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

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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

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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

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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

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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

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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

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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.

On the teaching philosophy of Euphemia Haynes

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I’m not an expert on her teaching philosophy but John Stigall is (the author of John Stigall, Euphemia Haynes’ philosophy of mathematics and mathematics education, Journal of Philosophy of Education, 2026). He pointed me to some of Euphemia’s work on the philosophy of teaching. The video below is inspired by, and based on, some of Dr. Haynes’ lectures, research and scholarly work. I could not have made it without AI (mostly Claude).

It’s titled “Euphemia Lofton Haynes – Mathematics as an open door”

Related posts:

Translating Phillip’s 1860 memoir on the mathematics of telling time

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In 1860, a French mining engineer named Édouard Phillips published a 54-page memoir in the Journal de mathématiques pures et appliquées that (among other things) solved a problem watchmakers had been attacking by trial and error for nearly two centuries: what shape should the ends of a balance spring have, so that a watch keeps the same time whether its mainspring is freshly wound or nearly spent? The answer — two clean geometric conditions on the center of gravity of the spring’s terminal curve — is still the design rule behind every Breguet overcoil made today, and its modern descendants are etched into silicon hairsprings.

As far as I can tell, the memoir has never been translated into English. It now has been, and this post describes both the mathematics and the somewhat instructive process of producing the translation.

The problem

Huygens invented the spiral balance spring in 1674 (built by the horologist Thuret, with priority disputed at the time by Hooke and by the abbé Hautefeuille). Paired with a balance wheel, it forms a torsional oscillator — the timekeeping heart of every mechanical watch. The trouble is isochronism: the period should not depend on the amplitude of oscillation, but in practice it does, because the spring does not behave as an ideal torsion element. Pierre Le Roy discovered empirically in the eighteenth century that isochronism could be coaxed out of a spring by choosing its endpoints suitably. Nobody knew why, and nobody had a theory: the exact equations of a deformed elastic spiral are hopeless to integrate directly.

Phillips — an elastician who came to horology from the theory of railway carriage springs — found the way around. His memoir was examined for the Académie des Sciences by a commission of Mathieu, Lamé, and Delaunay, whose report praised it for making it possible “to substitute simple rules for the trial-and-error by which the makers of chronometers seek to obtain the isochronism of the oscillations of the balance.”

What Phillips proved

The engine of the whole memoir is one identity. Clamp both ends of the spring (as the collet and stud in fact do), rotate the balance through an angle \alpha, and let M be the flexural rigidity of the blade, L its length. Cutting the spring at an arbitrary point and balancing moments, then integrating along the blade, Phillips obtains

M\alpha = GL + L(Yx_1 - Xy_1),

where G is the restoring couple, (X, Y) is the reaction force at the balance pivot, and (x_1, y_1) is the center of gravity of the spring. If the second term vanishes, the torque law is exactly linear, G = M\alpha/L, and the swing time comes out as

T = \pi\sqrt{AL/M}

(A = moment of inertia of the balance) — independent of amplitude. Isochronism, then, reduces to killing the term L(Yx_1 - Xy_1), and Phillips identifies two sufficient conditions: keep the spring’s center of gravity on the balance axis, or make the pivot reaction vanish. He then proves a lovely equivalence: the pivot reaction vanishes if and only if the change of curvature is uniform along the blade — circles deform to circles, the coils “breathe” concentrically, exactly what one observes in a good chronometer spring.

The centerpiece is the terminal-curve problem: shape the end curve of the spring so that, under this uniform-curvature deformation, the center of the body coils stays put for every working \alpha. After a page of integration by parts and a linearization in \alpha, four conditions collapse to two statements about the center of gravity G of the terminal curve alone. If the curve has developed length \ell and joins the coils (radius \rho_0) at the point C:

  1. OG \perp OC — the center of gravity lies on the radius perpendicular to the junction radius;
  2. OG = \rho_0^2/\ell — its distance from the axis is the third proportional to the curve’s length and the coil radius.

