Simple local trace formulas, revisited

Posted on 2026/11/13 by wdjoyner

In the late 1980s and early 1990s I was fascinated by Arthur’s local trace formula and by Waldspurger’s treatment of it. I wanted to learn the subject, and the way I chose to learn it was to write a paper on a relatively simple case. Two papers have now come out of that, and both are on GitHub:
https://github.com/wdjoyner/trace-formulas/tree/main/local-trace

The main paper, “Simple local trace formulas for unramified p-adic groups” is the result of that original effort. It appeared in the Pacific Journal of Mathematics 154 (1992), 103–129, and I circulated a corrected and extended version in 1997. The idea is to take a spherical function f and a function φ supported on the regular elliptic set of an unramified p-adic group, and to compute the trace pairing of the two by truncation. The geometric side is an integral of products of elliptic orbital integrals. The spectral side is an integral over the unramified unitary characters against the Plancherel measure.

That paper suffered from some sloppy writing and, since the topic itself is so beautiful, I want to correct any errors I made that could mar the presentation. Recently, I revised, corrected and expanded the 1997 version with AI assistance. The revised paper is currently in draft 2, and its authors are listed as David Joyner and Claude. While Claude drafted the expanded paper, ChatGPT acted as a referee (twice), providing valuable help. (The acknowledgments from 1997 still stand: I thank J.-L. Waldspurger for pointing out an error in an earlier version, and M. Assem for helpful discussions.)

The second paper, “The spectral measure of an elliptic orbital integral, via Arthur’s local trace formula” is supplementary. It reproves some of the results of the first paper by a different route. In place of the truncation argument, it specializes Arthur’s local trace formula to the case I had restricted to: one spherical function and one function supported on the elliptic set. This paper is new. It exists because of a suggestion by an AI agent! Currently at Draft 5 its authors are also listed as David Joyner and Claude. This time, Gemini (the Argon model) refereed and Claude, as before, drafted the papers.

Each paper comes with a revision log that records every change, the referee reports, and the responses to them. They can be found in the github repo mentioned above.

On quadratic residue codes and hyperelliptic curves, revisited

Posted on 2026/11/09 by wdjoyner

I’ve recently been revisiting older mathematical papers with the help of AI, checking proofs, correcting errors, and looking for conclusions that were not apparent when the papers were written. Years ago I worked on symmetric objects in the theory of error-correcting codes and had the opportunity to teach coding theory at the Naval Academy. But I fell in love with a different topic, symmetric objects in graph theory, and stopped thinking about codes as an object of research. I’m learning to fall back in love with those old objects again.

For quadratic residue codes, the relationship with hyperelliptic curves gives a good example of how one topic can connect with a totally different one – the unity of mathematics. The connection comes from curves of the form

$X_S:\quad y^2=\prod_{a\in S}(x-a),$

where $S$ is a nonempty subset of the finite field $\mathbb{F}_p$ . Remarkably, the number of rational points on these curves is closely related to the weights of codewords in certain binary linear codes, called quasi-quadratic residue (QQR) codes. This means that a problem about constructing good codes can be translated into a problem about counting solutions to polynomial equations.

About 20 years ago I wrote a paper, “On quadratic residue codes and hyperelliptic curves” (published in Discrete Mathematics & Theoretical Computer Science 10 (2008), pdf here) to dive into that topic more. I had more questions than answers. Recall Goppa’s conjecture is that the binary asymptotic Gilbert-Varshamov bound is best possible for any family of binary codes. The asymptotic GV bound states that the rate $R=dim(C)/length(C)$ of any binary linear block code $C$ is greater than or equal to $1-H(\delta)$ , where $H(\delta)=\delta-\delta\log_2(\delta)-(1-\delta)\log_2(1-\delta).$ A refinement of the Bazzi-Mitter conjecture is given below and is still open.

