Category Archives: Math.NT

Paper announcement: Quantitative HIT and Almost Prime Polynomial Discriminants

Theresa Anderson, Ayla Gafni, Robert Lemke Oliver, George Shakan, Ruixiang Zhang, and I have just uploaded a preprint to the arXiv called Quantitative Hilbert Irreducibility and Almost Prime Values of Polynomial Discriminants.

George has also written about this paper on his site.

This project began at an AIM workshop on Fourier analysis, arithmetic statistics, and discrete restriction.

Our guiding question was very open. For some nice local polynomial conditions, can we make sense of the Fourier transforms of these local conditions well enough to have arithmetic application?

This is partly inspired from Orbital exponential sums for prehomogeneous vector spaces by Takashi Taniguchi and Frank Thorne (preprint available on the arXiv). In this paper, Frank and Takashi algebraically compute Fourier transforms of a couple arithmetically interesting functions on prehomogeneous vector spaces over finite fields. It turns out that one can, for example, explicitly and completely compute the Fourier transform of the characteristic function of singular binary cubic forms over $\mathbb{F}_{q}$.

In a companion paper, Takashi and Frank combine those computations with sieves to prove that there are $\gg X / \log X$ cubic fields whose discriminant is squarefree, bounded above by $X$, and has at most $3$ prime factors. They also show there are $\gg X / \log X$ quartic fields whose discriminant is squarefree, bounded above by $X$, and has at most $8$ prime factors.

Results

We have two classes of result. Both rely on similar types of analysis, and are each centered on a study of a particular indicator-type function, its Fourier transform, and a sieve.

First, we prove a bound on the number of polynomials whose Galois group is a subgroup of $A_n$. For $H > 1$, define \begin{equation*} V_n(H) = \{ f \in \mathbb{Z}[x] : \mathrm{ht}(f) \leq H \} \end{equation*} and \begin{equation*} E_n(H, A_n) := \# \{ f \in V_n(H) : \mathrm{Gal}(f) \subseteq A_n \}. \end{equation*} We show that \begin{equation} E_n(H, A_n) \ll H^{n – \frac{2}{3} + O(1/n)}. \end{equation} This is an improvement on progress towards a conjecture of Van der Waerden and is a quantitative form of Hilbert’s Irreducibility Theorem, which shows (among other applications) that most monic irreducibile polynomials have full Galois group.

However I should note that Bhargava has announced a proof of a (slightly weakened form of) Van der Waerden’s conjecture, and his result is strictly stronger than our result.

Secondly, we prove that for any $n \geq 3$ and $r \geq 2n – 3$, we have \begin{equation} \# \{ f \in \mathbb{Z}[x] : \mathrm{ht}(f) \leq H, f \, \text{monic }, \omega(\mathrm{Disc}(f)) \leq r \} \gg_{n, r} \frac{H^n}{\log H}, \end{equation} where $\omega(\cdot)$ denotes the number of distinct prime divisors. Qualitatively, this says that there are lots of polynomials with almost prime discriminants.

As a corollary of this second result, we prove that for $n \geq 3$ and $r \geq 2n – 3$, \begin{equation} \# \{ F / \mathbb{Q} : [F \colon Q] = n, \mathrm{Disc}(F) \leq X, \omega(\mathrm{Disc}(F)) \leq r \} \gg_{n, r, \epsilon} X^{\frac{1}{2} + \delta_n – \epsilon} \end{equation} for explicit $\delta_n > 0$ and any $\epsilon > 0$. This shows that there are at least $X^{1/2}$ cubic fields whose discriminants are divisible by at most $3$ primes, or at least $X^{1/2}$ quartic fields whose discriminants are divisible by at most $5$ primes, for example. We guarantee fewer fields than Taniguchi and Thorne, but we guarantee fields with fewer prime factors and cover all degrees.

In the remainder of this post, I’ll describe a line of thinking that went towards proving our first result.

