MixedMath - explorations in math and programing

Paper: Sums of cusp forms coefficients along quadratic sequences

David Lowry-Duda — Fri, 27 Jan 2023 03:14:15 +0000

I am happy to announce that my frequent collaborators, Alex Walker and Chan Kuan, and I have just posted a preprint to the arxiv called Sums of cusp form coefficients along quadratic sequences.

Our primary result is the following.

Let $f(z) = \sum_{n \geq 1} a(n) q^n = \sum_{n \geq 1} A(n) n^{\frac{k-1}{2}} q^n$ denote a holomorphic cusp form of weight $k \geq 2$ on $\Gamma_0(N)$, possibly with nontrivial nebentypus. For any $h > 0$ and any $\epsilon > 0$, we have that \begin{equation} \sum_{n^2 + h \leq X^2} A(n^2 + h) = c_{f, h} X + O_{f, h, \epsilon}(X^{\eta(k) + \epsilon}), \end{equation} where \begin{equation*} \eta(k) = 1 - \frac{1}{k + 3 - \sqrt{k(k-2)}} \approx \frac{3}{4} + \frac{1}{32k - 44} + O(1/k^2). \end{equation*}

The constant $c_{f, h}$ above is typically $0$, but we weren't the first to notice this.

Context

We approach this by studying the Dirichlet series \begin{equation*} D_h(s) := \sum_{n \geq 1} \frac{r_1(n) a(n+h)}{(n + h)^{s}}, \end{equation*} where \begin{equation*} r_1(n) = \begin{cases} 1 & n = 0 \\ 2 & n = m^2, m \neq 0 \\ 0 & \text{else} \end{cases} \end{equation*} is the number of ways of writing $n$ as a (sum of exactly $1$) square.

This approach isn't new. In his 1984 paper Additive number theory and Maass forms, Peter Sarnak suggests that one could relate the series \begin{equation*} \sum_{n \geq 1} \frac{d(n^2 + h)}{(n^2 + h)^2} \end{equation*} to a Petersson inner product involving a theta function, a weight $0$ Eisenstein series, and a half-integral weight Poincaré series. This inner product can be understood spectrally, hence the spectrum of the half-integral weight hyperbolic Laplacian (and half-integral weight Maass forms) reflect the behavior of the ordinary-seeming partial sums \begin{equation*} \sum_{n \leq X} d(n^2 + h). \end{equation*}

A broader class of sums was studied by Blomer.Blomer. Sums of Hecke eigenvalues over values of quadratic polynomials. IMRN, 2008.

Let $q(x) \in \mathbb{Z}[x]$ denote any monic quadratic polynomial. Then Blomer showed that \begin{equation*} \sum_{n \leq X} A(q(n)) = c_{f, q}X + O_{f, q, \epsilon}(X^{\frac{6}{7} + \epsilon}). \end{equation*} Blomer already noted that the main term typically doesn't occur.

More recently, Templier and TsimermanTemplier and Tsimerman. Non-split sums of coefficients of $\mathrm{GL}(2)$-automorphic forms. Israel J. Math 2013

showed that $D_h(s)$ has polynomial growth in vertical strips and has reasonable polar behavior. This allows them to show that \begin{equation*} \sum_{n \geq 0} A(n^2 + h) g\big( (n^2 + h)/X \big) = c_{f, h, g} X + O_\epsilon(X^{\frac{1}{2} + \Theta + \epsilon}), \end{equation*} where $\Theta$ is a term that is probably $0$ coming from a contribution from potentially exceptional eigenvalues of the Laplacian and the Selberg Eigenvalue Conjecture, and where $g$ is a smooth function of sufficient decay.

Templier and Tsimerman approach their result in two different ways: one studies the Dirichlet series $D_h(s)$ with the same initial steps as outlined by Sarnak. The second way is more representation theoretic and allows greater flexibility in the permitted forms.

Placing our techniques in context

Broadly, our approach begins in the same way as Templier and Tsimerman — we study $D_h(s)$ through a Petersson inner product involving half-integral weight Poincaré series. The great challenge is to understand the discrete spectrum and half-integral weight Maass forms, and we deviate from Templier and Tsimerman sharply in our treatment of the discrete spectrum.

For each eigenvalue $\lambda_j$ there is an associated type $\frac{1}{2} + it_j$ and form \begin{equation*} \mu_j(z) = \sum_{n \neq 0} \rho_j(n) W_{\mathrm{sgn}(n) \frac{k}{2}, it_j}(4\pi \lvert n \rvert y) e(nx), \end{equation*} where $W$ is a Whittaker function and the coefficients $\rho_j(n)$ are very mysterious. We average Maass forms in long averages over the eigenvalues and types (indexed by $j$) and long averages over coefficients $n$. We base the former approach average on Blomer's work above, and for the latter we improve on (a part of) the seminar work of Duke, Friedlander, and Iwaniec.Duke, Friedlander, Iwaniec. The subconvexity problem for Artin $L$-functions. Inventiones. 2002.