Conjecture. There is a constant c < 2 such that $|X_s(\mathbb{F}_p)| \leq cp$ for infinitely many primes p ≡ ±3 (mod 8).
(a) This is equivalent to: the quaternary quadratic residue codes of length p have minimum distance $\geq \delta p$ for some fixed $\delta > 0$ , for infinitely many such p.
(b) If it holds with $c < 2(1 - H^{-1}(1/2))$ (which is about 1.779), then Goppa’s conjecture is false.

Now, 20 years later, I’d like to know what AI thinks of the questions and conjectures in that paper. The result is the AI-aided updated and revised version (2026 pdf here), all my questions were answered by Claude but the conjectures remain open.

Conjectural permutation decoding of some AG codes, revisited

In a related paper, “Conjectural permutation decoding of some AG codes” (ACM SIGSAM Bull. 39(1) (2005)26–32, pdf here), I tried to exploit the large automorphism group of the hyperelliptic curve $y^2=x^p-x$ over a finite field to construct good decoding methods for the associated Goppa code. The basic idea is that by permuting the coordinates of a received codeword message, with possible errors in transmission, to permute its error coordinates out of a chosen information set, making decoding simpler.

Open Question: How many errors can a given automorphism group guarantee to correct by permutation decoding, and how does the answer depend on the information set?

The original paper had two conjectures that addressed the above open question. (The conjectures are technical to state but see the original paper for details.) When I asked AI to referee this paper, ChatGPT and Claude showed both conjectures were false and provided specific counterexamples. They also corrected some sloppy writing on my part and the AI-aided revised paper is much better (pdf here). Again, it was a pleasure using one to referee the other while I was able to add or subtract material to improve readability and correctness. For more details, see the github README for that repo.

On the unitary dual of the metaplectic cover of SL(2), revisited

Posted on 2026/11/09 by wdjoyner

Over 25 years ago, I wrote a paper on the unitary dual of the n-fold metaplectic cover of $SL(2,F)$ , over a p-adic field (pdf of original), under some assumptions (such as p>2 and p does not divide n). It was mostly expository and never published, but I referenced it in other works. Although it boasted the overly pretentious title above, it does not give a full description of the unitary dual for arbitrary covering degree and wild residue characteristic.

I asked AI agents Argon-Gemini and Sol-ChatGPT to look it over and suggest corrections, each acting as referees. They found several errors, some serious, but they easily corrected them all. In addition, the paper was revised for readability, adding a literature review and glossary. The AI-aided corrected version is available (pdf here).

Here is an AI-generated summary of the corrected version:

This manuscript studies genuine representations of the $n$ -fold metaplectic cover $\widetilde{G}$ of $G=\mathrm{SL}(2,F)$ , where $F$ is a nonarchimedean local field containing the $n$ th roots of unity. The central complication is that the inverse image of the diagonal torus can be nonabelian. This affects its irreducible representations, orbital integrals, and principal-series characters.

Four main results

Representations of the covering torus. Write $\widetilde{A}$ for the covering torus, $Z_A$ for its center, and $d=n/\gcd(n,2)$ . In the tame setting, each genuine character $\lambda$ of $Z_A$ determines a unique irreducible genuine representation $\chi_\lambda$ of $\widetilde{A}$ , of dimension $d$ . Its character satisfies $\rm{tr}\chi_\lambda(a)=d\lambda(a)$ for $a\in Z_A$ , and $\rm{tr}\chi_\lambda(a)=0$ for $a\notin Z_A$ . Thus the trace vanishes away from the center of the covering torus.
Split orbital integrals. The corrected construction uses a subgroup that actually centralizes the element being integrated. For residue characteristic prime to $2n$ , the normalized split orbital-integral transform maps genuine compactly supported test functions onto the compactly supported, genuine, Weyl-invariant locally constant functions on $Z_A$ .
Principal-series character formula. For $\Pi=\rm{Ind}_{\widetilde B}^{\widetilde G}(\chi\otimes\delta_B^{1/2})$ , the character on regular split elements is $\Theta_\Pi(a)=\frac{\rm{tr}\chi(a)+\rm{tr}\chi^{\widetilde w}(a)}{|D(a)|^{1/2}}.$ The second term involves Weyl conjugation, not naive inversion. This distinction is essential for the character to transform correctly under the central roots of unity.
A restricted character transfer. Suppose $\gcd(p,2n)=1$ and either $n$ is odd or $4\mid n$ . A power map $\nu(h(t))=(h(t^d),1)$ gives an ordinary torus character $\eta_\lambda=\lambda\circ\nu$ . For test functions with matching normalized split orbital integrals, the corrected paper proves $\rm{tr}\Pi(\chi_\lambda)(\bar f)=\rm{tr}\rm{Ind}_{B}^{G}(\eta_\lambda\delta_B^{1/2})(f).$ It also establishes the existence of a split-matching function $f$ for each genuine test function $\bar f$ on the cover.

