Category Archives: math

New (review) paper: Recent progress on the Tate conjecture

My paper about the Tate conjecture for Bull. AMS is now available to view.

In it I survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely intertwined with other big problems in arithmetic and algebraic geometry, including the Hodge and Birch–Swinnerton-Dyer conjectures. I conclude by discussing the recent proof of the Tate conjecture for K3 surfaces over finite fields.

After returning the proofs to the AMS, it occurred to me that it could be helpful to readers if I recommended some available related videos. I was too slow for the AMS’s speedy production, however, so I make the recommendations here.

Videos

F. Charles. K3 surfaces over finite fields: insights from complex geometry (2015).

K. Madapusi Pera. Regular integral models for orthogonal Shimura varieties and the Tate conjecture for K3 surfaces in finite characteristic (2012).

D. Maulik. Finiteness of K3 surfaces and the Tate conjecture (2012).

Photo is from foldedspace.org.

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Now it’s about stacks (new paper)

I’ve posted an updated version of a paper on the arXiv, Hodge theory of classifying stacks (initial version posted 10 March 2017). We compute the Hodge and de Rham cohomology of the classifying space BG (defined as sheaf cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology. This is part of the grand theme of bringing topological ideas into algebraic geometry to build analogous organizing structures and machines.

In the first version of the paper, I defined Hodge and de Rham cohomology of classifying spaces by thinking of BG as a simplicial scheme. In the current version, I followed Bhargav Bhatt‘s suggestion to make the definitions in terms of the stack BG, with simplicial schemes as just a computational tool. Although I sometimes resist the language of algebraic stacks as being too abstract, it is powerful, especially now that so much cohomological machinery has been developed for stacks (notably in the Stacks Project). (In short, stacks are a flexible language for talking about quotients in algebraic geometry. There is a “quotient stack” [X/G] for any action of an algebraic group G on a scheme X.)

Using the language of stacks led to several improvements in the paper. For one thing, my earlier definition gave what should be considered wrong answers for non-smooth groups, such as the group scheme μp of pth roots of unity in characteristic p. Also, the paper now includes a description of equivariant Hodge cohomology for group actions on affine schemes, generalizing work of Simpson and Teleman in characteristic zero. There is a lot of room here for further generalizations (and calculations).

Photo: Susie the cat in Cambridge, November 2000.

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Very old paper: The cohomology ring of the space of rational functions

susie-papersThanks to Claudio Gonzales, who requested it, and to MSRI Librarian Linda Riewe, who found and scanned it, my 1990 MSRI preprint “The cohomology ring of the space of rational functions” is available on my webpage.

Abstract: We consider three spaces which can be viewed as finite-dimensional approximations to the 2-fold loop space of the 2-sphere, Ω2S2. These are Ratk(CP1), the space of based holomorphic maps S2→S2; Bβ2k, the classifying space of the braid group on 2k strings; and Ck(R2, S1), a space of configurations of k points in R2 with labels in S1. Cohen, Cohen, Mann, and Milgram showed that these three spaces are all stably homotopy equivalent. We show that these spaces are in general not homotopy equivalent. In particular, for all positive integers k with k+1 not a power of 2, the mod 2 cohomology ring of Ratk is not isomorphic to that of Bβ2k or Ck. There remain intriguing questions about the relation among these three spaces.

Since 1990, a few papers have built on this preprint, including:

J. Havlicek. The cohomology of holomorphic self-maps of the Riemann sphere. Math. Z. 218 (1995), 179–190.

D. Deshpande. The cohomology ring of the space of rational functions (2009). arXiv:0907.4412

Photo: Susie the cat in Cambridge c. 2001.

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Why I like the spin group / New paper: Essential dimension of the spin groups in characteristic 2

9df3daf31bf116de8c6d6bdddff66a7fjpg Mathematics is about rich objects as well as big theories. This post is about one of my favorite rich objects, the spin group, inspired by my new paper Essential dimension of the spin groups in characteristic 2. What I mean by “rich” is being simple enough to be tractable yet complicated enough to exhibit interesting behavior and retaining this characteristic when viewed from many different theoretical angles.

