Modular forms modulo p

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 [math]\displaystyle{ q=e^{2 \pi i z} }[/math]. So if [math]\displaystyle{ f }[/math] is a modular form, then there are coefficients [math]\displaystyle{ c(n) }[/math] such that [math]\displaystyle{ f(z) = \sum_{n \in \mathbb{N}} c(n)q^n }[/math]. To reduce modulo 2, consider the subspace of modular forms with coefficients of the [math]\displaystyle{ q }[/math]-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 [math]\displaystyle{ G_2 }[/math] and [math]\displaystyle{ G_3 }[/math]:..^[1] It is then possible to normalize [math]\displaystyle{ G_2 }[/math] and [math]\displaystyle{ G_3 }[/math] to [math]\displaystyle{ E_2 }[/math] and [math]\displaystyle{ E_3 }[/math], having integers coefficients in their [math]\displaystyle{ q }[/math]-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, [math]\displaystyle{ E_2 }[/math] and [math]\displaystyle{ E_3 }[/math] are just [math]\displaystyle{ 1 }[/math]. That is, a trivial reduction. To get a non-trivial reduction, mathematicians use the modular discriminant [math]\displaystyle{ \Delta }[/math]. It is introduced as a "priority" generator before [math]\displaystyle{ E_2 }[/math] and [math]\displaystyle{ E_3 }[/math]. Thus, modular forms are seen as polynomials of [math]\displaystyle{ E_2 }[/math],[math]\displaystyle{ E_3 }[/math] and [math]\displaystyle{ \Delta }[/math] (over the complex [math]\displaystyle{ \mathbb{C} }[/math] in general, but seen over integers [math]\displaystyle{ \mathbb{Z} }[/math] for reduction), once reduced modulo 2, they become just polynomials of [math]\displaystyle{ \Delta }[/math] over [math]\displaystyle{ \mathbb{F}_2 }[/math].

The modular discriminant modulo 2

The modular discriminant is defined by an infinite product:

[math]\displaystyle{ \Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24} = \sum_{n=1}^\infty \tau(n)q^n. }[/math]

The coefficients that matches are usually denoted [math]\displaystyle{ \tau }[/math], and correspond to the Ramanujan tau function. Results from Kolberg^[3] and Jean-Pierre Serre^[4] allows to show that modulo 2, we have: [math]\displaystyle{ \Delta(q) \equiv \sum_{m=0}^{\infty} q^{(2m+1)^2} \bmod 2 }[/math] i.e., the [math]\displaystyle{ q }[/math]-series of [math]\displaystyle{ \Delta }[/math] modulo 2 consists of [math]\displaystyle{ q }[/math] to powers of odd squares.

Hecke operators modulo 2

Hecke operators are commonly considered as the most important operators acting on modular forms. It is therefore justified to try to reduce them modulo 2.

The Hecke operators for a modular form [math]\displaystyle{ f }[/math] are defined as follows^[5] [math]\displaystyle{ T_nf(z) = n^{2k-1}\sum_{a \geq 1,\, ad=n,\, 0 \leq b \lt d} d^{-2k}f \left( \frac{az+b}{d} \right) }[/math] with [math]\displaystyle{ n \in \N }[/math].

Hecke operators may be defined on the [math]\displaystyle{ q }[/math]-series as follows:^[5] if [math]\displaystyle{ f(z) = \sum_{n \in \Z} c(n)q^n }[/math], then [math]\displaystyle{ T_nf(z) = \sum_{m \in \Z} \gamma(m)q^m }[/math] with

[math]\displaystyle{ \gamma(z) = \sum_{a | (n,m),\, a \geq 1} a^{2k-1} c\left( \frac{mn}{a^2} \right). }[/math]

Since modular forms were reduced using the [math]\displaystyle{ q }[/math]-series, it makes sense to use the [math]\displaystyle{ q }[/math]-series definition. The sum simplifies a lot for Hecke operators of primes (i.e. when [math]\displaystyle{ m }[/math] 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 [math]\displaystyle{ f }[/math] a modular form modulo 2 with [math]\displaystyle{ q }[/math]-representation [math]\displaystyle{ f(q) = \sum_{n \in \N} c(n)q^n }[/math], the Hecke operator [math]\displaystyle{ T_p }[/math] on [math]\displaystyle{ f }[/math] is defined by [math]\displaystyle{ \overline{T_p}|f(q) = \sum_{n \in \N} \gamma(n)q^n }[/math] where

[math]\displaystyle{ \gamma(n) = \begin{cases} c(np) & \text{ if } p \nmid n \\ c(np)+c(n/p) & \text{ if } p \mid n \end{cases} \quad \text{ and } p \text{ an odd prime}. }[/math]

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 [math]\displaystyle{ \mathbb{F}_2 }[/math].

