The importance of relations between automorphic forms and arithmetic functions

Question

As I understand things, one of the classical reasons to care about modular forms was their relation to interesting arithmetic functions/counting questions, i.e. on sums of squares and partitions. When I read Diamond and Shurman’s book([Diamond&Shurman]A First Course in Modular Forms), this point of view was briefly mentioned as an interesting application, but most of the book was focused on their role in the Modularity Theorem. I have now been working on Gelbart’s book on automorphic forms([Gelbart]Automorphic Forms on Adele Groups), which feels like it is moving even further off in this direction.

This point of view is certainly interesting and there’s obviously a lot of important mathematics here, but it leaves me wondering what happens to this when we move from modular forms to automorphic forms. Are there any interesting arithmetic functions leading to automorphic forms over groups besides $SL(2,\mathbb{Z})$? If not, what makes this particular group so special?

Answer 1

Let me just highlight one aspect that I find particularly interesting.

Without any doubt, one the most basic arithmetic functions is $\tau_k(n)$, which counts the number of ways $n$ can be written as a product of $k$ factors. It is the $k$-fold convolution of the constant $1$ function with itself, so it is multiplicative and its Dirichlet series is an $L$-function of degree $k$: $$ \sum_n \frac{\tau_k(n)}{n^s} = \prod_p\frac{1}{(1-p^{-s})^k},\qquad \Re s>1. $$ This $L$-function is apparently $\zeta(s)^k$, so it has a meromorphic continuation, functional equation, and the (analogue of the) Riemann hypothesis seems to be true for it.

A natural question is if $\tau_k(n)$ fits into some broader family of arithmetic functions which perhaps allows us to understand $\tau_k(n)$ better and gives insight into the mentioned properties of its $L$-function. A natural candidate for such arithmetic functions is the convolution of $k$ Dirichlet characters. Then the resulting $L$-function is the product of $k$ Dirichlet $L$-functions, which equally seems to satisfy the (analogue of the) Riemann hypothesis. It turns out that there is geometry behind these more general coefficients: they are Hecke eigenvalues of special Eisenstein series on the group $\mathrm{GL}_k$. But these Eisenstein series do not form a spectrally complete family: they are missing the more general Eisenstein series and the cusp forms. In these missing cases the Euler denominators $(1-p^{-s})^k$ are more general degree $k$ polynomials of $p^{-s}$, and the closer we are to a cusp form, the less we can canonically decompose them into lower degree polynomials. The $L$-functions of general Eisenstein series on $\mathrm{GL}_k$ decompose into a product of $L$-functions of cusp forms on $\mathrm{GL}_j$ with $j<k$ (and my first mentioned example corresponds to $j=1$ for each factor), while the cuspidal $L$-functions for $\mathrm{GL}_k$ are completely new for the degree $k$ and they do not factor into $L$-functions of smaller degrees.

So $\tau_k(n)$ and the convolutions $\chi_1\ast\dots\ast\chi_k$ of Dirichlet characters are just special Hecke eigenvalues, which come from one and the same geometrical and arithmetical object: $\mathrm{GL}_k$ over the adele ring of $\mathbb{Q}$. The automorphic forms corresponding to the various Hecke eigenvalue systems constitute a spectrally complete family, hence one can bring in methods of harmonic analysis and representation theory. In this way, for example, one can derive valuable results on the additive divisor problem or the closely related (hybrid) moment problem of $L$-functions (e.g. the fourth moment problem of the Riemann zeta function). The point is that the $L$-functions coming from $\mathrm{GL}_k$ are close relatives of each other, they include classical objects like $k$-fold products of Dirichlet $L$-functions, and one can study particular members of the family with the help of other members of the family. Last but not least, all these $L$-functions seem to satisfy the (analogue of the) Riemann hypothesis, and these are basically the only examples we know that seem to satisfy this hypothesis.