For this, it is necessary to establish certain uniform bounds for Whittaker functions.

To apply our bounds for the discrete spectrum, Maass forms, and Whittaker functions, we use that $f$ is holomorphic in an essential way. We decompose $f$ into a sum of finitely many holomorphic Poincaré series. This is done by Blomer as well. But in contrast, we study the resulting shifted convolutions whereas Blomer recollects terms into Kloosterman and Salié type sums.

Ultimately we conclude with a standard contour shifting argument.

Additional remarks

Continuous vs discrete spectra

The quality of our bound mirrors the quality of our understanding of the discrete spectrum. This is interesting in that the continuous and discrete spectra are typically of a similar calibre of size and difficulty.

But here we are examining a half-integral weight object into half-integral weight spectra. The continuous spectrum comes from Eisenstein series, and it turns out that the coefficients of real-analytic half-integral weight Eisenstein series are (essentially) Dirichlet $L$-functions — and these are relatively easy to understand.

Existing bounds for half-integral weight Maass forms are much weaker than corresponding bounds for full-integral weight Maass forms.

In principle, there is also a residual spectrum to consider here (in contrast to weight $0$ spectral expansions). But in practice this is perfectly handled by in the work of Templier and Tsimerman and presents no further difficulty.

Two too-brief summaries

One too-brief-to-be-correct summary of this paper is that by restricting to a smaller class of quadratic polynomials than Blomer, it is possible to prove a stronger result. In reality, we restrict to a class of quadratic polynomials that allows the corresponding Dirichlet series to be easily recognized as a Petersson inner product involving a standard theta function and a half-integral weight Poincaré series.

Another too-brief-to-be-correct summary of this paper is that examining Whittaker functions and Bessel functions even closer reveals that they control all of multiplicative number theory. Actually, this might be correct.As a corollary, I guess all multiplicative number theory is controlled by monodromy?

On Mathstodon

David Lowry-Duda — Sun, 01 Jan 2023 03:14:15 +0000

I've been on Mastodon for just over 6 weeks now, inspired by obvious events approximately two months ago. Specifically, I've been on the math-friendly server Mathstodon, which includes latex rendering.

In short: I like it.

Right now, Mathstodon is a kind place. The general culture is kind and inviting. I'm reminded a bit of young StackExchange sites, which are often so happy to come into existence that it seems like every new post is treasured.

Given the similarities to twitter, it is natural to compare and contrast. Twitter is not kind. Outrage evidently boosts engagement and snark generates retweets and likes. On Mathstodon, moderators maintain civility. I don't pay much attention to other Mastodon servers — they have different moderators and possibly different cultures.

But it's also true that Mastodon (and Mathstodon) are growing rapidly, and I don't know any social media sites that managed to maintain a positive culture while growing larger. Math.StackExchange¹ ¹ Which I've helped moderate for a decade and which is dear to me. used to be much friendlier than it is now, but I think it's impossible to bring that positivity back.² ² I have thought much about this. See my other posts Challenges facing cohesion at MSE, Ghosts of forums past, and Splitting MSE into NoviceMathSE is a bad idea for more.

Is the Mastodon system of having separate instances with separate cultures the answer? I don't know. Conceivably if Mathstodon starts to feel bad, I could pick and and move elsewhere — or of course run my own. Time will tell.

The Algorithm

I wasn't sure if I would like or not like the lack of the algorithm, the mysterious ordering system. But to my surprise, I miss essentially nothing about twitters algorithm.

Maintaining a reasonable signal-to-noise ratio on social media is a struggle. A deep problem I've faced on twitter is that I like to read things written by people who write about Hard Problems for a living. To ensure they get sufficient audience on twitter, it's necessary to post and repost the same essay/story (possibly with lightly different titles or formats). If any of these gets sufficient following, then it will bubble up to the feed.

This is too noisy for me.

The Mastodon system is closer to an interleaved set of RSS feeds.³ ³ The ActivityPub protocol that Mastodon implements is sort of like a souped up RSS/Atom protocol that allows more rapid updates. Everything is strictly in the order that they're made. I love RSS, so perhaps it is no surprise that I like this system.

And the system allows unimaginatively-named "lists", which are feeds containing various specific accounts.

At least at the moment, this has a great signal-to-noise ratio.

Lack of Trending

The notable exception is the lack of trending information. Mastodon does not have an answer to trending local content.

Concretely, I thought about this when the Boston MBTA screeched to a halt (as it has a recent tendency to do) a bit before Christmas. If I had opened twitter, I might have been able to find the source of the disruption — not because I follow any MBTA accounts, but because this type of kerfluffle would cause enough activity that it would have populated the feed.

