On 19 March I gave a talk at the 32nd Automorphic Forms Workshop, which ishosted by Tufts this year. The content of the talk concerned counting points on hyperboloids, and inparticular counting points on the three dimensional hyperboloid
$$\begin{equation} X^2 + Y^2 = Z^2 + h \end{equation}$$
for any fixed integer $h$. But thematically, I wanted to give another concrete example of using modularforms to compute some sort of arithmetic data, and to mention how the perhapsapparently unrelated topic of spectral theory appears even in such an arithmeticapplication.
Somehow, starting from counting points on $X^2 + Y^2 = Z^2 + h$ (which appearssimple enough on its own that I could probably put this in front of anelementary number theory class and they would feel comfortable experimentingaway on the topic), one gets to very scary-looking expressions like
$$\begin{equation} \sum_{t_j} \langle P_h^k, \mu_j \rangle \langle \theta^2 \overline{\theta} y^{3/4}, \mu_j \rangle + \sum_{\mathfrak{a}}\int_{(1/2)} \langle P_h^k, E_h^k(\cdot, u) \rangle \langle \theta^2 \overline{\theta} y^{3/4}, E_h^k(\cdot, u) \rangle du, \end{equation}$$
which is full of lots of non-obvious symbols and is generically intimidating.
Part of the theme of this talk is to give a very direct idea of how one gets tothe very complicated spectral expansion from the original lattice-countingproblem. Stated differently, perhaps part of the theme is to describe a simple-lookingnail and a scary-looking hammer, and show that the hammer actually works quitewell in this case.
The slides for this talk are available here.
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