MixedMath - explorations in math and programinghttps://davidlowryduda.comDavid's personal blog.en-usCopyright David Lowry-Duda (2022) - All Rights Reserved.admin@davidlowryduda.comadmin@davidlowryduda.comWed, 15 Nov 2023 22:01:31 +0000Wed, 15 Nov 2023 22:01:31 +0000mixedmathapp/generate_rss.py v0.1https://cyber.harvard.edu/rss/rss.htmlhttps://davidlowryduda.com/static/images/favicon-32x32.pngMixedMathhttps://davidlowryduda.comPaper: Towards a Classification of Isolated $j$-invariantshttps://davidlowryduda.com/paper-isolated-j-invariantsDavid Lowry-Duda<p>I'm happy to announce that a new paper, "Towards a classification of isolated
$j$-invariants", now appears <a href="https://arxiv.org/abs/2311.07740">on the arxiv</a>.
This was done with my collaborators Abbey Bourdon, Sachi Hashimoto, Timo
Keller, Zev Klagsbrun, Travis Morrison, Filip Najman, and Himanshu Shukla.</p>
<p>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).</p>
<p>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 <em>isolated</em> points, which are morally points on $X_1(N)$
that don't come from infinite families.</p>
<p>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 $d$.</p>
<p>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.</p>
<p>Roughly, we say that a closed point $x$ is <strong>isolated</strong> 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 <strong>sporadic</strong> if there are
only finitely many closed points of degree at most $d$.</p>
<p>If $x \in X_1(N)$ is an isolated point, we say $j(x) \in X_1(1) \cong
\mathbb{P}^1$ is an <strong>isolated $j$-invariant</strong>. In this paper, we seek to
answer a question of Bourdon, Ejder, Liu, Odumodo, and Viray.</p>
<blockquote>
<p>Question: Can one explicitly identify the (likely finite) set of isolated
$j$-invariants in $\mathbb{Q}$?</p>
</blockquote>
<p>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.</p>
<p>Stated differently, our algorithm has one-sided error. It either reports that
an element is not isolated, or it reports that it <em>might</em> be isolated and gives
a list of places containing the data where isolated points must come from.</p>
<p>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.</p>
<p>Concretely, we know that for any non-CM elliptic curve over $\mathbb{Q}$ with
an isolated $j(E) \in \mathbb{Q}$ and with</p>
<ul>
<li>conductor up to $500000$,</li>
<li>or with conductor that is $7$-smooth,</li>
<li>or of prime conductor $p < 3 \cdot 10^8$,</li>
</ul>
<p>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).<sup>1</sup>
<span class="aside"><sup>1</sup>An appendix to our paper by
Derickx and Mark van Hoeij shows that the last $j$-invariant is actually
isolated, not merely sporadic.</span></p>
<p>More generally, we are led to conjecture that these (and the CM $j$-invariants)
are all of the isolated points on $X_1(N)$.</p>
<p>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 collaborators<sup>2</sup>
<span class="aside"><sup>2</sup>Abbey proposed this problem at the workshop based on the
observation that her result might make things tenable. She was right!</span></p>
<p>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.</p>
<p>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
<a href="https://github.com/davidlowryduda/isolated_points">github.com/davidlowryduda/isolated_points</a>.</p>https://davidlowryduda.com/paper-isolated-j-invariantsWed, 15 Nov 2023 03:14:15 +0000Paper: Congruent number triangles with the same hypotenusehttps://davidlowryduda.com/paper-congruent-triangles-same-hypotenuseDavid Lowry-DudaThis 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.https://davidlowryduda.com/paper-congruent-triangles-same-hypotenuseTue, 14 Nov 2023 03:14:15 +0000Bringing back the blogrollhttps://davidlowryduda.com/bringing-back-the-blogrollDavid Lowry-Duda<p>It used to be <em>very</em> 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.</p>
<p>I stopped having a blogroll after I switched away from Wordpress and didn't
think too much about it.</p>
<p>But things changed. RSS worked<sup>1</sup>
<span class="aside"><sup>1</sup>RIP Google Reader, 2013. It died so that
Google+ could... also die a quiet death.</span>
and google search/Twitter were
semi-reliable ways to find new things.</p>
<p>Now Twitter is dead, email newsletters are awful<sup>2</sup>
<span class="aside"><sup>2</sup>though 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.</span>
, 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.<sup>3</sup>
<span class="aside"><sup>3</sup>And <a href="https://en.wikipedia.org/wiki/Goodhart%27s_law">Goodhart's
Law</a> strikes again!</span></p>
<p>The web is full of walled gardens, hiding the beautiful things within behind
large, bland, stone walls.</p>