What remains unresolved?

Despite its title, the revised paper does not claim a complete classification of the genuine unitary dual. It surveys known complementary-series and supercuspidal results, but does not establish a uniform complementary-series theorem in arbitrary degree. Nor does the transfer theorem cover $n\equiv2\pmod4$ (including the double cover), elliptic orbital integrals, or a full correspondence of all admissible representations.

The revision therefore identifies a precise, provable core of the original project while making clear where further work is needed.

“A note on gaps between zeros of L-functions,” revisited

Posted on 2026/11/08 by wdjoyner

Over 40 years ago I was fascinated by analytic number theory. I think it sprung from being gifted as a teen a copy of Ornstein Ore’s book, Number theory and it’s history. In any case, I was focused on learning as much about the Riemann zeta function $\zeta(s)$ as I could. It was amazing to me then and now how much number-theoretic information a single function like that could encode. The non-trivial zeros $\rho=\beta + i\gamma$ of $\zeta(s)$ are supposedly all on the line $\beta=1/2$ . Using arguments of A. Fujii (On the difference between r consecutive ordinates of the zeros of the Riemann zeta function, Proc. Japan Acad., 1975) and A. Ghosh (On the Riemann zeta function—mean value theorems and the distribution of |S(t)|, J. Number Theory, 1983), I wrote a preprint A note on gaps between zeros of L-functions (pdf here) to understand a conjecture of Hugh Montgomery. Years later I generalized the result from $\zeta(s)$ to any Dirichlet series with several properties analogous those of $\zeta(s)$ (such as an Euler product and so on), using the setup I had in the first book I wrote, Distribution theorems of L-functions (Pitman Research Notes,1986).

I don’t remember if I submitted it and it got rejected, or if I just posted it on my faculty webpage and forgot about it. In any case, it was never published. But, with the presence of AI as a math “paper referee”, I asked chatgpt and claude to be referees. They were asked to read it and suggest corrections, if any, to imprecise mathematical arguments, or revisions for readability. This process when through several revisions, with one or the other acting as the referee for various revisions. This is the AI-aided corrected version (pdf here).

Satake homomorphisms for the 2-fold cover of p-adic SL(2), revisited

Posted on 2026/11/06 by wdjoyner

In very vague, rough terms, the Langlands program gives a way of understanding certain types of maps between two specific types of groups that are related in a particular way. One way I like to think about it is that it gives mathematicians a template for conjectures of such “correspondences” which are likely to be true. Around 20-30 years ago, I was very interested specifically in the metaplectic covers of SL(2,F), where F is a p-adic field. I seems to have roped a number of victims into thinking and talking about such things with me. My colleagues Courtney Moen (who has retired from mathematics) and Jason Levy (who seems to have also left academia) are two examples.

Around 2004, with Jason, I wrote down a form of the correspondence that I thought the Langlands program template gave me as a conjectural “correspondence” between SL(2) over a p-adic field and the 2-fold cover. It’s very technical to write up, yes, but technically complicated and original are completely different things. I want to be clear, the notes/conversations had nothing original – we were writing down this template due to Robert Langlands in the case we wanted to investigate. I vaguely remember knowing that the conjectural guess was false, but it was just a template and I knew I’d have to sculpt it, like an artist, into a correct one. Sadly, my brain did not have the sharpness needed for the job, and I gave up.

