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 $q=e^{2\pi iz}$. So if $f$ is a modular form, then there are coefficients $c(n)$ such that $f(z)=\sum _{n\in \mathbb {N} }c(n)q^{n}$. To reduce modulo 2, consider the subspace of modular forms with coefficients of the $q$-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 $G_{2}$ and $G_{3}$:.. It is then possible to normalize $G_{2}$ and $G_{3}$ to $E_{2}$ and $E_{3}$, having integers coefficients in their $q$-series. This gives generators for modular forms, which may be reduced modulo 2. Note the Miller basis has some interesting properties Once reduced modulo 2, $E_{2}$ and $E_{3}$ are just $1$. That is, a trivial reduction. To get a non-trivial reduction, mathematicians use the modular discriminant $\Delta$. It is introduced as a "priority" generator before $E_{2}$ and $E_{3}$. Thus, modular forms are seen as polynomials of $E_{2}$,$E_{3}$ and $\Delta$ (over the complex $\mathbb {C}$ in general, but seen over integers $\mathbb {Z}$ for reduction), once reduced modulo 2, they become just polynomials of $\Delta$ over $\mathbb {F} _{2}$.

The modular discriminant modulo 2

The modular discriminant is defined by an infinite product:

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

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

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

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

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

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

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:.

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 $\mathbb {F} _{2}$.

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

That is, if ${\mathfrak {m}}(n)=\{T_{p_{1}}\cdot T_{p_{2}}\cdots T_{p_{k}}\mid p_{1},p_{2},\dots ,p_{k}\in \mathbb {P} ,k\geq 1\}$ is a sub-vector-space of ${\mathcal {F}}$, we get $A(n)=\mathbb {F} _{2}\oplus {\mathfrak {m}}(n)$.

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

The main property of the Hecke algebra $A$ is that it is generated by series of $T_{3}$ and $T_{5}$. That is: $A=\mathbb {F} _{2}\left[T_{p}\mid p\in \mathbb {P} \right]=\mathbb {F} _{2}\left[\left[T_{3},T_{5}\right]\right]$.

So for any prime $p\in \mathbb {P}$, it is possible to find coefficients $a_{ij}(p)\in \mathbb {F} _{2}$ such that: $T_{p}=\sum _{i+j\geq 1}a_{ij}(p)T_{3}^{i}T_{5}^{j}$