<p>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.</p>
<p>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 <a href="/blogroll">top of every
page</a>. Check it out!</p>
<p>If you have a site, bring out your blogroll. Help the small web flourish.</p>https://davidlowryduda.com/bringing-back-the-blogrollMon, 13 Nov 2023 03:14:15 +0000Blogroll (and interesting links)https://davidlowryduda.com/blogrollDavid Lowry-DudaThis 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.https://davidlowryduda.com/blogrollWed, 01 Nov 2023 03:14:15 +0000Paper: Murmurations of modular forms in the weight aspecthttps://davidlowryduda.com/paper-modular-murmurationsDavid Lowry-DudaThis 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.https://davidlowryduda.com/paper-modular-murmurationsSat, 21 Oct 2023 03:14:15 +0000Murmurations in Maass formshttps://davidlowryduda.com/maass-murmurationsDavid Lowry-DudaThis 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.https://davidlowryduda.com/maass-murmurationsWed, 19 Jul 2023 03:14:15 +0000Slides from a talk at Concordia Universityhttps://davidlowryduda.com/slides-from-concordia-2023David Lowry-Duda<p>Today, I'm giving a talk at QVNTS on computing Maass forms. My slides are
available <a href="/wp-content/uploads/2023/02/Concordia.pdf">here</a>.</p>
<p>Please let me know if there are any questions or comments.</p>https://davidlowryduda.com/slides-from-concordia-2023Thu, 23 Feb 2023 03:14:15 +0000Paper: Sums of cusp forms coefficients along quadratic sequenceshttps://davidlowryduda.com/paper-sums-of-coeffs-along-quadraticsDavid Lowry-Duda<p>I am happy to announce that my frequent collaborators, Alex Walker and Chan
Kuan, and I have just posted a <a href="https://arxiv.org/abs/2301.11901">preprint to the
arxiv</a> called <em>Sums of cusp form coefficients
along quadratic sequences</em>.</p>
<p>Our primary result is the following.</p>
<div class="theorem">
<p>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*}</p>
</div>
<p>The constant $c_{f, h}$ above is typically $0$, but we weren't the first to
notice this.</p>
<h2>Context</h2>
<p>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.</p>
<p>This approach isn't new. In his 1984 paper <em>Additive number theory and Maass
forms</em>, 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*}</p>
<p>A broader class of sums was studied by Blomer.<span class="aside">Blomer. <em>Sums of Hecke
eigenvalues over values of quadratic polynomials</em>. <strong>IMRN</strong>, 2008.</span></p>
<p>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.</p>
<p>More recently, Templier and Tsimerman<span class="aside">Templier and Tsimerman.
<em>Non-split sums of coefficients of $\mathrm{GL}(2)$-automorphic forms</em>.
<strong>Israel J. Math</strong> 2013</span></p>
<p>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.</p>
<p>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.</p>
<h2>Placing our techniques in context</h2>
<p>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.</p>
<p>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 $j$) <em>and</em> 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.<span class="aside">Duke, Friedlander, Iwaniec. <em>The subconvexity problem for
Artin $L$-functions</em>. <strong>Inventiones</strong>. 2002.</span></p>
<p>For this, it is necessary to establish certain uniform bounds for Whittaker functions.</p>
<p>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.</p>
<p>Ultimately we conclude with a standard contour shifting argument.</p>
<h2>Additional remarks</h2>
<h3>Continuous vs discrete spectra</h3>
<p>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.</p>
<p>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.</p>
<p>Existing bounds for half-integral weight Maass forms are much weaker than
corresponding bounds for full-integral weight Maass forms.</p>
<p>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.</p>
<h3>Two too-brief summaries</h3>
<p>One too-brief-to-be-correct summary of this paper is that <em>by restricting to a
smaller class of quadratic polynomials than Blomer, it is possible to prove a
stronger result</em>. 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.</p>
<p>Another too-brief-to-be-correct summary of this paper is that <em>examining
Whittaker functions and Bessel functions even closer reveals that they control
all of multiplicative number theory</em>. Actually, this might be
correct.<span class="aside">As a corollary, I guess all multiplicative number theory is
controlled by monodromy?</span></p>https://davidlowryduda.com/paper-sums-of-coeffs-along-quadraticsFri, 27 Jan 2023 03:14:15 +0000On Mathstodonhttps://davidlowryduda.com/on-mathstodonDavid Lowry-Duda<p>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 <a href="https://mathstodon.xyz/"><em>Mathstodon</em></a>, which includes latex rendering.</p>
