On the metaplectic analog of Kazhdan’s endoscopic lifting, revisited

Posted on 2026/11/15 by wdjoyner

In 1988, while living in Princeton just before moving to Annapolis for my first tenure-track job, I published a paper called “On the metaplectic analog of Kazhdan’s ‘endoscopic’ lifting”. I thought I knew the material well, but was overconfident and did a sloppy job writing it up.

I wonder if there is a rare psychological disorder called Metaplectic Moronicity, characterized by being around a lot of smart people who know a lot about metaplectic groups, the Langlands Program, and the beautiful number theory and function theory on such groups. This association, if it exists, thereby causes the afflicted person (me) to think they are also smart when they are really as thick as a brick. In any case, whether I had it or not, I think with such strong AI models available, I should do the honest thing and try to find corrections to the mathematical mistakes that crept into that paper.

In some sense, the Langlands Philosophy is a template for generating conjectures on the number theory and function theory of reductive algebraic groups. The hope back in the 1980s was to try to extend this to metaplectic covers (which are not algebraic), and I thought I could contribute to this maybe in some special cases, like SL(2) or GL(r). The above-mentioned paper was published in Isr. J. Math. 61 (1988)113-154. The AI-aided revision and correction is at this link (pdf).

True story: when work begun on the AI-aided revision I thought the paper had so many errors, it going to take a book to fix them! Not true. The original paper’s errors were patched (12 drafts, of going from one AI agent to referee the latest draft, another one to write a revision, and repeat) without serious difficulty. The mistake I made was in thinking an open problem mentioned at the end of the revised draft was actually a To Do item needed to complete the revision. No, it was just a related open problem that others can work on if they want (most of the AI-aided revisions I’ve been posting have such a section). Being confused, I actually kept the AI agents (Fable-claude, Carbon- and Argon-gemini, and Sol-chatgpt) working and working and they eventually turned it into a book!

To explain this very, very, very technical material in the book, I’m going to give some details. Sorry for this very long and technical blog (written with extensive help from the AI model Claude). It describes the above-mentioned paper and the book-length extension.

A metaplectic Jacquet–Langlands transfer: what is proved, what is open

The remainder of this post summarizes the mathematics in the book draft On a metaplectic Jacquet–Langlands transfer for division algebras of prime degree (Draft 3, November 2026), which is posted at github.com/wdjoyner/metaplectic-jl. It then compares the book with the 1988 Israel J. Math. paper, whose corrected 2026 version is the starting point of the book.

Every result in the book carries a status label: (P) proved here, (E) established in the literature, (O) proof outlined, (C) conjectural, and (P^c) proved modulo an exhaustive exact computation whose script is in the repository. I keep those labels below. The book does not claim the full metaplectic Jacquet–Langlands correspondence, and none of it has yet had independent human review.

The setting

Let $K$ be a totally imaginary number field containing the group $\mu_n$ of $n$ -th roots of unity, with $n$ odd. Let $r$ be a prime with $\gcd(n,r)=1$ , and let $D$ be a central division algebra of degree $r$ over $K$ , ramified at a finite set $S$ of places, with a cyclic maximal subfield $L$ .

The classical Jacquet–Langlands correspondence matches automorphic representations of $D^\times$ with those of $\mathrm{GL}(r)$ . The metaplectic question replaces both groups by $n$ -fold covers: the cover $\overline{G}\sb{D}(\mathbb{A}\sb{K})$ of $D^\times{\mathbb{A}}$ constructed in the 1988 paper, and the Kazhdan–Patterson cover of $\mathrm{GL}(r,{\mathbf{A}}_K)$ with twisting parameter $c_v$ . One then asks whether genuine representations (those on which $\mu_n$ acts through a fixed faithful character) still match. The twisting parameter is chosen so that $n \mid r-1+2rc_v$ . That condition does not make the cocycle exist (it is a cocycle for every $c_v$ ); it makes the whole preimage of the scalar matrices central.