Odd polynomials

We initially studied the Fourier transform of the odd-polynomial indicator function. We call a function $f(x) \in \mathbb{F}_p[x]$ odd if it has no repeated roots and the factorization type of $f$ corresponds to an odd permutation in the Galois group. That is, we can write $f$ as \begin{equation*} f(x) = f_1(x) f_2(x) \cdots f_r(x) \bmod p, \end{equation*} and there will be an element of the Galois group with cycle type $(\deg f_1) (\deg f_2) \cdots (\deg f_r)$. For odd $f$, this cycle must be an odd permutation.

A more convenient description of oddness is in terms of the Möbius function on $\mathbb{F}_p[x]$. A degree $n$ polynomial $f$ is odd precisely if $\mu_p(f) = (-1)^{n+1}$. Define $1^p_{sf}(f)$ to be the squarefree indicator function on $\mathbb{F}_p[x]$, and define $1^p_{odd, n}$ to be the odd indicator function on degree $n$ polynomials on $\mathbb{F}_p[x]$. Then \begin{equation*} 1^p_{odd, n}(f) = 1^p_n(f)\frac{(-1)^{n+1}\mu_p(f) + 1^p_{sf}(f)}{2}. \end{equation*} (Here, $1^p_n(f)$ keeps only the degree $n$ polynomials).

Fourier transform of odd indicator function: a first approach

We then studied the Fourier transform of $1^p_{odd, n}$. Identifying the vector space of polynomials of degree at most $n$ over $\mathbb{F}_p[x]$, which we denote at $V_n(\mathbb{Z}/p\mathbb{Z})$, as $(\mathbb{Z}/p\mathbb{Z})^{n+1}$, we can study the Fourier transform of a function $\psi:V_n(\mathbb{Z}/p\mathbb{Z}) \longrightarrow \mathbb{C}$, \begin{equation*} \widehat{\psi}(\mathbf{u}) = \frac{1}{p^{n+1}} \sum_{f \in V_n(\mathbb{Z}/p\mathbb{Z})} \psi(f) e_p(\langle f, \mathbf{u} \rangle). \end{equation*} Here, $e_p(x) = e^{2 \pi i x / p}$.

It is possible to understand this Fourier transform using ideas similar to those of Takashi and Thorne. $\mathrm{GL}(2)$ acts on these polynomials in a similar way as it acts on quadratic forms, and $1^p_{odd, n}$ is invariant under this action. As in Takashi and Thorne, one can study the sizes of the Fourier transform on each orbit. This leads to several classical polynomial counting problems.

But unlike the prehomogeneous vector space context of Takashi and Thorne, we can’t completely determine the Fourier transform. For general degree, there are too many other terms.

Ultimately, we intend to use the knowledge of this Fourier transform as an ingredient in a sieve. An old theorem of Dedekind shows that if $\mathrm{Gal}(f) \subseteq A_n$, then $f$ is never odd mod any prime $p$.

We could use a Selberg sieve in the following form. For a nonnegative weight function $\phi: V_n(\mathbb{R}) \longrightarrow \mathbb{R}$ (roughly supported on the box $[-1, 1]^{n+1}$). Then consider \begin{equation}\label{eq:basic_sieve} \sum_{f \in V_n(\mathbb{Z})} \phi(f/H) \Big(\sum_{d: f \bmod p \text{ is odd } \forall p \mid d} \lambda_d \Big)^2 \geq 0 \end{equation} for some real weights $\lambda_d$ to be chosen later, but where $\lambda_1 = 1$.

For $f$ with $\mathrm{Gal}(f) \subseteq A_n$, $f$ is never odd. Thus the sum of weights $\lambda_d$ is exactly $\lambda_1 = 1$ for those $f$, and we get that \eqref{eq:basic_sieve} is bounded below by \begin{equation}\label{eq:basic_sieve_LHS} \sum_{\substack{f \in V_n(\mathbb{Z}) \\\\ \mathrm{Gal}(f) \subseteq A_n}} \phi(f/H). \end{equation} On the other hand, \eqref{eq:basic_sieve} is equal to \begin{equation}\label{eq:basic_sieve_RHS} \sum_{d_1, d_2} \lambda_{d_1} \lambda_{d_2} \sum_{f \in V_n(\mathbb{Z})} \phi(f / H) \prod_{p \mid [d_1, d_2]} 1^p_{odd, n}(f). \end{equation} Thus we have that \eqref{eq:basic_sieve_LHS} $\leq$ \eqref{eq:basic_sieve_RHS}. To bound \eqref{eq:basic_sieve_RHS}, we use Poisson summation to transform the sum of $\phi 1^p_{odd, n}$ into a dualized sum of $\widehat{\phi} \widehat{1}^p_{odd, n}$ and use our understanding of the Fourier transform $1^p_{odd, n}$ to (try to) get good bounds. Then one plays a game of optimizing over the weights $\lambda_d$.

