Jacques Menard, author of Nicholas Bourbaki

The “anonymously” written exercise linked over on E. Kowalski’s blog must easily be the funniest mathematico-literary joke of all time.  A short excerpt:

“Menard (perhaps without wanting to) has enriched, by means of a new technique, the halting and rudimentary art of mathematical writing: this new technique is that of the deliberate anachronism and the erroneous attribution. This technique, whose applications are infinite, prompts us to read “The Theory of Functions” by Titchmarsh as if it were posterior to Donaldson’s “Riemann Surfaces” and the book “Locales” of Prof. V.I. Siletzsky as if it were by Prof. V.I. Siletzsky. This technique fills the most placid works with adventure…”

Go read it right now!

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A quote which captures a certain truth

“The sickness of the Internet age is that we’ve started to think we deserve to be taken seriously about something without putting a third of our lives into it.”

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More from California

I spent a pleasant hour in a cafe in Oakland reconstructing the elementary proof of the “supplementary law” (\frac{2}{p})=(-1)^{(p^2 -1)/8}.  This is exactly the kind of tricky elementary argument I have trouble remembering if I’ve only see it once before (which was the case).  Anyway, the proof goes as follows: let P = -1\cdot 2 \cdot -3 \cdot 4 \dots \pm\frac{p-1}{2}.  We’ll evaluate this mod p in two different ways.  On the one hand, we clearly have P=\prod_{i=1}^{(p-1)/2} (-1)^i i = \frac{p-1}{2}! \cdot (-1)^{\sum_{i=1}^{(p-1)/2} i} =\frac{p-1}{2}!\cdot (-1)^{(p^2 -1)/8}.  On the other hand, mod we have P = -1\cdot 2 \cdot -3 \cdot 4 \dots \pm\frac{p-1}{2} = (p-1) \cdot 2 \cdot (p-3) \cdot 4 \cdot \dots \cdot \frac{p \pm 1}{2} = 2 \cdot 4 \cdot 6 \cdot 8 \cdot \dots \cdot (p-3) \cdot (p-1) =\frac{p-1}{2}! \cdot 2^{(p-1)/2}.

Related puzzle: what is the value mod of f_p = 1\cdot 3 \cdot 5 \cdot \dots \cdot (p-4) \cdot (p-2)?  The first few values are misleading!

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God’s greatest creation?

I’m in California for 11 days, for (mostly) math.  Here’s something fun I just learned from Dick Gross.  Let G be a split simple algebraic group over \mathbf{Z}, and let V be (a \mathbf{Z}-lattice in) an irreducible algebraic representation of G.  It’s not hard to check that if V is minuscule, the representation V \otimes \mathbf{F}_p is an irreducible G \otimes \mathbf{F}_p-representation for every prime p.  Are there any other examples of V with this property?  It turns out there is exactly one more example: the adjoint representation of E_8.  When I said it was surprising to me that another such example existed, Dick explained that I shouldn’t be surprised, because after all, “E_8 is the greatest group that God ever created, so it’s not surprising that it has amazing properties like this”.

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Period morphisms in p-adic Hodge theory

In classical Hodge theory, variations of Hodge structure give rise to varying periods, and then to period morphisms.  In p-adic Hodge theory, there are plenty of periods but (until recently) not so many period morphisms.  It turns out these exist in some ridiculous generality, and they have some interesting applications – you can read more here! (The linked thing was originally meant to be a blog post.)

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Thoughts from Oberwolfach, Part I

Just back from a week-long workshop at Oberwolfach on “Nonarchimedean geometry and applications”. Some random comments:

  • During the workshop, Kiran Kedlaya started a nonarchimedean Scottish book, to record open problems in the foundations of nonarchimedean geometry, especially in the theory of adic and perfectoid spaces. (See here if you don’t understand the name.) There’s no short supply of problems!
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  • On a related note, Kiran convinced me (against my will) that given some Huber pair (A,A^+), the question of whether or not the structure presheaf on X=\mathrm{Spa}(A,A^+) is a sheaf depends a priori on the choice of A^+! The issue is that for two different choices A^+ \subseteq A'^+ with associated X' \subseteq X, rational subsets match up but coverings by rational subsets might not. Depressing.
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    On the other hand, every practical criterion for sheafyness (e.g. locally Noetherian, f.g. as an algebra over a Noetherian ring of definition, perfectoid, stably uniform, etc.) is independent of A^+.
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  • Werner Lutkebohmert is very nice.
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  • There’s a natural notion of a “perfectoid Deligne-Mumford stack”. To make this precise, let K/\mathbf{Q}_p be a nonarchimedean field.  Then (provisionally) a perfectoid DM stack over K is a stack in groupoids \mathcal{X} on \mathrm{Perf}^{\mathrm{proet}}_{/ \mathrm{Spd}\,K} whose diagonal \mathcal{X} \to \mathcal{X} \times_{\mathrm{Spd}\,K} \mathcal{X} is representable in diamonds and which admits a surjective and pro-etale 1-morphism \mathrm{Perf}^{\mathrm{proet}}_{/U}\to\mathcal{X} for some U\in\mathrm{Perf}_{/ \mathrm{Spd}\,K}.  (As is customary, we’ll abbreviate this last thing to “a morphism U \to \mathcal{X}“.)
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    One can make examples of these things by analytifying classical Deligne-Mumford stacks over \mathrm{Spec}\,K. Not very surprising (although it requires proof).  But here’s a much weirder example: Take K=\widehat{F^{nr}} for some finite F/\mathbf{Q}_p, and let D be the division algebra over F with invariant 1/n.  Then there’s a natural action of D^{\times} on \mathbf{P}^{n-1}_{K}, and the groupoid [\mathbf{P}^{n-1,\lozenge}_{K}/D^{\times}] is a perfectoid DM stack over K. Roughly speaking, the idea is that the infinite-level Lubin-Tate space gives a pro-etale and D^{\times}-equivariant \mathrm{GL}_n(F)-torsor \mathcal{M}_{\mathrm{LT},\infty} \to \mathbf{P}^{n-1}_{K}, so then quotienting we get [\mathcal{M}_{\mathrm{LT},\infty}^{\lozenge}/D^{\times}]\to [\mathbf{P}^{n-1,\lozenge}_{K}/D^{\times}] surjective and pro-etale. But \mathcal{M}_{\mathrm{LT},\infty}\cong\mathcal{M}_{\mathrm{Dr},\infty} matches up with Drinfeld space at infinite level, and then [\mathcal{M}_{\mathrm{LT},\infty}^{\lozenge}/D^{\times}]\cong[\mathcal{M}_{\mathrm{Dr},\infty}^{\lozenge}/D^{\times}]\cong \Omega^{n-1,\lozenge} is the diamond associated with the Drinfeld upper half-space. This gives the required U.  Anyway, I’ll try to write this up more carefully in the next version of this thing.
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    Nb. The notion of a perfectoid DM stack (and even of a perfectoid algebraic stack) will play a crucial role in Fargues’s geometrization of the local Langlands correspondence, so there’s ample motivation for considering these objects.
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  • Antoine Ducros takes very good photos.
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  • Prize for Best Dressed: Wiesława Nizioł, Fumiharu Kato.
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It’s good to be spectral

Here’s a nice little theorem in rigid analytic geometry.

Theorem. Let X=\mathrm{Sp}(A) be an affinoid rigid space, with Z=Z(f_1,\dots,f_n) \subset X a closed analytic subset and U \subseteq X an admissible open subset containing Z.  Then for some \varepsilon >0, the “tube” Z_{\varepsilon} = \left\{x \in X \mid |f_1(x)| \leq \varepsilon,\dots,|f_n(x)| \leq \varepsilon \right\} around Z is contained in U.

To quote from Conrad’s notes on nonarchimedean geometry: “This can be proved by the methods of rigid geometry, but the only proof along such lines which I know is long and complicated. A short proof was given by Kisin via Raynaud’s formal models…” Conrad then goes on to give a proof using Berkovich spaces.  Here we give a very short adic proof, with spectrality of qcqs adic spaces as the key ingredient.

Proof. WLOG U is qc (for any covering of U by affinoids, some finite subcover will contain all of Z). Passing to adic spaces, it clearly suffices to show that |Z_{\varepsilon}^{ad}| \cap |U^{ad}|^c = \emptyset for some \varepsilon > 0. Since |Z^{ad}| = \lim_{\leftarrow \varepsilon} |Z_{\varepsilon}^{ad}|, we clearly have \lim_{\leftarrow \varepsilon} |Z_{\varepsilon}^{ad}|\cap|U^{ad}|^c = |Z^{ad}|\cap|U^{ad}|^c=\emptyset. But |Z_{\varepsilon}^{ad}| \cap |U^{ad}|^c is a closed subset of the spectral space |Z_{\varepsilon}^{ad}|, so is spectral itself, and hence compact and Hausdorff in the constructible topology.  Therefore (putting the constructible topology on everything) \lim_{\leftarrow \varepsilon} |Z_{\varepsilon}^{ad}|\cap|U^{ad}|^c is an empty inverse limit of compact Hausdorff spaces, and thus one of the terms in the limit must be empty. (A reference for this last implication is e.g. Proposition 1.1.4 in this book.)

People often say things like “Berkovich spaces are great because they’re locally compact Hausdorff and so you can make nice topological arguments on them”; the moral here is that adic spaces have the same flexibility once one passes to the constructible topology.

(I don’t know Kisin’s proof, but it must be related somehow, since the topological spaces underlying the adic spaces come from the inverse limit of all formal models.)

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