ModularFormsModuloTwo.jl Documentation

This module contains usual routines to do computation on modular forms modulo two.

Introduction Modular Forms Modulo 2

Here is a very quick introduction to modular forms modulo two.

Modular Forms

Modular forms are holomorphic complex functions on the upper half plane.

For a real introduction on modular forms, you can read "A Course in Arithmetic" by J-P. Serre (1973), or the wikipedia page.

$\ q$-series of Modular Forms

Now, as they are holomorphic, they have a series expansion. For a modular form $\ f$, there is an expansion

$\ f(z) = f(q) = \sum_{n \in \Z} a_n(f) q^n \qquad q = e^{2\pi i z}$

Now, Victor Saul Miller has proved in his thesis that there exists a basis for modular forms such that all $\ a_n$ are integers.

Reduction Modulo 2

Using this basis, we can reduce a modular form $\ f$ by taking it's $\ q$-series and reducing all coefficients $\ a_n$ modulo 2. It may be proved that the resulting $\ f$ is composed only of powers of $\ \Delta$ with

$\ \Delta = \sum_{m=0}^{\infty} q^{(2m+1)^2}.$

Therefore, the space of modular forms modulo 2 is:

$\ \mathcal{F} = \left\langle \Delta^k | k \text{ odd} \right\rangle = \left\langle \Delta, \Delta^3, \Delta^5, \Delta^7, \dots \right\rangle$

Duality $\ q$-series and $\ \Delta$ polynomial

What make the study of modular forms modulo 2 interesting is the fact that for a modular form $\ f$, there are two representations:

• one as un infinite $\ q$-series
• one as a finite polynomial in terms of $\ \Delta$

The fact that we can switch from one representation is very useful. With the $\ q$-series, calculations (such as Hecke operators) are made easy, but we have to deal with an infinite series. On the other hand, the $\ \Delta$-polynomial is a finite representation, but calculations are harder.

Specific use of this library

The way usual computer programs would deal with such object would be to represent $\ 10^6$ $\ q$-coefficients of modular forms, and suppose it is enough to take conclusions. In this library, we try to take advantage of the fact that we have a duality $\ q-\Delta$ and that all coefficients are over $\ \mathbb{F}_2$ (so either $\ 0$ or $\ 1$). There are functions to_q(df, precalculated) and to_Δ(f, precalculated) (in the advanced operations section) to switch from one representation to an other. There is also a very special drop_error(f, precalculated, LENGTH) function which drops the numerical error from calculating a Hecke operator (in the advanced operations as well). (This is only possible by taking advantage of the fact that there is an equivalent finite representation in terms of $\ \Delta$.)