That would have ended it forever if not for the facts that (a) I typed up notes of the template conjecture (or my version of how I remembered them) and, (b) now in 2026 20-something years later, AI is being used to prove the quasi-Riemann hypothesis. So, can it fix the false template guess I made long ago?

Thanks to both Opus-Claude and Sol-ChatGPT, the answer is yes. The AI-written paper The Satake Isomorphism, the Fundamental Lemma, and Transfer of Orbital Integrals for the Metaplectic Double Cover of SL(2, F) is the result. Basically, I tag-teamed one AI agent against another, one acting as the researcher and the other as the referee, and I would be adding or subtracting material at each stage to increase readability, preserve topicality, and maintain correctness. This went through 7 rounds of revision and the pdf above is the result.

The real work was done by AI, not Jason, not me. I think the reason it was so much fun working with that AI team in this way was solely because it taught me something new and interesting. I don’t like to give up on a problem. No one does. This helped me finish the job I gave up on in 2004.

Arithmetic of characters of generalized symmetric groups, revisited

Posted on 2026/11/04 by wdjoyner

Around 25-30 years ago, I was fascinated by the group structure of the Rubik’s cube group (whose size is roughly 4.3×10^19). It turns out (see my book on the Rubik’s cube for details) to be simply described using generalized symmetric groups. As a result, I wanted to know more about their representations and this lead to the question, what is the smallest field over which you can faithfully represent the generalized symmetric group (as matrices)? This lead to some lecture notes in 1998 and then a more refined preprint in 2002. After revisions, this was published in Archiv der Mathematik, vol 81 (2003)113–120.

What did AI think of this paper? While it found no serious errors (the main results and proofs were correct), Opus-Claude found an incorrect statement and that I’d mis-stated a result due to J.P. Serre. So I asked AI to write a more readable corrected version, adding a Glossary, Further directions paragraph, Literature review, and a Summary section. The corrected revision is here (11 page pdf). Hope you enjoy reading it as much as I did writing it!

Zeta functions and their associated operators, revisited

Posted on 2026/11/03 by wdjoyner

In 1981, as an undergraduate at Georgia Tech, I wrote a typewritten manuscript called “Zeta functions and their associated operators.” I’m not going to post it here. I think only Johan Belinfante (my professor at Georgia Tech) and John Benedetto (my PhD advisor at Univ Maryland) read it, and I hope to keep it that way! The notation is poor, the statements are poorly formulated, and the organization is worse.

But I remember what I was trying to say, and AI (mostly Opus-Claude, but also a lot of chatGPT) has the skills I lacked. This new paper, “A Grosswald–Schnitzer Invariance Theorem for Arithmetic Summation Operators,” is here: pdf.

I have to say, this co-working with AI was a very satisfying experience. I was able to take a paper I wrote as an undergraduate, which I still think has cool ideas, and put it into the form I like. Rereading that paper, I’m frustrated by my lack of skill as a writer and a disciplined thinker. It’s a huge relief to have something to show for all the thinking I did on it back in 1980-1981. The only thing I would not have dreamed of writing back then is (thanks to Opus-Claude) itemized in the “Future Directions” section at the end. Speculations of possible results that could be accomplished by going deeper with some of the ideas. I’ll come back to that later, but I’d also be happy if someone beats me to it.