These are the Phillips conditions. Along the way he proves that a circular arc can never serve as a terminal curve (explaining why the old practice of simply using a piece of the last coil had been abandoned as “vicious”), and he shows the conditions are independent of the blade’s cross-section, of the total spring length, and of the relative placement of the two end curves — which is precisely what leaves Le Roy’s length-adjustment method available as a final regulating trim.

The memoir then keeps giving. The terminal-curve conditions automatically place the center of gravity of the entire spring on the axis, by an elegant moment cancellation (m = m' = 2\rho_0^2\sin\tfrac{\beta}{2} for every opening angle \beta). Pushing to second order yields two further integral conditions, which Phillips converts — via the theorem of Pappus–Guldin — into statements about volumes of revolution that a draftsman can check from a drawing: a numerical-verification protocol, vintage 1860. There is a fully practical chapter on constructing compliant curves graphically, an analysis of the flat (watch) spring showing it can be isochronous only for small arcs unless terminated by a curve brought back toward the center — the mathematics of the Breguet overcoil — a proof that the work of deforming the spring depends only on its volume and the imposed strains, a temperature chapter showing the conditions are preserved under thermal dilation (and that thermal stress at the attachments vanishes if the terminal curve springs from the axis itself), and a closing analysis of Coulomb friction proving the beautiful classical result: friction shortens the outgoing half-swing and lengthens the return by exactly equal amounts, so the period between equilibrium passages is unchanged — friction decays the amplitude (linearly, a hallmark of dry friction) without perturbing the rate.

My favorite passages are the last ones, where Phillips confronts the theory with the workshop literature — Moinet, Saunier, the Tribune chronométrique — and shows the accumulated craft lore falling out as corollaries. A chronometer-maker in Dieppe, M. Jacob, found that one of Phillips’ theoretical curves was identical to a curve he had reached by trial and error a dozen years earlier and had used in instruments delivered to the navy. Theory recovering craft, then extending it with drawable rules.

The translation, and what it taught me about AI and old mathematics

The memoir is freely available on Numdam, but only in French, and only as a scan whose OCR text layer mangles the mathematics badly — 1860s typography, accents, braces spanning multiple lines. The translation was produced in collaboration with Anthropic’s Claude (the Fable 5 model), and the workflow turned out to be a case study in what such tools can and cannot do.

What the AI did well: the prose translation (Phillips’ mathematical French is formal and regular), the LaTeX typesetting, and — importantly — rederivation. Where the OCR was hopeless, the model reconstructed equations from the surrounding argument and verified the reconstructions symbolically (in SymPy); several nontrivial checks, such as the exact reduction of the two brace-grouped expressions for the moment (m) at equation (46), or the endpoint conditions on \int s^2 \cos(\theta_0'' - \theta_0)\,ds, were confirmed this way.

What it could not do, even when told the OCR was wrong: faithfully read a multiline displayed equation from the original. Every reconstruction was mathematically coherent, but coherence is not fidelity — the model substitutes derivation for transcription, and the result can differ from what Phillips actually printed (a missing equation here, a sign convention there, a coefficient absorbed into the wrong factor). In this project, essentially every displayed equation spanning more than one line required checking against the page images and manual retyping, and several equations present in the original had to be restored outright. The division of labor that finally worked: machine for derivation, symbolic verification, and typesetting; human eyes on the actual page for everything the page says. If you are contemplating a similar rescue of an old mathematical text, budget for that human pass — it is not optional.

The current version of the translation runs to about 36 pages: all fourteen sections of the memoir, all 89 numbered equations under their original numbers, the original figures, translator’s notes where conventions need flagging, and an appendix with annotated diagrams decoding Phillips’ notation (his accents mark evaluation at the endpoints, not derivatives; the letters A, G, and D each do double duty).