Problem

There is a major problem with this approach. As we’re unable to completely determine the Fourier transform, it’s necessary to determine where it’s large and small and to handle the regions where it’s large well. Let’s look again at the expression \begin{equation*} 1^p_{odd, n}(f) = 1^p_n(f)\frac{(-1)^{n+1}\mu_p(f) + 1^p_{sf}(f)}{2}. \end{equation*} The Fourier transform of $\mu_p$ is expected to behave very well away from $0$. But the Fourier transform of $1^p_{sf}$ can be shown to have large Fourier coefficients away from $0$, strongly affecting the resulting bounds.

Graded indicator function: a second approach

Instead of studying the indicator function $1^p_{odd, n}$, we chose to study a sort of graded indicator function \begin{equation*} \psi_p(f) = \frac{(-1)^{n+1}1^p_n(f)\mu_p(f) + 1}{2}. \end{equation*} This is $1$ if $f$ is odd and squarefree, $0$ if $f$ is squarefree and even, and $1/2$ if $f$ is not squarefree.

On the Fourier transform side, we completely understand the Fourier transform of $1$ and we can hope to have good understanding of the Möbius function. So we should expect much better bounds.

But on the other side, this is not as clean of an indicator function as $1^p_{odd, n}$. In comparison to the basic sieve inequality \eqref{eq:basic_sieve_LHS} $\leq$ \eqref{eq:basic_sieve_RHS}, the product of indicator functions on the right hand side now becomes much messier, and the basic setup no longer applies.

Instead, in \eqref{eq:basic_sieve}, we replace $\big( \sum \lambda_d \big)^2$ by a positive semidefinite quadratic form in $\lambda_{d_1}, \lambda_{d_2}$ to get a modified Selberg sieve inequality similar to \eqref{eq:basic_sieve_LHS} $\leq$ \eqref{eq:basic_sieve_RHS}. The tail of the argument remains largely the same. Instead of bounding \eqref{eq:basic_sieve_RHS}, we bound

\begin{equation*} \sum_{d_1, d_2} \lambda_{d_1} \lambda_{d_2} \sum_{f \in V_n(\mathbb{Z})} \phi(f / H) \prod_{p \mid [d_1, d_2]} \psi_p(f). \end{equation*}

After Poisson summation, the goal becomes controlling $\widehat{\psi_p}(f)$, which essentially boils down to understanding $\widehat{\mu_p}(f)$.

In explicit coordinates, this is the task of understanding \begin{equation*} \widehat{\mu_p}(u_0, \ldots, u_n) = \frac{1}{p^{n+1}} \sum_{t_i \in \mathbb{F}_p} \mu_p(t_n x^n + \cdots + t_0) e_p(u_n t_n + \cdots + u_0 t_0). \end{equation*} This is a $\mathbb{F}_p[x]$-analogue of the classical question of bounding \begin{equation*} \sum_{n \leq x} \mu(n) e(n\theta) \end{equation*} for some real $\theta$. Baker and Harman have proved that GRH implies that\begin{equation*} \Big \lvert \sum_{n \leq x} \mu(n) e(n\theta) \Big \rvert \ll x^{\frac{3}{4} + \epsilon}, \end{equation*} and Porritt has proved the analogous result holds over function fields (where RH is known).

Applying this bound in our modified form of the Selberg sieve is what allows us to prove our first theorem.

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Slides from a talk on Visualizing Modular Forms

Yesterday I gave a talk at the University of Oregon Number Theory seminar on Visualizing Modular Forms. This is a spiritual successor to my paper on Visualizing modular forms that is to appear in Simons Symposia volume Arithmetic Geometry, Number Theory, and Computation.

