Modular forms modulo p

In mathematics, modular forms are functions on the Upper half-plane. Their study is part of complex analysis. However, once reduced modulo 2, modular forms create a completely different theory, which may be classified as Algebraic number theory.

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 the possible to normalize $G_{2}$ and $G_{3}$ to $E_{2}$ and $E_{3}$, having integers coeffieints 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 taht 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.