Following Serre and Nicolas's notations from^[7] [math]\displaystyle{ \mathcal{F} = \left\langle \Delta^k \mid k \text{ odd} \right\rangle }[/math], i.e. [math]\displaystyle{ \mathcal{F} = \left\langle \Delta, \Delta^3, \Delta^5, \Delta^7, \Delta^9, \dots \right\rangle }[/math]. Writing [math]\displaystyle{ \mathcal{F}(n) = \left\langle \Delta, \Delta^3, \Delta^5, \dots, \Delta^{2n-1} \right\rangle }[/math] so that [math]\displaystyle{ \dim(\mathcal{F}(n)) = n }[/math], define [math]\displaystyle{ A(n) }[/math] as the [math]\displaystyle{ \mathbb{F}_2 }[/math]-subalgebra of [math]\displaystyle{ \text{End}\left(\mathcal{F}(n)\right) }[/math] given by [math]\displaystyle{ \mathbb{F}_2 }[/math] and [math]\displaystyle{ T_p }[/math].

That is, if [math]\displaystyle{ \mathfrak{m}(n) = \{T_{p_1} \cdot T_{p_2} \cdots T_{p_k} \mid p_1, p_2, \dots, p_k \in \mathbb{P}, k\geq 1\} }[/math] is a sub-vector-space of [math]\displaystyle{ \mathcal{F} }[/math], we get [math]\displaystyle{ A(n) = \mathbb{F}_2 \oplus \mathfrak{m}(n) }[/math].

Finally, define the Hecke algebra [math]\displaystyle{ A }[/math] as follows: Since [math]\displaystyle{ \mathcal{F}(n) \subset \mathcal{F}(n+1) }[/math], one can restrict elements of [math]\displaystyle{ A(n+1) }[/math] to [math]\displaystyle{ \mathcal{F} }[/math] to obtain an element of [math]\displaystyle{ A(n) }[/math]. When considering the map [math]\displaystyle{ \phi_n: A(n+1) \to A(n) }[/math] as the restriction to [math]\displaystyle{ \mathcal{F}(n) }[/math], then [math]\displaystyle{ \phi_n }[/math] is a homomorphism. As [math]\displaystyle{ A(1) }[/math] is either identity or zero, [math]\displaystyle{ A(1) \cong \mathbb{F}_2 }[/math]. Therefore, the following chain is obtained: [math]\displaystyle{ \dots \to A(n+1) \to A(n) \to A(n-1) \to \dots \to A(2) \to A(1) \cong \mathbb{F}_2 }[/math]. Then, define the Hecke algebra [math]\displaystyle{ A }[/math] to be the projective limit of the above [math]\displaystyle{ A(n) }[/math] as [math]\displaystyle{ n \to \infty }[/math]. Explicitly, this means [math]\displaystyle{ A = \varprojlim_{n \in \N} A(n) = \left\lbrace T_{p_1} \cdot T_{p_2} \cdots T_{p_k} | p_1, p_2, \dots, p_k \in \mathbb{P}, k\geq 0 \right\rbrace }[/math].

The main property of the Hecke algebra [math]\displaystyle{ A }[/math] is that it is generated by series of [math]\displaystyle{ T_3 }[/math] and [math]\displaystyle{ T_5 }[/math].^[7] That is: [math]\displaystyle{ A = \mathbb{F}_2\left[ T_p \mid p \in \mathbb{P} \right] = \mathbb{F}_2 \left[\left[ T_3, T_5 \right]\right] }[/math].

So for any prime [math]\displaystyle{ p \in \mathbb{P} }[/math], it is possible to find coefficients [math]\displaystyle{ a_{ij}(p) \in \mathbb{F}_2 }[/math] such that: [math]\displaystyle{ T_p = \sum_{i+j \geq 1} a_{ij}(p) T_3^iT_5^j }[/math]

References

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