I’ve worked with modular forms for almost 10 years now, but I’ve only known what a modular form looks like for about 2 years. In this talk, I explored visual representations of modular forms, with lots of examples.

The slides are available here.

I’ll share one visualization here that I liked a lot: a visualization of a particular Maass form on $\mathrm{SL}(2, \mathbb{Z})$.

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Slides from a talk on Half Integral Weight Dirichlet Series

On Thursday, 18 March, I gave a talk on half-integral weight Dirichlet series at the Ole Miss number theory seminar.

This talk is a description of ongoing explicit computational experimentation with Mehmet Kiral, Tom Hulse, and Li-Mei Lim on various aspects of half-integral weight modular forms and their Dirichlet series.

These Dirichlet series behave like typical beautiful automorphic L-functions in many ways, but are very different in other ways.

The first third of the talk is largely about the “typical” story. The general definitions are abstractions designed around the objects that number theorists have been playing with, and we also briefly touch on some of these examples to have an image in mind.

The second third is mostly about how half-integral weight Dirichlet series aren’t quite as well-behaved as L-functions associated to GL(2) automorphic forms, but sufficiently well-behaved to be comprehendable. Unlike the case of a full-integral weight modular form, there isn’t a canonical choice of “nice” forms to study, but we identify a particular set of forms with symmetric functional equations to study. There are several small details that can be considered here, and I largely ignore them for this talk. This is something that I hope to return to in the future.

In the final third of the talk, we examine the behavior and zeros of a handful of half-integral weight Dirichlet series. There are plots of zeros, including a plot of approximately the first 150k zeros of one particular form. These are also interesting, and I intend to investigate and describe these more on this site later.

The slides for this talk are available here.

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A balancing act in “Uniform bounds for lattice point counting”

I was recently examining a technical hurdle in my project on “Uniform bounds for lattice point counting and partial sums of zeta functions” with Takashi Taniguchi and Frank Thorne. There is a version on the arxiv, but it currently has a mistake in its handling of bounds for small $X$.

In this note, I describe an aspect of this paper that I found surprising. In fact, I’ve found it continually surprising, as I’ve reproven it to myself three times now, I think. By writing this here and in my note system, I hope to perhaps remember this better.

Landau’s Method

In this paper, we revisit an application of “Landau’s Method” to estimate partial sums of coefficients of Dirichlet series. We model this paper off of an earlier application by Chandrasakharan and Narasimhan, except that we explicitly track dependence of the several implicit constants and we prove these results uniformly for all partial sums, as opposed to sufficiently large partial sums.

The only structure is that we have a Dirichlet series $\phi(s)$, some Gamma factors $\Delta(s)$, and a functional equation of the shape $$ \phi(s) \Delta(s) = \psi(s) \Delta(1-s). $$ This is relatively structureless, and correspondingly our attack is very general. We use some smoothed approximation to the sum of coefficients, shift lines of integration to pick up polar main terms, apply the functional equation and change variables so work with the dual, and then get some collection of error terms and error integrals.

It happens to be that it’s much easier to work with a $k$-Riesz smoothed approximation. That is, if $$
\phi(s) = \sum_{n \geq 1} \frac{a(n)}{\lambda_n^s}
$$
is our Dirichlet series, and we are interested in the partial sums $$
A_0(s) = \sum_{\lambda_n \leq X} a(n),
$$
then it happens to be easier to work with the smoothed approximations $$
A_k(X) = \frac{1}{\Gamma(k+1)}\sum_{\lambda_n \leq X} a(n) (X – \lambda_n)^k a(n),
$$
and to somehow combine several of these smoothed sums together.

This smoothed sum is recognizable as $$
A_k(X) =
\frac{1}{2\pi i}\int_{c – i\infty}^{c + i\infty} \phi(s)
\frac{\Gamma(s)}{\Gamma(s + k + 1)} X^{s + k}ds
$$
for $c$ somewhere in the half-plane of convergence of the Dirichlet series. As $k$ gets large, these integrals become better behaved. In application, one takes $k$ sufficiently large to guarantee desired convergence properties.