$\begin{center} \renewcommand{\arraystretch}{1.3} \begin{tabular}{@{}>{\raggedright}p{0.44\textwidth}>{\raggedright\arraybackslash}p{0.48\textwidth}@{}} \toprule \textbf{Grosswald--Schnitzer Theorem} & \textbf{Operator analog}\\ \midrule perturbed primes $q_n$ & summation operator $E_q g(x)=\sum_\nu g(\nu x)$\\ the function $\zeta_q(s)$ & the operator $E_q$, via \eqref{eq:bridge}\\ factorization $\zeta_q=\varphi\,\zeta$ & factorization $E_q=\Phi\circ E_p$\\ $\varphi$ holomorphic and zero-free on $\Re s>0$ & $\Phi$ continuous and invertible on $\cS$\\ $\zeta_q$ and $\zeta$ have the same zeros & $\Ran E_q=\Ran E_p$\\ the zeros of $\zeta$ in $\Re s>0$ & the eigenvalues of $\Theta$ on $\cS/\Ran E_p$\\ a zero $\rho$ of multiplicity $m$ & a Jordan block of size $m$ for $\Theta$ at $\rho$\\ $1/\zeta_q=\sum_\nu\mu_q(\nu)\nu^{-s}$ & the M\"obius operator $W_q$, inverse of $E_q$ on $\Ran E_q$\\ the region $\Re s>0$ & the space $\cS$, built over $\Re s>0$\\ \bottomrule \end{tabular} \end{center}$

Odd king tours on even chessboards, revisited

Posted on 2026/10/31 by wdjoyner

Almost 30 years ago this year, a former student Michael Fourte and I wrote a combinatorics paper (original paper, 1997) with the title above, eventually published in the now defunct Journal of Recreational Mathematics 31(2003)173–177.That paper was a pleasure to write, but was it correct?

With AI tackling open math problems left and right, I asked AI to review our paper and correct it, if any errors were found. In fact, I expected no errors but it turns out the main theorem was not as carefully stated as it should be, and some conjectural statements were not formulated in the best way (though our proofs were correct). A corrected version, made with the help of AI, can be found here: 2026 corrected version. An AI-written report on the paper with some possible directions for further research is here: 2026 AI-written report/review.

“Invariant distributions on the n-fold metaplectic covers of GL(r,F), F p-adic”, revisited

Posted on 2026/10/30 by wdjoyner

Over 25 years ago I wrote a paper with the pretentious title “Invariant distributions on the n-fold metaplectic covers of GL(r,F), F p-adic” (J. Fourier Anal. and Appl. 7(2001)343-358.). In fact, it was on such invariant distributions restricted to a very special class of functions where everything was much simpler to describe.

Inspired by a blog post of Professor Alonso Castillo-Ramirez, I asked AI to look over this paper. Here, by AI I mean using both chatGPT-Sol and claude-Fable or claude-Opus, teaming one up against the other (so they had to find my errors and, if any, also those of the other AI agent). Apparently, my “simple” argument was a bit confusing to the AI I used. This in turn confused me, until I started to remember the argument. Then I explained the argument to AI, and it got back on track. None-the-less, I’m happy for the diversion because it helped AI look into generalizations. Anything leading to more interesting mathematics is a positive thing, IMHO.

The long story short here is that I was sloppy with some definitions and some hypotheses, and I apologize for that. The usual excuse is that family and work obbligations are to blame but in fact, I love hunting down new and interesting mathematical arguments but I’m less enthusiastic about double-checking details. So, it’s really my fault. However, the overall results are more-or-less correct and even appear (if you believe AI) to generalize significantly. I have not pursued these generalizations. Nor will I, as I’ve been retired and out of that field for some time and no longer have access to the needed literature. For anyone interested, available are the original 2000 paper, the AI-aided corrected paper (2026), and AI-generated notes on generalizing these results are also available. Just email me.

Zeros of some self-reciprocal polynomials, revisited

Posted on 2026/10/26 by wdjoyner

Inspired by a blog post of Professor Alonso Castillo-Ramirez, I asked AI to look over an old survey paper I wrote in 2012. Long story short, ChatGPT-Sol (OpenAI) and Claude-Fable (Anthropic) found and corrected several errors. Embarrassing. (For the full mea culpa, see the abstract of the pdf of the corrected version.) AI even proved a conjecture I made in the paper and pointed out that another conjecture I made was a consequence of an Annals of Math paper published in 2020 . It’s embarrassing making a mistake, especially in a survey paper (apologies to the original authors!), but there is a rainbow at the end of the road.