I'm planning to give a short and introductory talk about modular forms, and I'm going to say that modular forms' Fourier coefficients are arithmetic, which is certainly not a mathematically well-defined sentence but (at least for me) shows the importance of modular form in number theory. Three basic examples I'll introduce is the following:
Eisenstein series
Discriminant function
Theta function
The Fourier coefficients of Eisenstein series are divisor power sum functions, and Fourier coefficients of (power of) theta functions are related to the number of representing a given number as sum of squares. However, it is hard for me to explain arithmetic significance of $\tau$ function. Here's a list of facts that I know about $\tau(n)$:
Multiplicative since $\Delta(z)$ is a Hecke eigenform
$|\tau(p)| \leq 2 p^{11/2}$ (Ramanujan-Petersson conjecture, which has automorphic form version that is only known for a few cases)
- Can be expressed as combination of divisor sum functions, by using the identity between $\Delta(z)$ and Eisenstein series
- several congruences in wikipedia
and this list doesn't seem to be the arithmetic heart of the $\tau(n)$. Could anyone explain the arithmetic essence of $\tau(n)$?