The process of taking several of these smoothed approximations for large $k$ together, studying them through basic functional equation methods, and combinatorially combining these smoothed approximations via finite differencing to get good estimates for the sharp sum $A_0(s)$ is roughly what I think of as “Landau’s Method”.

Application and shape of the error

In our paper, as we apply Landau’s method, it becomes necessary to understand certain bounds coming from the dual Dirichlet series $$
\psi(s) = \sum_{n \geq 1} \frac{b(n)}{\mu_n^s}.
$$
Specifically, it works out that the (combinatorially finite differenced) between the $k$-smoothed sum $A_k(X)$ and its $k$-smoothed main term $S_k(X)$ can be written as $$
\Delta_y^k [A_k(X) – S_k(X)] = \sum_{n \geq 1}
\frac{b(n)}{\mu_n^{\delta + k}} \Delta_y^k I_k(\mu_n X),\tag{1}
$$
where $\Delta_y^k$ is a finite differencing operator that we should think of as a sum of several shifts of its input function.

More precisely, $\Delta_y F(X) := F(X + y) – F(X)$, and iterating gives $$
\Delta_y^k F(X) = \sum_{j = 0}^k (-1)^{k – j} {k \choose j} F(X + jy).
$$
The $I_k(\cdot)$ term on the right of $(1)$ is an inverse Mellin transform $$
I_k(t) = \frac{1}{2 \pi i} \int_{c – i\infty}^{c + i\infty}
\frac{\Gamma(\delta – s)}{\Gamma(k + 1 + \delta – s)}
\frac{\Delta(s)}{\Delta(\delta – s)} t^{\delta + k – s} ds.
$$
Good control for this inverse Mellin transform yields good control of the error for the overall approximation. Via the method of finite differencing, there are two basic choices: either bound $I_k(t)$ directly, or understand bounds for $(\mu_n y)^k I_k^{(k)}(t)$ for $t \approx \mu_n X$. Here, $I_k^{(k)}(t)$ means the $k$th derivative of $I_k(t)$.

Large input errors

In the classical application (as in the paper of CN), one worries about this asymptotic mostly as $t \to \infty$. In this region, $I_k(t)$ can be well-approximated by a $J$-Bessel function, which is sufficiently well understood in large argument to give good bounds. Similarly, $I_k^{(k)}(t)$ can be contour-shifted in a way that still ends up being well-approximated by $J$-Bessel functions.

The shape of the resulting bounds end up being that $\Delta_y^k I_k(\mu_n X)$ is bounded by either

  • $(\mu_n X)^{\alpha + k(1 – \frac{1}{2A})}$, where $A$ is a fixed parameter that isn’t worth describing fully, and $\alpha$ is a bound coming from the direct bound of $I_k(t)$, or
  • $(\mu_n y)^k (\mu_n X)^\beta$, where $\beta$ is a bound coming from bounding $I_k^{(k)}(t)$.

In both, there is a certain $k$-dependence that comes from the $k$-th Riesz smoothing factors, either directly (from $(\mu_n y)^k$), or via its corresponding inverse Mellin transform (in the bound from $I_k(t)$). But these are the only aspects that depend on $k$.

At this point in the classical argument, one determines when one bound is better than the other, and this happens to be something that can be done exactly, and (surprisingly) independently of $k$. Using this pair of bounds and examining what comes out the other side gives the original result.

Small input errors

In our application, we also worry about asymptotic as $t \to 0$. While it may still be true that $I_k$ can be approximated by a $J$-Bessel function, the “well-known” asymptotics for the $J$-Bessel function behave substantially worse for small argument. Thus different methods are necessary.

It turns out that $I_k$ can be approximated in a relatively trivial way for $t \leq 1$, so the only remaining hurdle is $I_k^{(k)}(t)$ as $t \to 0$.

We’ve proved a variety of different bounds that hold in slightly different circumstances. And for each sort of bound, the next steps would be the same as before: determine when each bound is better, bound by absolute values, sum together, and then choose the various parameters to best shape the final result.

But unlike before, the boundary between the regions where $I_k$ is best bounded directly or bounded via $I_k^{(k)}$ depends on $k$. Aside from choosing $k$ sufficiently large for convergence properties (which relate to the locations of poles and growth properties of the Dirichlet series and gamma factors), any sufficiently large $k$ would suffice.

