Lagmental Vicfred

The Modular Discriminant Defines Ramanujan's Tau Function by Vicfred

tags: mathematics, modular forms, q-series, partitions

The cusp form Delta is an infinite product whose Fourier coefficients are the tau values. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Statement

A modular form \(f\) on the upper half-plane transforms predictably under fractional linear maps and has a Fourier expansion in \(q=e^{2\pi i\tau}\). Its coefficients often encode arithmetic.

$$ \Delta(\tau)=q\prod_{n\ge1}(1-q^n)^{24} $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \Delta(\tau)=\sum_{n\ge1}\tau(n)q^n $$

Worked algebra

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \Delta=\frac{E_4^3-E_6^2}{1728}=q-24q^2+252q^3-1472q^4+\cdots $$

Conceptual compression

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \Delta(\tau)=q\prod_{n\ge1}(1-q^n)^{24},\\[5pt] \mathsf{C}\;&:\quad \Delta(\tau)=\sum_{n\ge1}\tau(n)q^n. \end{aligned} $$

Caveat

Weight, level, character, and cusp conditions are part of the definition. A formal \(q\)-series is not automatically a modular form.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \Delta(\tau)=\sum_{n\ge1}\tau(n)q^n \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Fri 17 June 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.