MixedMath - explorations in math and programing

Paper: Towards a Classification of Isolated $j$-invariants

David Lowry-Duda — Wed, 15 Nov 2023 03:14:15 +0000

I'm happy to announce that a new paper, "Towards a classification of isolated $j$-invariants", now appears on the arxiv. This was done with my collaborators Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, Travis Morrison, Filip Najman, and Himanshu Shukla.

There are many collaborators because this was borne out of a workshop at CIRM from earlier this year, and we all attacked this problem together. This is the first time I have collaborated with any of these collaborators (though several of us are now involved in another project that will eventually appear).

The modular curve $X_1(N)$ is moduli space and an algebraic curve (defined over $\mathbb{Q}$ for us) whose points parametrize elliptic curves with a point of order $N$. We study isolated points, which are morally points on $X_1(N)$ that don't come from infinite families.

Perhaps the simplest form of infinite families comes come from a rational map $f: X_1(N) \longrightarrow \mathbb{P}^1$ of degree $d$. By Hilbert's irreducibility theorem, $f^{-1}(\mathbb{P}^1(\mathbb{Q}))$ contains infinitely many closed points of degree $d$.

Similarly, to any closed point $x$ of degree $d$ one can associate the rational divisor $P_1 + \cdots + P_d$, where $P_j$ are the points in the Galois orbit associated to $x$. This gives a natural map $\Phi_d: X_1(N)^{(d)} \longrightarrow \mathrm{Jac}(X_1(N))$. If $\Phi_d(x) = \Phi_d(y)$ for some point $y$, one can show that there exists a nonconstant function $f : X_1(N) \longrightarrow \mathbb{P}^1$ of degree $d$ again. Thus positive rank abelian subvarieties of the Jacobian also give infinite families of points.

Roughly, we say that a closed point $x$ is isolated if it doesn't come from either of the two constructions ($\mathbb{P}^1$, which we call $\mathbb{P}^1$ isolated — and abelian subvariety, which we call AV-isolated) above. Further, a closed point $x$ of degree $d$ is called sporadic if there are only finitely many closed points of degree at most $d$.

If $x \in X_1(N)$ is an isolated point, we say $j(x) \in X_1(1) \cong \mathbb{P}^1$ is an isolated $j$-invariant. In this paper, we seek to answer a question of Bourdon, Ejder, Liu, Odumodo, and Viray.

Question: Can one explicitly identify the (likely finite) set of isolated $j$-invariants in $\mathbb{Q}$?

Our main result is decision algorithm. Given a $j$-invariant, our algorithm produces a finite list of potential (level, degree) pairs such that one only needs to verify that degree $\mathrm{degree}$ points on $X_1(\mathrm{level})$ are not isolated.

Stated differently, our algorithm has one-sided error. It either reports that an element is not isolated, or it reports that it might be isolated and gives a list of places containing the data where isolated points must come from.

In principle, this sounds like it might be insufficient. But we ran our algorithm on every elliptic curve in the LMFDB and the outputs of our algorithm are always the empty set — except for $4$ exceptions where we know the $j$-invariants are isolated.

Concretely, we know that for any non-CM elliptic curve over $\mathbb{Q}$ with an isolated $j(E) \in \mathbb{Q}$ and with

conductor up to $500000$,
or with conductor that is $7$-smooth,
or of prime conductor $p < 3 \cdot 10^8$,

then $j(E) \in \{ -140625/8, -9317, 351/4, -162677523113838677 \}$. These latter correspond to $\mathbb{P}^1$ isolated points on $X_1(21), X_1(37), X_1(28)$, and $X_1(37)$ (respectively).¹ ¹An appendix to our paper by Derickx and Mark van Hoeij shows that the last $j$-invariant is actually isolated, not merely sporadic.

More generally, we are led to conjecture that these (and the CM $j$-invariants) are all of the isolated points on $X_1(N)$.

Broadly, our algorithm works by first considering the Galois image of elliptic curves (thanks to the code of David Zywina and the efforts of David Roe and others for making this broadly accessible). An earlier result of Bourdon and her collaborators² ²Abbey proposed this problem at the workshop based on the observation that her result might make things tenable. She was right!

allows one to radically narrow the focus of the Galois image to a minimal set of points of interest. We do this narrowing. We also show that no elliptic curve with adelic image of genus $0$ gives an isolated point, which allows us to ignore many potential leafs of computation.

The details are interesting and we try to be as explicit as possible. The code for this project is also available, and can be found on my github at github.com/davidlowryduda/isolated_points.

Paper: Congruent number triangles with the same hypotenuse

David Lowry-Duda — Tue, 14 Nov 2023 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

Bringing back the blogroll

David Lowry-Duda — Mon, 13 Nov 2023 03:14:15 +0000

It used to be very common for sites to have a page that said "blogroll" at the top and it linked to a bunch of blogs and sites that the author liked. People used to find other sites like this, and these sorts of links powered early search engines.

I stopped having a blogroll after I switched away from Wordpress and didn't think too much about it.

But things changed. RSS worked¹ ¹RIP Google Reader, 2013. It died so that Google+ could... also die a quiet death. and google search/Twitter were semi-reliable ways to find new things.

Now Twitter is dead, email newsletters are awful² ²though almost all of them were almost always awful. The medium is inferior to RSS in almost every way except a very important one: it's easier to monetize. , and Google search has lost its utility to the combination of earnest actors paying top dollar to appear at the top and SEO optimzer spam flooding the web.³ ³And Goodhart's Law strikes again!

The web is full of walled gardens, hiding the beautiful things within behind large, bland, stone walls.

The fundamental problem of internet content discovery is hard again! It's now rather hard to find places with consistently good content. I love longform media. It requires thought and intention. And it requires time.

So I'm bringing back my blogroll (and more generally, a list of sites, links, resources, and books that I find interesting). It is at the top of every page. Check it out!

If you have a site, bring out your blogroll. Help the small web flourish.

Paper: Murmurations of modular forms in the weight aspect

David Lowry-Duda — Sat, 21 Oct 2023 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

Murmurations in Maass forms

David Lowry-Duda — Wed, 19 Jul 2023 03:14:15 +0000

This post is larger than 10000 bytes, which is above the limit for this RSS feed. Perhaps it is long or has embedded images or code. Please view it directly at the url.

Slides from a talk at Concordia University

David Lowry-Duda — Thu, 23 Feb 2023 03:14:15 +0000

Today, I'm giving a talk at QVNTS on computing Maass forms. My slides are available here.

Please let me know if there are any questions or comments.

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.

MixedMath - explorations in math and programing

Paper: Towards a Classification of Isolated $j$-invariants

Paper: Congruent number triangles with the same hypotenuse

Bringing back the blogroll

Blogroll (and interesting links)

Paper: Murmurations of modular forms in the weight aspect

Murmurations in Maass forms

Slides from a talk at Concordia University

Paper: Sums of cusp forms coefficients along quadratic sequences

Context

Placing our techniques in context

Additional remarks

Continuous vs discrete spectra

Two too-brief summaries

On Mathstodon

The Algorithm

Lack of Trending

Implementation notes on modular curve visualizations