totallydisconnected

Three ideas for new journals

Accepta Mathematica.
You submit your paper to the editors like normal. It is scrupulously refereed by the usual process. If it is eventually accepted by the editorial board, it will be listed as accepted on the journal’s website. No issues are ever published, either digitally or in print.

Rejecta Mathematica.
A perfectly normal journal in every respect, except at the time of submission you must prove your paper has already been rejected by at least two other journals.

Viscount Journal of Mathematics.
Only publishes papers which have been rejected by Duke.

It’s been real. wait, that doesn’t sound right.

In all likelihood, there won’t be any more substantial mathematical posts on this blog. I just don’t have the time or inclination for it anymore. There will probably still be occasional posts about events I am organizing, hiring opportunities, and the work of my students.

See you around!

p-adic workshop in Singapore this November!

In the week of November 18-22, 2024, we will host a workshop on p-adic geometry here at NUS. The main focus of this workshop will be a series of four lecture courses, taught in person by Pierre Colmez, Shizhang Li, Lucas Mann, and Wiesława Nizioł. We especially hope this event will be helpful to early career researchers in the area, and that it will help foster new connections between the European and Asian arithmetic geometry communities. We will have some funding available for junior participants, but all mathematicians are welcome to attend!

More detailed information will be available soon. For now, mark your calendars!

The evolution of a concept

Bonus points if you can identify the first paper immediately!

.com is out, .net is in

My old website is gone because I’m useless with technology. My new website is davidrenshawhansen.net. It’s mostly broken right now but I’ll fix it in the next day or two. If you really need one of my papers before then, use the wayback machine.

Bontje, 8/25/2007-5/8/2023

Our beloved dog Bontje died yesterday. During her long and joyous life, she travelled widely, lived in countless cities on three continents, ate like royalty, barked at many eminent mathematicians, charmed strangers everywhere with her huge personality, and was deeply loved by J. and myself. We miss her terribly.

Distinguished affinoids

Fix a complete nonarchimedean field $K$ equipped with a fixed norm, with residue field $k$ . Let $A$ be a $K$ -affinoid algebra in the sense of classical rigid geometry. Here’s a funny definition I learned recently.

Definition. A surjection $\alpha : T_{n,K} \twoheadrightarrow A$ is distinguished if the associated residue norm $|\cdot|_\alpha$ equals the supremum seminorm $|\cdot|_{\mathrm{sup}}$ . A $K$ -affinoid algebra $A$ is distinguished if it admits a distinguished surjection from a Tate algebra.

Being distinguished imposes some obvious conditions on $A$ : since the supremum seminorm is a norm iff $A$ is reduced, it certainly it implies
1) $A$ is reduced.
Since any residue norm takes values in $|K|$ , it also implies
2) $|A|_{\mathrm{sup}} = |K|$ .

If $K$ is stable (which holds if $K$ is discretely valued or algebraically closed), then the converse is true: 1) and 2) imply that $A$ is distinguished. Since 2) is automatic for $K$ algebraically closed, we see that any reduced $K$ -affinoid is distinguished if $K$ is algebraically closed. It is also true that if $\alpha: T_{n,K} \to A$ is a distinguished surjection, then $\alpha^\circ: T_{n,K}^{\circ} \to A^\circ$ is surjective. Moreover, if $A$ satisfies 2), or $K$ is not discretely valued, then a surjection $\alpha: T_{n,K} \to A$ is distinguished iff $\alpha^\circ$ is surjective. Either way, if $A$ is distinguished then $A^\circ$ is a tft $K^\circ$ -algebra.

All of this can be found in section 6.4.3 of BGR.

Question 1. If $A$ is reduced, is there a finite extension $L/K$ such that $A\otimes_K L$ is distinguished as an $L$ -affinoid algebra?

This should be easy if it’s true. I didn’t think much about it.