But I don't have twitter on my devices, and I don't currently see myself reinstalling twitter. In principle, the various MBTA twitter posts could have appeared on Mastodon as well — but I wouldn't see them. I suppose I could look for the tag #MBTA, or maybe #Boston? These aren't sufficient yet.

One can debate the merits of having twitter be the de facto place for random civic and political organizations to post news, but this is common. And this is another place where Mastodon is currently lacking.

Implementation notes on modular curve visualizations

David Lowry-Duda — Wed, 07 Dec 2022 03:14:15 +0000

The LMFDB will soon have a new section on modular curves. And as with modular forms, each curve will have a portrait or badge that gives a rough approximation to some of the characteristics of the curve.

I wrote a note on some of the technical observations and implementation details concerning these curves. This note can be found here. I've also added a link to it in the unpublished notes section of my research page.

Instead of going into details here, I'll refer to the details in the note. I'll give the core idea.

Each modular curve comes from a subgroup $H \subset \mathrm{GL}(2, \mathbb{Z}/N\mathbb{Z})$ for some $N$ called the level. To form a visualization, we compute cosets for $H \cap \mathrm{SL}(2, \mathbb{Z}/N\mathbb{Z})$ inside $\mathrm{SL}(2, \mathbb{Z}/N\mathbb{Z})$, lift these to nice elements in $\mathrm{SL}(2, \mathbb{Z})$, and then translate the standard fundamental domain of $\mathrm{SL}(2, \mathbb{Z}) \backslash \mathcal{H}$ by these cosets.

We show this on the Poincaré disk, to give a badge format similar to what we did for modular forms.

This is not a perfect representation, but it captures some of the character of the curve.

Here are a few of the images that we produce.

I had studied how to produce space efficient SVG files as well, though I did not go in this direction in the end. But I think these silhouettes are interesting, so I include them too.

Slides on Maass forms, nearing completion

David Lowry-Duda — Fri, 02 Dec 2022 03:14:15 +0000

I'm giving an update today on my project to compute Maass forms. Today, I describe the final steps about how to make the computation rigorous. This complements a talk I gave two years ago about how to implement a heuristic evaluation.

The slides for my talk today are available here.

I hope to have a preprint describing this algorithm and its implementation shortly. I also hope to have a beta update to LMFDB with this information by the next meeting of the collaboration.

Slides on Modular Forms and the L-functions, a talk at the Simons Center for Geometry and Physics

David Lowry-Duda — Wed, 09 Nov 2022 03:14:15 +0000

I'm giving a talk today on modular forms and their $L$-functions at the Simons Center for Geometry and Physics. The slides for this talk are available here.

I refer to many things that I have done before in this talk.

For references and proofs of the various aspects of Dirichlet series associated to half-integral weight modular forms, I wrote a series of notes stemming from this post.

For more information on computing and working with Maass forms, see notes from another talk I gave.

And finally, many of the objects discussed are in the LMFDB.

Supplement to Langlands survey

David Lowry-Duda — Thu, 03 Nov 2022 03:14:15 +0000

Today, I gave an introductory survey on the Langlands program to a group of mathematicians and physicists at the Simons Center for Geometry and Physics at Stonybrook University.

Many of these connections are well-illustrated on the LMFDB. I highly suggest poking around and clicking on things — on many pages there are links to related objects.

Part of the Langlands program includes the modularity conjecture for elliptic curves over $\mathbb{Q}$. On the LMFDB, this means that we can go look at elliptic curves over $\mathbb{Q}$, take some arbitrary elliptic curve like \begin{equation*} y^2 + xy + y = x^3 - 113x - 469, \end{equation*} (which has this homepage in the LMFDB), and then see that this corresponds to this modular form on the LMFDB. And they have the same L-function.

During the talk, Brian gave an example or an Artin representation of the symmetric group $S_3$ on three symbols. For reference, he pulled data from this representation page on the LMFDB.

From one perspective, the Langlands program is a monolithic wall of imposing, intimidating mathematics. But from another perspective, the Langlands program is most interesting because it organizes and connects seemingly different phenomena.

It's not necessary to understand each detail — instead it's interesting to note that there are many fundamentally different ways of producing highly structured data (like $L$-functions). And remarkably we think that every such $L$-function will behave beautifully, including satisfying their own Riemann Hypothesis.

Notes from a survey talk by Jeff Lagarias on the Collatz problem

David Lowry-Duda — Wed, 02 Nov 2022 03:14:15 +0000

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Paul R. Halmos -- Lester R. Ford Award

David Lowry-Duda — Mon, 17 Oct 2022 03:14:15 +0000

This is an update with (unexpected) good news. My collaborator Miles Wheeler and I were given the Paul R. Halmos – Lester R. Ford Award for our paper Perturbing the mean value theorem: implicit functions, the morse lemma, and beyond¹ ¹The arxiv version of this paper is called When are there continuous choices for the mean value abscissa, and is slightly longer than the final published form.

