there would be many cancellations modulo 2, and the legitimacy of the process would be doubtable. Thus, Hecke operators modulo 2 are usually defined only for primes numbers.
With
a modular form modulo 2 with
-representation
, the Hecke operator
on
is defined by
\overline{Tp}|f(q)=\sumn\gamma(n)qn
where
\gamma(n)=\begin{cases}
c(np)&ifp\nmidn\\
c(np)+c(n/p)&ifp\midn
\end{cases} andpanoddprime.
It is important to note that Hecke operators modulo 2 have the interesting property of being nilpotent.Finding their order of nilpotency is a problem solved by Jean-Pierre Serre and Jean-Louis Nicolas in a paper published in 2012:.[6]
The Hecke algebra modulo 2
The Hecke algebra may also be reduced modulo 2.It is defined to be the algebra generated by Hecke operators modulo 2, over
.
Following Serre and Nicolas's notations,
l{F}=\left\langle\Deltak\midkodd\right\rangle
, i.e.
l{F}=\left\langle\Delta,\Delta3,\Delta5,\Delta7,\Delta9,...\right\rangle
.
[7] Writing
l{F}(n)=\left\langle\Delta,\Delta3,\Delta5,...,\Delta2n-1\right\rangle
so that
, define
as the
-subalgebra of
given by
and
.
That is, if
ak{m}(n)=
⋅
…
\midp1,p2,...,pk\inP,k\geq1\}
is a sub-vector-space of
, we get
.
Finally, define the Hecke algebra
as follows:Since
, one can restrict elements of
to
to obtain an element of
.When considering the map
as the restriction to
, then
is a homomorphism.As
is either identity or zero,
.Therefore, the following chain is obtained:
...\toA(n+1)\toA(n)\toA(n-1)\to...\toA(2)\toA(1)\congF2
.Then, define the Hecke algebra
to be the projective limit of the above
as
.Explicitly, this means
A=\varprojlimnA(n)=\left\lbrace
⋅
…
|p1,p2,...,pk\inP,k\geq0\right\rbrace
.
The main property of the Hecke algebra
is that it is generated by series of
and
.
[7] That is:
A=F2\left[Tp\midp\inP\right]
=F2\left[\left[T3,T5\right]\right]
.
So for any prime
, it is possible to find coefficients
such that
.
Notes and References
- Book: Stein, William . 2007 . Modular Forms, a Computational Approach . Theorem 2.17 . Graduate Studies in Mathematics . 978-0-8218-3960-7.
- Book: Stein, William . 2007 . Modular Forms, a Computational Approach . Lemma 2.20 . Graduate Studies in Mathematics . 978-0-8218-3960-7.
- Kolberg . O. . 1962 . Congruences for Ramanujan's function
. Årbok for Universitetet i Bergen Matematisk-naturvitenskapelig Serie . 11 . 0158873.
- Book: Serre, Jean-Pierre . 1973 . A course in arithmetic . Springer-Verlag, New York-Heidelberg . 96 . 978-1-4684-9884-4 .
- Book: Serre, Jean-Pierre . 1973 . A course in arithmetic . Springer-Verlag, New York-Heidelberg . 100 . 978-1-4684-9884-4 .
- Nicolas . Jean-Louis . Serre . Jean-Pierre . 2012 . Formes modulaires modulo 2: l'ordre de nilpotence des opérateurs de Hecke . Comptes Rendus Mathématique. 350 . 7–8 . 343–348 . 10.1016/j.crma.2012.03.013 . 1204.1036 . 2012arXiv1204.1036N . 117824229 . 1631-073X .
- Nicolas . Jean-Louis . Serre . Jean-Pierre . 2012 . Formes modulaires modulo 2: structure de l'algèbre de Hecke . Comptes Rendus Mathématique. 350 . 9–10 . 449–454 . 10.1016/j.crma.2012.03.019 . 1204.1039 . 2012arXiv1204.1039N . 119720975 . 1631-073X .