Now suppose $A$ is distinguished, and let $\tilde{A} = A^{\circ} / A^{\circ \circ}$ be its reduction to a finite type $k$ -algebra. As usual we have the specialization map $\mathrm{sp}: \mathrm{Sp}A \to \mathrm{Spec} \tilde{A}$ . It is not hard to see that if $D(f) \subset \mathrm{Spec} \tilde{A}$ is a principal open, then $\mathrm{sp}^{-1}D(f)$ is a Laurent domain in $\mathrm{Sp}A$ . Much less obvious is that for any open affine $U \subset \mathrm{Spec} \tilde{A}$ , the preimage $\mathrm{sp}^{-1}U$ is an affinoid subdomain such that $A_U=\mathcal{O}(\mathrm{sp}^{-1}U)$ is distinguished and $\widetilde{A_U} = \mathcal{O}_{\mathrm{Spec} \tilde{A}}(U)$ . This is buried in a paper of Bosch.

Loosely following Bosch, let us say an affinoid subdomain $V \subset \mathrm{Sp}A$ is formal if it can be realized as $\mathrm{sp}^{-1}U$ for some open affine $U \subset \mathrm{Spec} \tilde{A}$ . Now let $X$ be a reduced quasicompact separated rigid space over $K$ . Let us say a finite covering by open affinoids $U_1=\mathrm{Sp}A_1,\dots,U_n= \mathrm{Sp}A_n \subset X$ is a formal cover if
1) all $A_i$ are distinguished, and
2) for each $(i,j)$ , the intersection $\mathrm{Sp}A_{ij}=U_{ij} := U_i \cap U_j$ , which is automatically affinoid, is a formal affinoid subdomain in $U_i$ and in $U_j$ .

This is a very clean kind of affinoid cover: we can immediately build a formal model for $X$ by gluing the tft formal affines $\mathrm{Spf}(A_i^\circ)$ along their common formal affine opens $\mathrm{Spf}(A_{ij}^\circ)$ . Moreover, the special fiber of this formal model is just the gluing of the schemes $\mathrm{Spec}\widetilde{A_i}$ along the affine opens $\mathrm{Spec}\widetilde{A_{ij}}$ .

Question 2. For $X$ a reduced qc separated rigid space over $K$ , is there a finite extension $L/K$ such that $X_L$ admits a formal affinoid cover?

When is it supercuspidal?

Today I want to talk about an amazing theorem of Moeglin, which I learned from WTG.

Let $F/\mathbf{Q}_p$ be a finite extension, $G=\mathrm{SO}_{2n+1}$ the split odd special orthogonal group over $F$ , $G'$ its unique inner form. By work of Arthur and Moeglin, there is a natural bijection between discrete series representations of $G$ or $G'$ , and pairs $(\phi,\chi)$ where $\phi: W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n}$ is a discrete L-parameter and $\chi$ is a character of the centralizer group $A_\phi$ . In this setting, $\phi$ is discrete if it is the sum of $m$ pairwise-distinct irreducible representations $\phi_i = \sigma_i \boxtimes [d_i] : W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{2n_i}$ with $\sum_{1 \leq i \leq m} n_i = n$ . Here $[d]:\mathrm{SL}_2 \to \mathrm{SL}_d$ is the usual d-1st symmetric power representation. The associated centralizer group $A_\phi$ is of the form $\{ \pm 1\}^m$ , and has a canonical basis indexed by the irreducible summands $\phi_i$ . Given $(\phi,\chi)$ , let $\pi(\phi,\chi)$ be the associated discrete series representation. Note that $\pi(\phi,\chi)$ is a representation of $G$ if $\chi$ is trivial on the evident subgroup $\{ \pm 1 \} = Z(\mathrm{Sp}_{2n}) \subset A_\phi$ , and is a representation of $G'$ otherwise. This splits the representations up evenly: for $\phi$ fixed, there are $2^m$ possible $\chi$ ‘s, and we get a Vogan L-packet $\Pi_\phi = \Pi_\phi(G) \cup \Pi_\phi(G')$ where $\Pi_\phi(G)$ and $\Pi_\phi(G')$ each contain $2^{m-1}$ elements.

