Subject: Re: two postings From: Gerd Laures Date: Fri, 01 Jun 2007 15:19:10 +0200 On Fri, 2007-06-01 at 08:28 -0400, Don Davis wrote: >> Two postings: A question and a conference update..........DMD >> _______________________________________________ >> >> Subject: possible question for the list >> From: Jack Morava >> Date: Fri, 1 Jun 2007 08:02:30 -0400 (EDT) >> >> I suspect the answer to this question is known (but not by me!): >> >> What is the (Ochanine-Landweber-Ravenel-Stong) elliptic genus of >> the quaternionic projective space HP_n? [Its A^-genus HP_n vanishes >> (if n > 0), for example because it admits a nontrivial circle action.] >> >> Help, go-to's, or references would be very much appreciated. >> >> Best, >> Jack It is the appropriate power of epsilon if n is even and 0 else. You can find it in Hirzebruch, Berger, Jung: manifolds and modular forms on page 29. Regards, Gerd ____________________________________________________________________ Subject: RE: two postings From: "John Greenlees" Date: Fri, 1 Jun 2007 17:17:07 +0100 Reference for Jack: MR1990835 (2004k:58035) Gálvez, Imma(4-SHEF-SM); Tonks, Andrew(4-LNDMU-CTM) Differential operators and the Witten genus for projective spaces and Milnor manifolds. Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 1, 123--131. 58J26 (55P60 57R20 57R75 58J20) ____________________________________________________________________ Subject: Re: two postings From: Peter Landweber Date: Fri, 1 Jun 2007 14:45:36 -0400 On Jun 1, 2007, at 2:32 PM, Peter Landweber wrote: > Dear Jack, > > Here's the answer to your question about the original elliptic genus of HP^n. > > Of course, one knows that for HP2 one gets epsilon (where delta and epsilon are the usual parameters; epsilon is also viewed as a modular form of weight 4). It says so on pages 4 and 56 of Springer LNM 1326. > > Now continue to page 57, to learn that the elliptic genus vanishes on HP^n when n is odd, while if n=2m its value is epsilon^m. I could dig this out from an unpublished paper, but I believe Hirzebruch found a nicer way to prove it: > > Hirzebruch, F. and Slodowy, P., Elliptic genera, involutions, and homogeneous spin manifolds, Geom. Dedicata 35 (1990), 309--343 > > Best wishes, > Peter > ________________________________________________________________________ Subject: Great! From: Jack Morava Date: Fri, 1 Jun 2007 14:50:29 -0400 (EDT) To: Peter Landweber Dear Peter, It's great to hear from you! Gerd Laures also wrote me about this; he points out that it's in Hirzebruch's book (p. 29). In fact I'm not very familiar with that -- I'd only seen it in German before a couple of weeks ago, and I think some vague memory of that must have prompted my question. Anyway here's a note I sent around after encountering the book: Hope things go well with you, thanks again & all best, (:+{)} Jack ---------- Forwarded message ---------- Date: Mon, 30 Apr 2007 19:20:10 -0400 (EDT) From: Jack Morava To: Michael J. Hopkins , Mark Mahowald Cc: Nora Ganter , MCKAY john , Haynes Miller Subject: cube root of Hirzebruch's problem? Hi folks: I'm using this bulk e-mail to record some folklore I learned last weekend at John McKay's birthday conference in Montreal, which might deserve wider dissemination: In {\sc Manifolds and modular forms} (which I'd only seen in German before the meeting) Hirzebruch et al construct an eight-dimensional (string) manifold $M_8$ with Witten genus $j(q3)^{1/3}$ (I think!). I believe this is the first in the sequence $M_{8k}$ of $8k$-dimensional manifolds you construct in \S 2 of your paper. It seems to be common knowledge in the Moonshine community that $j(q3)^{1/3}$ is the McKay-Thompson series associated to an element of order three in the Monster. The centralizer of such an element is [I'm told] Thompson's simple group, which constitutes a big chunk of $E_8({\mathbb F}_3)$. The existence of an interesting action of the Thompson group on (some manifold string cobordant to ?) $M_8$ would apparently be a corollary to Hirzebruch's prize problem. It occurred to me to wonder if $M_8$ might have a model which is an algebraic variety, maybe defined over $\mathbb{Z}$, whose mod three reduction might admit an action of $E_8(\mathbb{F}_3)$. [I guess it would have to be Calabi-Yau.] I don't have any feel for these things, this is really just a shot in the dark; but this corollary to H's question looks like it might be more accessible, so I thought you might be interested. The conference was great fun. All best, (:+{)} Jack