- Algebraists eat corn row by row, analysts eat corn circle by circle. Yep, I eat down the rows like a typewriter. Why? Because it is the right way.
- This short paper by Johan de Jong and Wei Ho addresses an interesting question I’d never thought about; does a Brauer-Severi variety over a field K contain a genus-1 curve defined over K? They show the answer is yes in dimensions up to 4 (3 and 4 being the new cases.) In dimension 1, this just asks about covers of Brauer-Severi curves by genus 1 curves; I remember this kind of situation coming up in Ekin Ozman’s thesis, where certain twists of modular curves end up being covers of Brauer-Severi curves. Which Brauer-Severi varieties are split by twisted modular curves?
- Always nice to see a coherent description of the p-adic numbers in the popular press; and George Musser delivers the goods in Scientific American, in the context of recent work in cosmology by Harlow, Shenker, Stanford, and Susskind. Two quibbles: first, if I understood Susskind’s talk on this stuff correctly, the point is to model things by an infinite regular tree. The fact that when the out-degree is a prime power this happens to look like the Bruhat-Tits tree is in some sense tangential, though very useful for getting an intuitive picture of what’s going on — as long as your intuition is already p-adic, of course! Second quibble is that Musser seems to suggest at the end that p-adic distances can’t get arbitrarily small:

On top of that, distance is always finite. There are no p-adic infinitesimals, or infinitely small distances, such as the dx and dy you see in high-school calculus. In the argot, p-adics are “non-Archimedean.” Mathematicians had to cook up a whole new type of calculus for them.

Prior to the multiverse study, non-Archimedeanness was the main reason physicists had taken the trouble to decipher those mathematics textbooks. Theorists think that the natural world, too, has no infinitely small distances; there is some minimal possible distance, the Planck scale, below which gravity is so intense that it renders the entire notion of space meaningless. Grappling with this granularity has always vexed theorists. Real numbers can be subdivided all the way down to geometric points of zero size, so they are ill-suited to describing a granular space; attempting to use them for this purpose tends to spoil the symmetries on which modern physics is based.

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The corn-eating remark is hilarious. Unfortunately, I don’t remember how I eat corn to see if I’ m supposed to be algebraist or analytist…

The big question is: how do Bombieri or Mumford eat corn?

The Bruhat–Tits tree of SL_2 is regular with out-degree one more than the cardinality of the residue field. I think it’s still open whether there exist large Ramanujan graphs of degree 7.

Well, I eat corn the more efficient way, circle by circle. I guess I’m just a maverick.

I am a bit skeptical about splitting (generic) Severi-Brauer varieties of higher, prime order by the function field of a genus 1 curve C. This problem seems close to (but a bit easier than) the problem of splitting Severi-Brauer varieties by a cyclic extension, cf. Section 1 of “Open Problems on Central Simple Algebras”, http://arxiv.org/abs/1006.3304. Cyclic splitting fields also exist for small index, but these are expected to not exist for high index. Since C comes from a torsor for Pic^0(C)[n] for some integer n, then at least when the base field contains the algebraic closure of the prime field, the genus 1 curve has a rational point after passing to the compositum of two cyclic, degree n extensions. By Merkurjev-Suslin, the element in H^2(\mu_n) corresponding to the Severi-Brauer variety is a sum of “pure tensors”, i.e., elements in the image of the cup-product map H^1(Z/nZ) x H^1(\mu_n) –> H^2(\mu_n). The cyclic extension problem roughly asks if the element is a pure tensor, and the genus 1 problem “roughly” asks if the element is a sum of two pure tensors.

Jason, this requires some care — the extension generated by the p-torsion points has Galois group GL_2(F_p), not (Z/pZ)^2. So I’m not sure it’s that close to cyclic splitting fields. Even the question of whether the class [C] in H^1(G_K, Pic^0(C)) is split by a solvable extension is quite hard; Ciperiani and Wiles give a positive answer (mod local conditions) in the number field case here:

Click to access solvable.pdf

Dear Jordan,

I see your point — I thought the hypothesis about containing the algebraic closure of the prime field would be enough, but now I see it is not that simple.

… on the other hand, I guess it is clear that the cyclicity conjecture (which people actually expect to fail for higher order) would imply the existence of a morphism from an (isotrivial) genus 1 curve to the associated Severi-Brauer variety. So those few cases where cyclicity is known also give cases where the genus 1 curves also exist.

I saw that corn article, too; the thing is, I’m pretty sure that I eat row circle by circle. And, if I’m an analyst, those tendencies are buried pretty deep…