A place $v$ of $S$ is tame if its residual characteristic does not divide $n$ , and wild if it does. The wild places are where most of the difficulty lives.

The running example. $K=\mathbb{Q}(\zeta_3)$ , $n=3$ , $r=2$ , $L=K(i)$ , and the quaternion algebra $D=(-1,4-\zeta_3)_K$ . Since $4-\zeta_3=(1-\zeta_3)(3+\zeta_3)$ , the algebra $D$ ramifies exactly at the prime $\mathfrak{p}_3$ above 3 (wild) and a prime $\mathfrak{p}_7$ above 7 (tame). Every chapter tests its constructions on this one example.

What the book proves

Chapter 4: one ambient cover, regular matching, and a conditional transfer.

Both covers sit inside a single cover of $\mathrm{GL}(r,\mathbf{A}_L)$ . This settles an identification problem that comes before any harmonic analysis: an isomorphism of central extensions is only determined up to a character valued in $\mu_n$ , so comparing “the same” genuine function on two differently built covers needs a canonical choice. (P)
Theorem A: for every maximal subfield $E\subset D_v$ the two covered elliptic tori have the same commutator pairing and the same good elements, namely the preimage of $F^\times E^{\times n}$ , of index $|n|_F^{-(r-1)}$ . (P) In the running example the covered tori are non-abelian at $\mathfrak{p}_3$ (index 9, irreducibles of dimension 3) and abelian at $\mathfrak{p}_7$ .
Theorem B: genuine test functions supported on the regular set match, with the sign $(-1)^{r-1}$ . (P)
Theorem C: a genuine automorphic representation of $\overline{G}_D(\mathbb{A}_K)$ with a supercuspidal component at some auxiliary place $w_0\notin S$ transfers to a genuine cuspidal representation of the cover of $\mathrm{GL}(r)$ with the same components outside $S$ , provided its packet satisfies a non-cancellation hypothesis $(\mathrm{Hyp}_D)$ . That hypothesis holds when the packet is finite. (P under the hypothesis)

Chapter 5: matching of arbitrary functions.

Theorem D: at every finite place, including the wild ones, a canonical $n$ -th root near the unipotent set gives a section that transports genuine orbital integrals near the center to linear germ expansions. The germs are the Shalika germs of $\mathrm{GL}(r,F)$ ; on the $D$ side only the trivial germ occurs. (P)
Theorem E: every genuine test function on either side has a matching function on the other, including functions that do not vanish near the identity. The matching function on the $\mathrm{GL}(r)$ side has the form $\tilde f_D(1),\tilde f^{\mathrm{St}}+\tilde f_{\mathrm{reg}}$ , a Steinberg-type piece plus a regular-supported piece. (P, using two linear inputs from the literature)
Theorem C’: the global transfer now needs elliptic support at only one place of $S$ , under a weaker hypothesis $(\mathrm{Hyp}_D')$ . That hypothesis fails exactly when the multiplicities at $S$ are proportional to the Plancherel weights, and it holds whenever the levels at $S$ are bounded. (P under the hypothesis)

Chapter 6: matching individual representations on the division-algebra side.

The chapter builds a unitary transform $L_D$ from class functions on $D_v^\times$ to genuine class functions on its cover, with normalizing constant $b_D=|n|_F^{-(r-1)/2}$ forced by unitarity. The question is whether $L_D$ sends irreducible characters to irreducible genuine characters (“integrality”).

At tame places it does, bijectively and preserving dimensions. Composing with the classical Jacquet–Langlands correspondence and Zou’s corrected Flicker–Kazhdan correspondence gives the local character relation in the range where that correspondence is available. (P)
At wild places the tame argument breaks down for a concrete reason. The cover no longer splits over the norm-one group, and every genuine irreducible has dimension divisible by $b_D$ . If integrality held, each irreducible $\rho$ would go to an irreducible $\tau$ with $\dim\tau=c_D\dim\rho$ , where $c_D=|n|_F^{-(r^2-1)/2}$ . At $\mathfrak{p}_3$ that means $b_D=3$ and $c_D=27$ , so the trivial representation would have to go to an irreducible genuine representation of dimension 27. Wild integrality in general is (C).