Question. When is $\pi(\phi,\chi)$ a supercuspidal representation?

For $n=2$ I had previously memorized the (already complicated) answer to this question, so you can imagine my pleasure when I learned that Moeglin found a simple conceptual criterion which works in general! To state her theorem, we need a little vocabulary.

Definition. A discrete parameter $\phi=\oplus_i \phi_i$ is without gaps if for every $\sigma \boxtimes [d]$ occurring among the $\phi_i$ ‘s with $d \geq 3$ , then also $\sigma \boxtimes [d-2]$ occurs among the $\phi_i$ ‘s.

Definition. Suppose $\phi$ is without gaps. A character $\chi$ of the component group is alternating if for every pair $\sigma \boxtimes [d]$ and $\sigma \boxtimes [d-2]$ (with $d \geq 3$ ) occurring among the $\phi_i$ ‘s, $\chi(\sigma \boxtimes [d]) = - \chi (\sigma \boxtimes [d-2])$ . Moreover we require that on every summand of the form $\sigma \boxtimes [2]$ , we have $\chi(\sigma \boxtimes [2])=-1.$

Theorem (Moeglin). The representation $\pi(\phi,\chi)$ is supercuspidal iff $\phi$ is without gaps and $\chi$ is alternating.

Example 0. By definition, $\phi$ is supercuspidal if $d_i =1$ for all summands. In this case, $\phi$ is (vacuously) without gaps and every $\chi$ is (vacuously) alernating, so $\Pi_\phi$ consists entirely of supercuspidal representations. The converse – if $\Pi_\phi$ consists only of supercuspidals then necessarily $\phi$ is supercuspidal – is also immediate!

Example 1. Let $\sigma_2, \sigma_2':W_F \to \mathrm{SL}_2$ be distinct supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_2' \oplus \sigma_2' \boxtimes [3]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{10}$ is a discrete parameter without gaps, with component group of size 8. It is easy to see that four of the possible $\chi$ ‘s are alternating, and two of these are trivial on the center of $\mathrm{Sp}_{10}$ . Thus, the packets $\Pi_\phi(\mathrm{SO}_{11})$ and $\Pi_\phi(\mathrm{SO}_{11}')$ each contain four elements, two of which are supercuspidal and two of which are non-supercuspidal.

Example 2. Let $\sigma_2:W_F \to \mathrm{SL}_2$ and $\sigma_3:W_F \to \mathrm{O}_3$ be supercuspidal parameters. Then $\phi = \sigma_2 \oplus \sigma_3 \boxtimes [2] \oplus \sigma_3 \boxtimes [4]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{20}$ is a discrete parameter without gaps, with component group again of size 8. Now only two of the possible $\chi$ ‘s are alternating, and one of these is trivial on the center of $\mathrm{Sp}_{20}$ . Thus, the packets $\Pi_\phi(\mathrm{SO}_{21})$ and $\Pi_\phi(\mathrm{SO}_{21}')$ each contain four elements, one of which is supercuspidal and three of which are non-supercuspidal.

Example 3. Let $\tau:W_F \to \{ \pm 1 \}$ be a nontrivial character. Then $\phi = 1 \boxtimes [2] \oplus \tau \boxtimes [2]:W_F \times \mathrm{SL}_2 \to \mathrm{Sp}_{4}$ is a discrete parameter (vacuously) without gaps, with component group of size 4. Now only one of the possible $\chi$ ‘s is alternating, and it is trivial on the center of $\mathrm{Sp}_{4}$ . Thus, the packets $\Pi_\phi(\mathrm{SO}_{5})$ and $\Pi_\phi(\mathrm{SO}_{5}')$ each contain two elements, with $\Pi_\phi(\mathrm{SO}_{5})$ containing one supercuspidal and $\Pi_\phi(\mathrm{SO}_{5}')$ containing no supercuspidals.

More generally, if $\phi$ is without gaps and all $d_i$ ‘s are even, then only one $\chi$ is alternating, so the packet $\Pi_\phi$ contains a single supercuspidal representation (which may be a representation of $G$ or $G'$ – both possibilities occur) swimming in a sea of discrete series representations.