Limiting behavior gives a heuristic region

After I step away from this paper and argument for a while and come back, I wonder about the right way to choose the balancing error. That is, I rework when to use bounds coming from studying $I_k(t)$ directly vs bounds coming from studying $I_k^{(k)}(t)$.

But it turns out that there is always a reasonable heuristic choice. Further, this heuristic gives the same choice of balancing as in the case when $t \to \infty$ (although this is not the source of the heuristic).

Making these bounds will still give bounds for $\Delta_y^k I_k(\mu_n X)$ of shape

  • $(\mu_n X)^{\alpha + k(1 – \frac{1}{2A})}$, where $A$ is a fixed parameter that isn’t worth describing fully, and $\alpha$ is a bound coming from the direct bound of $I_k(t)$, or
  • $(\mu_n y)^k (\mu_n X)^\beta$, where $\beta$ is a bound coming from bounding $I_k^{(k)}(t)$.

The actual bounds for $\alpha$ and $\beta$ will differ between the case of small $\mu_n X$ and large $\mu_n X$ ($J$-Bessel asymptotics for large, different contour shifting analysis for small), but in both cases it turns out that $\alpha$ and $\beta$ are independent of $k$.

This is relatively easy to see when bounding $I_k^{(k)}(t)$, as repeatedly differentiating under the integral shows essentially that $$
I_k^{(k)}(t) =
\frac{1}{2\pi i}
\int \frac{\Delta(s)}{(\delta – s)\Delta(\delta – s)}
t^{\delta – s} ds.
$$
(I’ll note that the contour does vary with $k$ in a certain way that doesn’t affect the shape of the result for $t \to 0$).

When balancing the error terms $(\mu_n X)^{\alpha + k(1 – \frac{1}{2A})}$ and $(\mu_n y)^k (\mu_n X)^\beta$, the heuristic comes from taking arbitrarily large $k$. As $k \to \infty$, the point where the two error terms balance is independent of $\alpha$ and $\beta$.

This reasoning applies to the case when $\mu_n X \to \infty$ as well, and gives the same point. Coincidentally, the actual $\alpha$ and $\beta$ values we proved for $\mu_n X \to \infty$ perfectly cancel in practice, so this limiting argument is not necessary — but it does still apply!

I suppose it might be possible to add another parameter to tune in the final result — a parameter measuring deviation from the heuristic, that can be refined for any particular error bound in a region of particular interest.

But we haven’t done that.

In fact, we were slightly lossy in how we bounded $I_k^{(k)}(t)$ as $t \to 0$, and (for complicated reasons that I’ll probably also forget and reprove to myself later) the heuristic choice assuming $k \sim \infty$ and our slighly lossy bound introduce the same order of imprecision to the final result.

More coming soon

We’re updating our preprint and will have that up soon. But as I’ve been thinking about this a lot recently, I realize there are a few other things I should note down. I intend to write more on this in the short future.

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Slides from a talk at AIM

I’m currently at an AIM workshop on Arithmetic Statistics, Discrete Restriction, and Fourier Analysis. This morning (AIM time)/afternoon (USEast time), I’ll be giving a talk on Lattice points and sums of Fourier Coefficients of modular forms.

The theme of this talk is embodied in the statement that several lattice counting problems like the Gauss circle problem are essentially the same as very modular-form-heavy problems, sometimes very closely similar and sometimes appearing slightly different.

In this talk, I describe several recent adventures, successes and travails, in my studies of problems related to the Gauss circle problem and the task of producing better bounds for the sum of the first several coefficients of holomorphic cuspforms.

Here are the slides for my talk.

I’ll note that various parts of this talk have appeared in several previous talks of mine, but since it’s the pandemic era this is the first time much of this has appeared in slides.

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Slides from a talk on computing Maass forms

Yesterday, I gave a talk on various aspects of computing Maass cuspforms at Rutgers.

Here are the slides for my talk.

Unlike most other talks that I’ve given, this doesn’t center on past results that I’ve proved. Instead, this is a description of an ongoing project to figure out how to rigorously compute many Maass forms, implement this efficiently in code, and add this data to the LMFDB.