Other objects in mathematics are rich in this way. In algebraic geometry, K3 surfaces come to mind, and rich objects live at various levels of sophistication: the Leech lattice, the symmetric groups, E8, the complex projective plane,…. I’d guess other people have other favorites.

Back to spin. The orthogonal group is a fundamental example in mathematics: much of Euclidean geometry amounts to studying the orthogonal group O(3) of linear isometries of R3, or its connected component, the rotation group SO(3). The 19th century revealed the striking new phenomenon that the group SO(n) has a double covering space which is also a connected group, the spin group Spin(n). That story probably started with Hamilton’s discovery of quaternions (where Spin(3) is the group S3 of unit quaternions), followed by Clifford’s construction of Clifford algebras. (A vivid illustration of this double covering is the Balinese cup trick.)

In the 20th century, the spin groups became central to quantum mechanics and the properties of elementary particles. In this post, though, I want to focus on the spin groups in algebra and topology. In terms of the general classification of Lie groups or algebraic groups, the spin groups seem straightforward: they are the simply connected groups of type B and D, just as the groups SL(n) are the simply connected groups of type A. In many ways, however, the spin groups are more complex and mysterious.

One basic reason for the richness of the spin groups is that their smallest faithful representations are very high dimensional. Namely, whereas SO(n) has a faithful representation of dimension n, the smallest faithful representation of its double cover Spin(n) is the spin representation, of dimension about 2n/2. As a result, it can be hard to get a clear view of the spin groups.

For example, to understand a group G (and the corresponding principal G-bundles), topologists want to compute the cohomology of the classifying space BG. Quillen computed the mod 2 cohomology ring of the classifying space BSpin(n) for all n. These rings become more and more complicated as n increases, and the complete answer was an impressive achievement. For other cohomology theories such as complex cobordism MU, MU*BSpin(n) is known only for n at most 10, by Kono and Yagita.

In the theory of algebraic groups, it is especially important to study principal G-bundles over fields. One measure of the complexity of such bundles is the essential dimension of G. For the spin groups, a remarkable discovery by Brosnan, Reichstein, and Vistoli was that the essential dimension of Spin(n) is reasonably small for n at most 14 but then increases exponentially in n. Later, Chernousov and Merkurjev computed the essential dimension of Spin(n) exactly for all n, over a field of characteristic zero.

Even after those results, there are still mysteries about how the spin groups are changing around n = 15. Merkurjev has suggested the possible explanation that the quotient of a vector space by a generically free action of Spin(n) is a rational variety for small n, but not for n at least 15. Karpenko’s paper gives some evidence for this view, but it remains a fascinating open question. The spin groups are far from yielding up all their secrets.

Image is a still from The Aristocats (Disney, 1970). Recommended soundtrack: Cowcube’s Ye Olde Skool.

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WAGS @ Colorado State, 15–16 October

Cat_TailThe Fall 2016 edition of the Western Algebraic Geometry Symposium (WAGS) will be held at Colorado State University on the weekend of 15–16 October. In addition to an excellent program of talks, there will be a lively poster session.

Speakers are:
Enrico Arbarello, Sapienza Universita di Roma/Stony Brook
Emily Clader, San Francisco State University
Luis Garcia, University of Toronto
Diane Maclagan, University of Warwick
Sandra Di Rocco, KTH
Brooke Ullery, University of Utah

For more information and to register, see the WAGS Fall 2016 site.

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Our friend the Tate elliptic curve

Cat discovers genusRigid analytic spaces are all the rage these days, thanks to the work of Peter Scholze and his collaborators on perfectoid spaces. In this post, I want to briefly describe the example that inspired the whole subject of rigid analytic spaces: the Tate elliptic curve. Tate’s original 1959 notes were not published until 1995. (My thanks to Martin Gallauer for his explanations of the theory.)

