mixedmath

Explorations in math and programming
David Lowry-Duda



I'm in San Diego, and it's charming here. (It's certainly much nicer outside than the feet of snow in Boston. I've apparently brought some British rain with me, though).

Today I give a talk on counting lattice points on one-sheeted hyperboloids. These are the shapes described by $$ X_1^2 + \cdots + X_{d-1}^2 = X_d^2 + h,$$ where $h > 0$ is a positive integer. The question is: how many lattice points $x$ are on such a hyperboloid with $| x |^2 \leq R$; or equivalently, how many lattice points are on such a hyperboloid and contained within a ball of radius $\sqrt R$ centered at the origin?

I describe my general approach of transforming this into a question about the behavior of modular forms, and then using spectral techniques from the theory of modular forms to understand this behavior. This becomes a question of understanding the shifted convolution Dirichlet series $$ \sum_{n \geq 0} \frac{r_{d-1}(n+h)r_1(n)}{(2n + h)^s}.$$ Ultimately this comes from the modular form $\theta^{d-1}(z) \overline{\theta(z)}$, where $$ \theta(z) = \sum_{m \in \mathbb{Z}} e^{2 \pi i m^2 z}.$$

Here are the slides for this talk. Note that this talk is based on chapter 5 of my thesis.


Leave a comment

Info on how to comment

To make a comment, please send an email using the button below. Your email address won't be shared (unless you include it in the body of your comment). If you don't want your real name to be used next to your comment, please specify the name you would like to use. If you want your name to link to a particular url, include that as well.

bold, italics, and plain text are allowed in comments. A reasonable subset of markdown is supported, including lists, links, and fenced code blocks. In addition, math can be formatted using $(inline math)$ or $$(your display equation)$$.

Please use plaintext email when commenting. See Plaintext Email and Comments on this site for more. Note also that comments are expected to be open, considerate, and respectful.

Comment via email