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Talk on computing Maass forms

In a remarkable coincidence, I’m giving two talks on Maass forms today (after not giving any talks for 3 months). One of these was a chalk talk (or rather camera on pen on paper talk). My other talk can be found at https://davidlowryduda.com/static/Talks/ComputingMaass20/.

In this talk, I briefly describe how one goes about computing Maass forms for congruence subgroups of $\mathrm{SL}(2)$. This is a short and pointed exposition of ideas mostly found in papers of Hejhal and Fredrik Strömberg’s PhD thesis. More precise references are included at the end of the talk.

This amounts to a description of the idea of Hejhal’s algorithm on a congruence subgroup.

Side notes on revealjs

I decided to experiment a bit with this talk. This is not a TeX-Beamer talk (as is most common for math) — instead it’s a revealjs talk. I haven’t written a revealjs talk before, but it was surprisingly easy.

It took me more time than writing a beamer talk, most likely because I don’t have a good workflow with reveal and there were several times when I wanted to use nontrivial javascript capabilities. In particular, I wanted to have a few elements transition from one slide to the next (using the automatic transition capabilities).

At first, I had thought I would write in an intermediate markup format and then translate this into revealjs, but I quickly decided against that plan. The composition stage was a bit more annoying.

But I think the result is more appealing than a beamer talk, and it’s sufficiently interesting that I’ll revisit it later.

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Notes from a talk at Dartmouth on the Fibonacci zeta function

I recently gave a talk “at Dartmouth”1. The focus of the talk was the (odd-indexed) Fibonacci zeta function:
$$ \sum_{n \geq 1} \frac{1}{F(2n-1)^s},$$
where $F(n)$ is the nth Fibonacci number. The theme is that the Fibonacci zeta function can be recognized as coming from an inner product of automorphic forms, and the continuation of the zeta function can be understood in terms of the spectral expansion of the associated automorphic forms.

This is a talk from ongoing research. I do not yet understand “what’s really going on”. But within the talk I describe a few different generalizations; firstly, there is a generalization to other zeta functions that can be viewed as traces of units on quadratic number fields, and secondly there is a generalization to quadratic forms recognizing solutions to Pell’s equation.

I intend to describe additional ideas from this talk in the coming months, as I figure out how pieces fit together. But for now, here are the slides.

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Pictures of equidistribution – the line

In my previous note, we considered equidistribution of rational points on the circle $X^2 + Y^2 = 2$. This is but one of a large family of equidistribution results that I’m not particularly familiar with.

This note is the first in a series of notes dedicated to exploring this type of equidistribution visually. In this note, we will investigate a simpler case — rational points on the line.

(more…)

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Points on X^2 + Y^2 = 2 equidistribute with respect to height

When you order rational points on the circle $X^2 + Y^2 = 2$ by height, these points equidistribute.

Stated differently, suppose that $I$ is an arc on the circle $X^2 + Y^2 = 2$. Then asymptotically, the number of rational points on the arc $I$ with height bounded by a number $H$ is equal to what you would expect if $\lvert I\rvert /2\sqrt{2}\pi$ of all points with height up to $H$ were on this arc. Here, $\lvert I\rvert /2\sqrt{2}\pi$ the ratio of the arclength of the arc $I$ with the total circumference of the circle.

This only makes sense if we define the height of a rational point on the circle. Given a point $(a/c, b/c)$ (written in least terms) on the circle, we define the height of this point to be $c$.

In forthcoming work with my frequent collaborators Chan Ieong Kuan, Thomas Hulse, and Alexander Walker, we count three term arithmetic progressions of squares. If $C^2 – B^2 = B^2 – A^2$, then clearly $A^2 + C^2 = 2B^2$, and thus a 3AP of squares corresponds to a rational point on the circle $X^2 + Y^2 = 2$. We compare one of our results to what you would expect from equidistribution. From general principles, we expected such equidistribution to be true. But I wasn’t sure how to prove it.

With helpful assistance from Noam Elkies, Emmanuel Peyre, and John Voight (who each immediately knew how to prove this), I learned how to prove this fact.

The rest of this note contains this proof.

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