Let {\bf C}_p be the completion of the algebraic closure of the p-adic numbers {\bf Q}_p. The difficulty in defining analytic spaces over {\bf C}_p, by analogy with complex analytic spaces, is that {\bf C}_p is totally disconnected, and so there are too many locally analytic (or even locally constant) functions. Tate became convinced that it should be possible to get around this problem by his discovery of the Tate elliptic curve. Namely, by explicit power series, he argued that some elliptic curves X over {\bf Q}_p could be viewed as a quotient of the affine line minus the origin as an analytic space: {\bf Q}_p^*/\langle q^{\bf Z}\rangle \cong X({\bf Q}_p).

Trying to make sense of the formulas led Tate to his definition of rigid analytic spaces. In short, one has to view a rigid analytic space not just as a topological space, but as a space with a Grothendieck topology — that is, a space with a specified class of admissible coverings. So, for example, the closed unit disc \{ z: |z| \leq 1\} acts as though it is connected, because its covering by the two disjoint open subsets \{ z: |z| < 1\} and \{ z: |z| = 1\} is not an admissible covering. (“Affinoids,” playing the role of compact open sets, include closed balls such as |z|\leq a for any real number a, but not the open ball |z|<1. An admissible covering of an affinoid such as \{ z: |z| \leq 1\} is required to have a refinement by finitely many affinoids.)

Tate’s formulas for the p-adic analytic map G_m \rightarrow X, modeled on similar formulas for the Weierstrass p-function, are as follows.

Theorem. Let K be a complete field with respect to a non-archimedean absolute value, and let q \in K^* have 0<|q|<1. Then the following power series define a isomorphism of abelian groups K^*/q^{\bf Z}\cong X(K), for the elliptic curve X below:

x(w)=\sum_{m\in {\bf Z}}\frac{q^m w}{(1-q^mw)^2} -2s_1

y(w)=\sum_{m\in {\bf Z}}\frac{q^{2m} w}{(1-q^mw)^2} +s_1,

where s_l=\sum_{m\geq 1}\frac{m^lq^m}{1-q^m} for positive integers l. The corresponding elliptic curve X in {\bf P}^2 is defined in affine coordinates by y^2+xy=x^3+Bx+C, where B=-5s_3 and C=(5s_3+7s_5)/12. Its j-invariant is j(q)=1/q+744+196884q+\cdots. For every element j\in K with |j|>1 (corresponding to an elliptic curve over K that does not have potentially good reduction), there is a unique q\in K with j(q)=j.

It is worth contemplating why the formulas for x(w) and y(w) make sense, for w\in K^*. The series both have poles when w is an integer power of {q}, just because these points map to the origin of the elliptic curve, which is at infinity in affine coordinates. More important, these formulas make it formally clear that x(qw)=x(w) and y(qw)=y(w), but the series do not obviously converge; the terms are small for m \rightarrow \infty, but they are large for m\rightarrow -\infty.

To make sense of the formulas, one has to use the identity of rational functions \frac{w}{(1-w)^2} = \frac{w^{-1}}{(1-w^{-1})^2}. As a result, the series for x(w) (for example) can be written as

x(w)=\frac{w}{(1-w)^2}+\sum_{m\geq 1}\big(\frac{q^mw}{(1-q^mw)^2}+\frac{q^mw^{-1}}{(1-q^mw^{-1})^2} -2\frac{q^m}{(1-q^m)^2}\big),

which manifestly converges. One checks from this description that the series x(w) satisfies x(qw)=x(w), as we want.

References:

S. Bosch, U. Güntzer, R. Remmert. Non-Archimedean Analysis. Springer (1984).

B. Conrad. Several approaches to non-Archimedean geometry. P-adic Geometry, 9–63, Amer. Math. Soc. (2008).

W. Lütkebohmert. From Tate’s elliptic curve to abeloid varieties. Pure and Applied Mathematics Quarterly 5 (2009), 1385–1427.

