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

The modular discriminant modulo 2

The modular discriminant is defined by an infinite product:

${\displaystyle \mathrm{\Delta }(q)=q\prod _{n=1}^{\mathrm{\infty }}(1-q^n{)}^{24}=\sum _{n=1}^{\mathrm{\infty }}\tau (n)q^n.}$

The coefficients that matches are usually denoted ${\displaystyle \tau }$, and correspond to the Ramanujan tau function. Results from Kolberg^[3] and Jean-Pierre Serre^[4] allows to show that modulo 2, we have: ${\displaystyle \mathrm{\Delta }(q)\equiv \sum _{m=0}^{\mathrm{\infty }}{q}^{(2m+1{)}^2}mod2}$ i.e., the ${\displaystyle q}$-series of ${\displaystyle \mathrm{\Delta }}$ modulo 2 consists of ${\displaystyle q}$ 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 ${\displaystyle f}$ are defined as follows^[5] ${\displaystyle T_{n}f(z)=n^{2k-1}\sum _{a\ge 1,ad=n,0\le b<d}{d}^{-2k}f\left(\frac{az+b}{d}\right)}$ with ${\displaystyle n\in \text{N}}$.

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

${\displaystyle \gamma (z)=\sum _{a|(n,m),a\ge 1}{a}^{2k-1}c\left(\frac{mn}{a^2}\right).}$

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

${\displaystyle \gamma (n)=\{\begin{array}{ll}c(np)& \text{ if }p\nmid n\\ c(np)+c(n/p)& \text{ if }p\mid n\end{array}\text{ and }p\text{ an odd prime}.}$

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 ${\displaystyle {\mathbb{F}}_2}$.

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

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

Finally, define the Hecke algebra ${\displaystyle A}$ as follows: Since ${\displaystyle \mathcal{F}(n)\subset \mathcal{F}(n+1)}$, one can restrict elements of ${\displaystyle A(n+1)}$ to ${\displaystyle \mathcal{F}}$ to obtain an element of ${\displaystyle A(n)}$. When considering the map ${\displaystyle {\varphi }_n:A(n+1)\to A(n)}$ as the restriction to ${\displaystyle \mathcal{F}(n)}$, then ${\displaystyle {\varphi }_n}$ is a homomorphism. As ${\displaystyle A(1)}$ is either identity or zero, ${\displaystyle A(1)\cong {\mathbb{F}}_2}$. Therefore, the following chain is obtained: ${\displaystyle \cdots \to A(n+1)\to A(n)\to A(n-1)\to \cdots \to A(2)\to A(1)\cong {\mathbb{F}}_2}$. Then, define the Hecke algebra ${\displaystyle A}$ to be the projective limit of the above ${\displaystyle A(n)}$ as ${\displaystyle n\to \mathrm{\infty }}$. Explicitly, this means ${\displaystyle A=\underset{n\in \text{N}}{\underleftarrow{\lim }}A(n)=\{T_{p_1}\cdot T_{p_2}\cdots T_{p_k}|p_1,p_2,\dots ,p_k\in \mathbb{P},k\ge 0\}}$.

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

So for any prime ${\displaystyle p\in \mathbb{P}}$, it is possible to find coefficients ${\displaystyle a_{ij}(p)\in {\mathbb{F}}_2}$ such that: ${\displaystyle T_p=\sum _{i+j\ge 1}{a}_{ij}(p)T_3^i{T}_5^j}$

References

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