References:

Colette Moeglin, Classification des series discretes pour certains groupes classiques p-adiques, 2006

Bin Xu, On the cuspidal support of discrete series for p-adic quasisplit Sp(N) and SO(N), 2015

Report from Oberwolfach

Recently returned from a workshop on “Arithmetic of Shimura varieties.”

The organizers did a great job choosing the speakers. For one thing, the overlap between speakers this time and speakers at the previous edition of this event was a singleton set, which I think is a reasonable choice. Moreover, the majority of the speakers were junior people, which is also totally reasonable. It was great to hear what everyone is doing.
Best talks: Ana Caraiani, Teruhisa Koshikawa, Keerthi Madapusi, Sug Woo Shin, Joao Lourenco
Best chaotic talk with amazingly strong theorems: Ian Gleason
SWS has advised a very disproportionate number of Alexanders.
The usual hike wasn’t possible, due to snow in the mountains. Ah well. Instead we hiked along a path parallel to the road. But there was still cake.
The food was slightly better than usual: they didn’t serve the notorious bread casserole, and one dinner (the polenta thing) was actually really good.
“I mean, you know Ben. He’s pretty unflappable. But, yeah… [redacted], uh… flaps him.”
During the workshop, LM and I hit upon a conceptual explanation for Bernstein-Zelevinsky duality, which works both for group representations and for sheaves on $\mathrm{Bun}_G$ , even when $\ell=p$ ! More on this later.
“What was the motivation for this conjecture?” “The motivation was that it is true.”
Some young people have extremely weird expectations for how the postdoc job market should work.
The notion of “genericity” in various guises, and its relevance for controlling the cohomology of local and global Shimura varieties, was very much in the air. This came up in Caraiani and Koshikawa’s talks, and also in my (prepared but undelivered, see the first bullet above) talk. My handwritten notes are here, and may be of some interest. Conjectures 3 and 5, in particular, seem quite fun.
Had some interesting conversations with VL about nearby cycles and related topics. Here’s a concrete question: can the results in this paper be adapted to etale cohomology? There are definite obstructions in positive characteristic related to Artin-Schreier sheaves, but in characteristic zero it should be ok.
During the workshop, Ishimoto posted a beautiful paper completing Arthur’s results for inner forms of odd special orthogonal groups, at least for generic discrete parameters. I was vaguely sure for several years that this was the (only) missing ingredient in proving compatibility of the Fargues-Scholze LLC and the Arthur(-Ishimoto) LLC for $\mathrm{SO}_{2n+1}$ and its unique inner form. After reading this paper, and with some key assists from SWS and WTG, I now see how to prove this compatibility (at least over unramified extensions $F/\mathbf{Q}_p$ with $p>2$ ). It shouldn’t even take many pages to write down!
On a related note, shortly before the workshop, Li-Huerta posted his amazing results comparing Genestier-Lafforgue and Fargues-Scholze in all generality!

As always, Oberwolfach remains one of my favorite places to do mathematics. Thank you to the organizers for putting together a wonderful workshop!

Postdoc position at NUS

I’m looking to hire a postdoc here at NUS! This position is for two years, with the possibility of renewal for a third year, and carries no teaching duties. Ideally you will collaborate with me, but the job comes with near-total freedom to pursue your research. I also want to emphasize that Singapore is a beautiful country, with friendly people and amazing food, and it’s hard to imagine anyone regretting coming here for a few years.

Serious applicants, whose research interests are compatible with mine, are encouraged to apply via Mathjobs here. Although there is a quasi-official deadline of Jan. 31, in reality the position will stay open until I hire a suitable candidate, so late applications are welcome too.

I’m happy to answer further questions about the position via email.

	Arun Soor on Distinguished affinoids
	Sorry if I am wrong,… on When is it supercuspidal?
	totally disconnected on When is it supercuspidal?
	Sorry I could be wro… on When is it supercuspidal?
	vigneras on Questions on mod-p representat…