Chapter 7: toral positivity.

For a quadratic extension $E/F$ of odd residual characteristic and any odd $n$ with $\mu_n\subset F$ , the discriminant ratio $R_E(t)=\Delta(t)/\Delta(t^n)$ is a permutation character of a finite quotient of $E^\times/F^\times$ . So its Fourier coefficients are nonnegative integers, even at wild places. (P) The chapter also shows that wild integrality is equivalent to every transferred character being positive definite. Positivity on each torus does not imply positivity on the whole nonabelian cover; the book gives small counterexamples showing the gap is real. The chapter ends with a conditional global character comparison.

Chapter 8: the wild place of the running example.

At $F=\mathbb{Q}_3(\zeta_3)$ , with $n=3$ and $r=2$ :

Target A: $L_D\mathbf{1}$ is the character of a single genuine irreducible representation of dimension 27, induced from an explicit genuine character of a congruence subgroup. (P^c) The proof combines a Brauer reduction to the pro-3 Sylow subgroup, a finite design problem on $\mathbb{F}_3^5$ with a unique solution up to sign, and an explicit Hilbert-symbol computation.
Target B: for every irreducible $\rho$ trivial on the third congruence subgroup $K^3$ (levels 0, 1 and 2), and every norm twist of one, $L_D\Theta_\rho$ is a genuine irreducible character of dimension $27\dim\rho$ . (P^c)
Levels 3 and higher are open.

Open problems, and how feasible they look

What follows is my assessment, not a claim made in the book.

Level 3 at $\mathfrak{p}_3$ . The first unresolved case is an irreducible representation trivial on $K^4$ but not on $K^3$ . The local structure is explicit, so a computational attack in the style of Chapter 8 is plausible. But the book stopped at level two deliberately: the finite design grows quickly, and the authors judged that an inductive mechanism in the level is needed rather than one more case. Feasibility: within reach for a single level with more computation; not a route to the general case.
Wild integrality in general. This means every wild place, every prime degree $r$ and every admissible odd $n$ . Chapter 8 notes that its Brauer reduction should be uniform but its finite design is not. Even toral positivity, which is the necessary first step, is open for $r\geq 3$ . The book reformulates the problem as the statement that an explicit kernel is the character of a representation, which would then have a Howe-duality-type multiplicity-one property. Feasibility: the hardest problem in the book; it probably needs a structural idea, not more computation.
Removing the auxiliary supercuspidal place and the packet hypotheses. Theorems C and C’ fix an auxiliary place $w_0$ with a supercuspidal component and assume $(\mathrm{Hyp}_D)$ or $(\mathrm{Hyp}_D')$ . The natural tool is W.-W. Li’s invariant trace formula for covering groups. Feasibility: the tools exist, but it is long and technical work. A proof that packets are finite would remove the non-cancellation hypothesis on its own and may be the quicker target.
Split and archimedean places. A complete global comparison also needs local character comparison at the finite places outside $S$ (formulated conditionally in Chapter 7) and at the complex places. Feasibility: the complex places should be the easier part, since the cover splits there. The split finite places tie into the corrected Flicker–Kazhdan correspondence and inherit its current range.
Composing with the 1988 lifting. This is the step that would finish the original program: torus, then the cover of $D^\times$ , then the cover of $\mathrm{GL}(r)$ , then $\mathrm{GL}(r)$ itself, landing on automorphic induction. Feasibility: mostly bookkeeping once 3 and 4 are done, but it also depends on how much of Flicker–Kazhdan’s global correspondence survives Zou’s corrections.

