Danylo Radchenko and Campbell Wheeler have posted an extraordinary new paper in which they prove that Stark units for real quadratic fields are algebraic numbers. (Not yet a Shimura reciprocity law that shows these generate abelian extensions but hey, this is only part I.) This, to me, is arguably the most significant mathematical result of the past month. There is a lot of rich mathematics going on here, and I apologize for not having the time to even summarize it. But I didn’t want it to go unmentioned!
Update: There is now a “part II” as it were, posted here.
Yowza!
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Reading their article; somehow, the mathematics that seem to be involved feel more futuristic than any of the recent breakthroughs announced by OpenAI or Anthropic. What a cool approach taken to prove that certain varieties are zero-dimensional!
Such a cool paper! Even without any of the number theory applications, this solves one of my very favorite open problems in fusion categories.
Kinda kicking myself for not asking the right questions about https://arxiv.org/pdf/1002.0168 which we were writing at roughly the same time as our paper with you. There we were trying to make precise the sense in which certain fusion categories were “exotic” (i.e. unrelated to quantum groups at roots of unity) and noticed that one exotic category’s minimal field of definition was a non-abelian extension of Q, which makes it strikingly different. But the right thing to have noticed would have been that it is an abelian extension of Q(\sqrt{13}) and then try to use some already known generalizations of this construction to get other such extensions. In particular, up to a well-understood construction, the Haagerup example there is essentially the same thing as a Z/3 x Z/3 Izumi near-group category, and following Radchenko-Wheeler it’s exactly the category predicted to give the narrow ray class field of Q(\sqrt{13}) for (3).
(Of course this still wouldn’t get you close to what R-W have, since you can’t prove that all the relevant Izumi categories exist without the finite dilogarithm part of the paper. But with more thought it could have gotten to the idea that Izumi categories give an engine for producing abelian extensions of real quadratic fields.)
Your line « This is arguably the most significant mathematical result of the past month” made me laugh out loud. To me this announcement seems like one of the most striking things I’ve seen in my career. We live in astonishing times…
It might be worth pointing out that Gannon–Schopieray–Yadav obtained very similar results around the same time (https://arxiv.org/pdf/2609.25185), both constructing certain fusion categories using quantum dilogarithms (aka double sine functions) and observing that this gives algebraicity of special values of quantum dilogs (Remark 2.6 of their paper).
This work did use some AI, as disclosed by the authors. Yet another similar result was obtained by Huang, also with AI collaboration https://arxiv.org/pdf/2609.15986. It’s quite interesting to observe that similar results are obtained by different groups at the same time, some of whom are using AI and some of whom are not. I think something similar happened with Feige’s conjecture a while ago (if people still remember that).
I think they’re more similar from a tensor category viewpoint than they are from a number theory viewpoint?
Haagerup-Izumi categories (i.e. the ones constructed in GSY and H) are equivariantizations of Z/n x Z/n near-group categories. But just having the Z/n x Z/n near-group categories does not get you very close to Hilbert’s 12th. (AI tells me that Z/n x Z/n groups account for half the extensions when the negative Pell’s equation is solvable and none when it’s not.)
The timing is a bit odd, I wonder whether the Huang preprint pushed R-W to post faster than intended? But they’re very clear that they didn’t use AI substantially, and that’s consistent with both how their paper reads and the way that they handled their follow-up AI posting today. Perhaps also some of this was “in the air,” for example I’m going to an ICERM conference next summer whose abstract anticipates a connection between tensor categories and the Stark conjectures but predates all these papers https://icerm.brown.edu/program/topical_workshop/tw-27-dgs
Have you seen this? https://www.danrad.net/papers/stark-reciprocity.pdf