This was an unexpected honor. Partly, this is due to the fact that we first wrote this paper years ago and it was accepted in 2019. But the publication backlog meant that it wasn't published until January of 2021, and thus eligible for the 2022 award. But also it's always sort of a nice surprise to hear that people read what I write.

I described this paper before, and wrote an additional note on how we chose our functions and made the figures.

When my wife learned that this paper won an award, she asked if this "was that paper you really liked that you wrote wtih Miles during grad school"? Yes! It is that paper. I really do like this paper.

Origin story

The string of ideas leading to this paper began when I first began to TA calculus at Brown. I was becoming aware of the fact that I would soon be teaching calculus courses, and I began to really think about why we structured the courses the way we do.² ²I don't have a completely satisfactory explanation for everything, especially for the "integration bag of tricks" or the "series convergence bag of tricks" portion. But upon reflection, I can understand the purpose of most portions of the calculus sequence.

One of my least favorite questions was the verify the mean value theorem for the function $f(x)$ on the interval $[1, 4]$ by... sort of question. The problem is that this question is really just a way to check that one understands the statement of the mean value theorem — and this statement feels very unimportant.

But it turns out that the mean value theorem is extremely important. The mean value theorem and intermediate value theorems are the two sneaky abstractions that encapsulate underlying topological ideas that we typically brush aside in introductory calculus courses.

We don't do calculus on real valued functions over the rationals

We illustrate this with two examples in the analogous case of functions \begin{equation*} f: \mathbb{Q} \longrightarrow \mathbb{R}. \end{equation*} I would expect that many introductory students would think these functions feel intuitively about the same as functions from $\mathbb{R}$ to $\mathbb{R}$. But in fact both the intermediaet value theorem and mean value theorem are false for real-valued functions defined on the rationals.

The intermediate value theorem is false here. Consider the function \begin{align*} f(x): \mathbb{Q} &\longrightarrow \mathbb{R}\\ x &\mapsto x^2 - 2. \end{align*} On the interval $[0, 2]$, for example, we see that $f(0) = -2$ and $f(2) = 2$. But there is no $x \in \mathbb{Q} \cap [0, 2]$ such that $f(x) = 0$.

The mean value theorem is false too. Consider the function \begin{align*} f(x): \mathbb{Q} &\longrightarrow \mathbb{R} \\ x &\mapsto \begin{cases} 0 & \text{if } x^2 > 2, \\ 1 & \text{if } x^2 > 2. \end{cases} \end{align*} For this function, $f'(x) = 0$ everywhere, even though the function isn't constant. (But it is locally constant). We try to avoid pathological examples initially.³ ³It is a funny thing. Initially, we pretend that everything is as nice as possible to build intuition. Then we see that there are pathological counterexamples and use these to sharpen intuition. Seeing these too early is potentially misleading. Implicit within the order is an idea of what usually hapens and what behavior is exceptional.

It turns out that the topological space the functions are defined on really matter.

Mean value theorem as abstraction

I don't talk about this in a typical calculus class because topological concerns are almost entirely ignored. We work with the practical case of real-valued functions defined over the reals. And the key tools we use to hide the underlying topological details are the intermediate and mean value theorems.

I didn't fully realize this until I wondered whether we could teach calculus without covering these two theorems.

It might be possible. But I decided that it was a bad idea.

Instead, I think it's a good idea to give students a better idea of how these two theorems are two of the most important ideas in calculus. I kept this idea in mind when I wrote An intuitive introduction to calculus for my students in 2013.

And more broadly, I began to investigate just how many things we could reduce, as directly as possible, to the mean value theorem. Miles realized we could investigate the continuity of the mean value abscissa with a low dimnsional implicit function theorem and baby version of Morse's lemma, and the rest is history.

I've collected so many random facts and questions tangentially related to the mean value theorem over the years. But I could always collect more!

MathFest

I attended MathFest for the first time to receive this award in person. MathFest was an extraordinary and interesting experience. I was particularly impressed by how many sessions and talks focused on actionable ways to improve math education in the classroom now.

Slides from a talk at Maine-Quebec

David Lowry-Duda — Sat, 15 Oct 2022 03:14:15 +0000

This weekend I'm at the Maine-Québec number theory conference. This year, it's in Québec!

I'm giving a talk surveying ideas from Improved bounds on number fields of small degree, joint work with Anderson, Gafni, Hughes, Lemke Oliver, Thorne, Wang, and Zhang.

The slides for this talk are available here.

Initial thoughts on visualizing number fields

David Lowry-Duda — Fri, 22 Jul 2022 03:14:15 +0000

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