Links. The original: J. Math. Pures Appl. (2) 5 (1860), 313–366, on Numdam. The translation: PDF and a 12 page summary summary of Phillip’s memoir, including many details left out of this blog post.

A dice game of the Arapaho Indians as a coin-tossing game

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In Stewart Culin’s Games of the North American Indians, one of the games he describes is an Arapaho women’s basket dice game. The original game uses five two-sided dice-like pieces tossed in a basket. A simple mathematical way to understand it is to model it instead as a coin-tossing game with 2 nickels and 3 dimes.

This is not meant to say that the Arapaho players literally used coins. They did not. Rather, the coin model is a convenient combinatorial translation of the game. Two of the original pieces are of one type, and three are of another type. That makes the game mathematically equivalent to flipping two coins of one denomination and three of another denomination, then scoring according to the pattern of heads and tails.

A historical game made modern

Culin describes the game as using a small yucca-fiber bowl and five “dice” made of bone or plum stones. Two of the five must match each other in shape and marking, while the other three form another matching set.

The coin model

Suppose we replace the five original pieces by:

  • 2 nickels
  • 3 dimes

Let heads mean “marked side up” and tails mean “blank side up.” Then one throw of the Arapaho basket becomes one simultaneous flip of the 2 nickels and the 3 dimes.

Define:

  • N = the number of nickels showing heads, so N can be 0, 1, or 2
  • D = the number of dimes showing heads, so D can be 0, 1, 2, or 3

So each flip produces a pair (N, D). The important point is that the score depends on the distribution of heads between the nickels and dimes, not merely on the total number of heads. That is why this is not just a five-coin game with all coins interchangeable.

Scoring rule

A convenient mathematical version of the scoring is:

  • 8 points if both nickels and all three dimes are heads: (2,3)
  • 3 points if:
    • both nickels are heads and the dimes are not all heads: (2,0), (2,1), or (2,2), or
    • both nickels are tails and all three dimes are heads: (0,3)
  • 1 point if:
    • all five are tails: (0,0), or
    • exactly one nickel is heads and all three dimes are heads: (1,3)
  • 0 points in all other cases

If a flip scores, the player continues flipping. If a flip scores 0, the turn ends and the basket passes. As Culin says, a successful throw earns another throw, a failed throw passes, and a game is commonly played to 100 points.

The full outcome table

Nickels (N)Dimes (D)ProbabilityScore
001/321
013/320
023/320
031/323
102/320
116/320
126/320
132/321
201/323
213/323
223/323
231/328

Probability distribution of the score

Assuming the coin model uses fair independent coins, the probabilities are:

  • P(score = 8) = 1/32
  • P(score = 3) = 8/32 = 1/4
  • P(score = 1) = 3/32
  • P(score = 0) = 20/32 = 5/8

So the probability that a single flip scores anything at all is:

P(score > 0) = 12/32 = 3/8.

That means a player succeeds on 37.5% of flips and fails on 62.5% of flips.

Expected score

The expected score on a single flip is:

E(score) = 8(1/32) + 3(8/32) + 1(3/32) = 35/32 ≈ 1.094.

Because a player keeps flipping after a successful result, the expected score in one uninterrupted turn is larger. If we let T be the expected total score of one turn, then:

T = E(score) / (1 – P(score > 0)) = (35/32) / (1 – 3/8) = 7/4 = 1.75.

So in this idealized coin-tossing model, the average value of a single turn is 1.75 points.

References

  1. Stewart Culin, Games of the North American Indians (1907), Bureau of American Ethnology. Internet Archive edition: https://archive.org/details/gamesofnorthamer00culirich
  2. Text version of Culin’s work, including the discussion of dice games and the Arapaho women’s basket dice game: https://archive.org/stream/in.ernet.dli.2015.31459/2015.31459.Games-Of-The-North-American-Indians_djvu.txt
  3. This blog post draft was prepared with assistance from ChatGPT by OpenAI: https://openai.com/chatgpt