J. Tate. A review of non-Archimedean elliptic functions. Elliptic Curves, Modular Forms, & Fermat’s Last Theorem (Hong Kong, 1993), 162–184. Int. Press (1995).

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Book review: The Serre–Tate correspondence

This is not Serre's catFor the past month I’ve been punctuating my life by reading the correspondence between Jean-Pierre Serre and John Tate, recently published in two volumes. Anyone interested in the development of number theory and algebraic geometry will find something to enjoy here.

The book was presumably suggested by the success of the Grothendieck–Serre correspondence, published by the Société Mathématique de France in 2001 and in English translation by the American Mathematical Society in 2003. The Grothendieck–Serre correspondence, beyond its outstanding mathematical interest, has the additional personal fascination of Grothendieck’s story. At first a complete outsider to algebraic geometry, he becomes the master builder of the subject in the 1960s, before rejecting mathematics and, by the end, the rest of humanity.

By comparison, Serre and Tate are reasonable men. The attraction of their correspondence lies in the mathematical ideas that they gradually develop, over the years from 1956 to 2009. Some of the key topics are Galois cohomology (essentially created by Serre and Tate), Tate’s notion of rigid analytic spaces, the Tate conjecture on algebraic cycles, Tate’s invention of p-adic Hodge theory, and Serre’s work on the image of Galois representations, for example for elliptic curves.

Serre usually writes in French, and Tate in English; but both writers make occasional use of the other language for the fun of it.

One running theme is Tate’s reluctance to write up or publish some of his best work. Serre encourages Tate and edits Tate’s papers, but sometimes has to concede defeat. Mazur and Serre started to prepare the publication of Tate’s Collected Papers in about 1990, which would include letters and unpublished work; sadly, nothing has appeared. Serre reports that the AMS has revived the project, and concludes: “I cross my fingers.”

A major topic of the correspondence starting in the 1970s is the relation between modular forms and Galois representations. Deligne and Serre showed in 1974 that a modular form of weight 1 determines a Galois representation with image a finite subgroup of PGL(2,C). At that time, however, it was a serious computational problem to give any example at all of a modular form of weight 1 for which the image is an “interesting” subgroup (that is, A4, S4, or A5, not a cyclic or dihedral group). Tate and a group of students found the first example on June 21, 1974. Soon Tate becomes fascinated with the HP25 programmable calculator as a way to experiment in number theory.

Both Serre and Tate are strongly averse to abstract theories unmoored to explicit examples, especially in number theory. This is a very attractive attitude, but it had one unfortunate effect. One of Serre’s best conjectures, saying that odd Galois representations into GL(2) of a finite field come from modular forms, was formulated in letters to Tate in 1973. But for lack of numerical evidence, Serre ended up delaying publication until 1987. The conjecture played a significant role in the lines of ideas leading to Wiles’s proof of Fermat’s last theorem. Serre’s Conjecture was finally proved by Khare and Wintenberger.

Finally, the correspondence has its share of mathematical gossip. One memorable incident is the Fields Medals of 1974. Tate is on the Fields Medal committee, and Serre suggests “Manin-Mumford-Arnold” as not a bad list, with Arnold as the strongest candidate outside number theory and algebraic geometry. In the event, the award went only to two people, Bombieri and Mumford. At least in the case of Arnold, it seems clear (compare this MathOverflow question) that this was a disastrous result of official anti-Semitism in the USSR, with the Soviet representative to the International Mathematical Union, Pontryagin, refusing to allow the medal to go to Arnold.

I hope that some mathematical readers will go on from the Serre–Tate correspondence to Serre’s Collected Papers. Serre took the idea of cohomology from topology into algebraic geometry and then into number theory. He is one of the finest writers of mathematics. I recommend his papers without reservation.

Correspondance Serre–Tate, 2 volumes. Editée par Pierre Colmez et Jean-Pierre Serre. Société Mathématique de France (2015).

Photo was from the Cambridge branch of Cats Protection, but a different cat is now featured.

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