Comparison with the 1988 paper (corrected 2026)

What the 1988 paper does. It proves a metaplectic analog of Kazhdan’s lifting, from the covered torus $\overline{T}(\mathbb{A}\sb{K})$ attached to $L$ to the cover of $D^{\times}\sb{\mathbb{A}}$ . Under the same standing assumptions as the book (and with the same cover), every genuine automorphic representation $\chi$ of the covered torus whose components at $S$ are not Galois-conjugate to themselves lifts to an $\varepsilon$ -invariant genuine automorphic representation $\overline{\pi}$ , where $\varepsilon$ is the order- $r$ character attached to $L$ . The unramified components are explicit: at split places, the unramified constituent of a genuine principal series; at inert places, the unique $\varepsilon$ -invariant unramified genuine representation with the given central character. There is a converse. The local inputs are:

twisted orbital integrals vanish at elements that are not good;
a metaplectic unit-element identity deduced from Kazhdan’s, using $\gcd(n,r)=1$ ;
uniqueness of $\varepsilon$ -invariant unramified representations;
the metaplectic fundamental lemma for the whole spherical Hecke algebra, proved directly at split places and by a globalization argument at inert places.

Different comparisons. The 1988 paper compares an $\varepsilon$ -twisted trace formula on the cover of $D^\times$ with the trace formula of a torus, the analog of automorphic induction. The book compares untwisted trace formulas of two different groups, the cover of $D^\times$ and the cover of $\mathrm{GL}(r)$ , the analog of Jacquet–Langlands. The 1988 paper names this second step as the missing piece and sketches it in eight steps (its Remark 10.3); the book is a sustained attack on that sketch.

The trace formulas. In the 1988 paper both sides have compact quotient, so the trace formula is simple with no conditions on the test functions. In the book the $\mathrm{GL}(r)$ side is not compact. That is why the book’s global theorems need an auxiliary supercuspidal place and a packet hypothesis, while the 1988 main theorem needs neither.

Separating representations. The 1988 paper separates its spectral identity using automorphic induction and strong multiplicity one for the linear group $\mathrm{GL}(r)$ ; no metaplectic multiplicity-one theorem is needed. The book cannot borrow those tools for the cover of $\mathrm{GL}(r)$ , and the packet non-cancellation hypotheses take their place.

Where the local difficulty sits. In the 1988 paper the hard local work is at unramified places (the fundamental lemma). At the ramified places of $D$ , transfer is easy, because twisted orbital integrals on a compact-mod-center group vanish near the center. In the book it is the reverse: unramified places are routine, and the ramified places carry all the weight, including matching across the center via germs and the wild places above $n$ .

The 1988 roadmap against the book. Remark 10.3 expected steps (1), (3) and (6) to be the hard ones, and the rest routine. In the book:

Step (1), local matching at $S$ , is fully done (Theorems A, B, D, E).
Step (3), the trace formulas, is done only with an auxiliary supercuspidal place.
Step (5), the spectral comparison, gives Theorems C and C’ under packet hypotheses.
Step (6), local character relations, is done at tame places in the range of the corrected Flicker–Kazhdan correspondence, and at the wild place of the running example for levels at most two.
Steps (7) and (8), composing with Flicker–Kazhdan and identifying the result with automorphic induction, are still open.

The main surprise relative to the 1988 sketch is the wild place. The remark expected a local Jacquet–Langlands correspondence via orthogonality relations, as in the linear case. The book shows that at wild places the covered side has a forced dimension factor ( $b_D$ , and $c_D$ in dimensions), so the correspondence cannot be a naive bijection of characters there. Making it work even for the trivial representation required the explicit 27-dimensional construction of Chapter 8.

Status. The 1988 paper’s main theorem is presented as proved, with one caveat recorded in the paper: the inert-place fundamental lemma rests on its global argument, and an independent explicit check is still wanted. The book’s local theorems are proved, but its global theorems are conditional, and its wild results depend on exact computations (P^c) that have not yet been rerun independently.

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.