<p>In short: I like it.</p>
<p>Right now, <em>Mathstodon</em> 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.</p>
<p>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 <em>Mathstodon</em>, moderators maintain civility. I don't pay
much attention to other Mastodon servers — they have different moderators
and possibly different cultures.</p>
<p>But it's also true that Mastodon (and <em>Mathstodon</em>) are growing rapidly, and I
don't know any social media sites that managed to maintain a positive culture
while growing larger. Math.StackExchange<sup>1</sup>
<span class="aside"><sup>1</sup> Which I've helped moderate for
a decade and which is dear to me.</span>
used to be much friendlier than it is
now, but I think it's impossible to bring that positivity back.<sup>2</sup>
<span class="aside"><sup>2</sup> I have
thought much about this. See my other posts <a href="/challenges-facing-community-cohesion-and-math-stackexchange-in-particular/">Challenges facing cohesion at
MSE</a>,
<a href="/ghosts-of-forums-past/">Ghosts of forums past</a>, and <a href="/splitting-mathse-into-novicemathse-is-a-bad-idea/">Splitting MSE into
NoviceMathSE is a bad idea</a>
for more.</span></p>
<p>Is the Mastodon system of having separate instances with separate cultures the
answer? I don't know. Conceivably if <em>Mathstodon</em> starts to feel bad, I could
pick and and move elsewhere — or of course run my own. Time will tell.</p>
<h2>The Algorithm</h2>
<p>I wasn't sure if I would like or not like the lack of <strong>the algorithm</strong>, the
mysterious ordering system. But to my surprise, I miss essentially nothing
about twitters algorithm.</p>
<p>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.</p>
<p>This is too noisy for me.</p>
<p>The Mastodon system is closer to an interleaved set of RSS feeds.<sup>3</sup>
<span class="aside"><sup>3</sup> The
ActivityPub protocol that Mastodon implements is sort of like a souped up
RSS/Atom protocol that allows more rapid updates.</span>
Everything is strictly
in the order that they're made. I love RSS, so perhaps it is no surprise that I
like this system.</p>
<p>And the system allows unimaginatively-named "lists", which are feeds containing
various specific accounts.</p>
<p>At least at the moment, this has a great signal-to-noise ratio.</p>
<h2>Lack of Trending</h2>
<p>The notable exception is the lack of <strong>trending</strong> information. Mastodon does
not have an answer to trending local content.</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>https://davidlowryduda.com/on-mathstodonSun, 01 Jan 2023 03:14:15 +0000Implementation notes on modular curve visualizationshttps://davidlowryduda.com/modcurvevizDavid Lowry-Duda<p>The LMFDB will soon have a new section on modular curves. And as with modular
forms, each curve will have a <em>portrait</em> or <em>badge</em> that gives a rough
approximation to some of the characteristics of the curve.</p>
<p>I wrote a note on some of the technical observations and implementation details
concerning these curves. This note can be <a href="/wp-content/uploads/2022/12/VisualizingModularCurves.pdf">found
here</a>. I've also
added a link to it in the unpublished notes section of my <a href="/research">research
page</a>.</p>
<p>Instead of going into details here, I'll refer to the details in the note. I'll
give the core idea.</p>
<p>Each modular curve comes from a subgroup $H \subset \mathrm{GL}(2,
\mathbb{Z}/N\mathbb{Z})$ for some $N$ called the <em>level</em>. 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 <em>nice</em> 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.</p>
<p>We show this on the Poincaré disk, to give a badge format similar to what we
did for modular forms.</p>
<p>This is not a perfect representation, but it captures some of the character of
the curve.</p>
<p>Here are a few of the images that we produce.</p>
<figure class="center shadowed">
<img src="/wp-content/uploads/2022/12/mcportrait.8.24.1.13.png"
width="400px" />
</figure>
<figure class="center shadowed">
<img src="/wp-content/uploads/2022/12/mcportrait.10.18.0.1.png"
width="400px" />
</figure>
<figure class="center shadowed">
<img src="/wp-content/uploads/2022/12/mcportrait.10.72.1.1.png"
width="400px" />
</figure>
<p>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.</p>
<figure class="center shadowed">
<img src="/wp-content/uploads/2022/12/mc8.24.1.13.svg"
width="400px" />
</figure>
<figure class="center shadowed">
<img src="/wp-content/uploads/2022/12/mc10.72.1.1.svg"
width="400px" />
</figure>https://davidlowryduda.com/modcurvevizWed, 07 Dec 2022 03:14:15 +0000