Persiflage

Galois Representations for non self-dual forms, Part II

Posted on April 21, 2013 by Persiflage

(Now with updates!)

Let’s recap from part I. We have a Shimura variety $Y$, a minimal projective compactification $X$, and a (family of) smooth toroidal compactifications $W$. We also have Galois representations of the correct shape associated to eigenclasses in

$H^0(X^{\mathrm{ord}},\xi_{\rho}).$

So at this point (well, not only at this point) there is some confusion. In the construction above, I am imagining that we are working with the rigid analytic space corresponding to the ordinary locus. But now there are some remarks in my notes about dagger spaces. Here is what I am imagining is going on. For any sufficiently small radius, we may consider the rigid analytic space $Y[\nu]$ which corresponds (on the moduli level) to the appropriate abelian varieties $A$ (with polarization and level structure and enomorphisms, blah blah) together with a canonical subgroup which (under some measure) is close to being ordinary. Then there is a “dagger space” $Y^{\dagger}$ which is the limit of all such spaces. The issue (for me) is that I don’t really know anything about dagger spaces, but since this is probably not the main point, I will (again) elide the issue here. Of course, the goal is to realize the eigenvalues of the Eisenstein series $\Pi$ inside this cohomology. Let’s assume that $\Pi$ actually has good reduction at $p$. Then it is probably going to be true that $\Pi$ actually has finite slope, and so it lives inside the cohomology of some overconvergent neighbourhood of $X^{\mathrm{ord}}$. So there’s some flexibility with exactly what spaces one is working with. Perhaps working with finite slope eigenforms might help to get local-global compatibility at $p$.

(update: it’s most natural to work with the dagger spaces (whose cohomology is as described above) since that most naturally corresponds to the rigid cohomology groups occurring below.)

OK, so, we may take the direct limit over all compact subgroups $U$ of the cohomology above, and we want to realize the Eisenstein series $\Pi$ as a p-adic cusp form inside this space.

To this end, one introduces the following cohomology groups:

$ H^*_{c,\partial}(\overline{X}^{\mathrm{ord}}) := \mathbb{H}^{*}(W^{\mathrm{ord}}, \Omega^{\bullet}_{W^{\mathrm{ord}}}(\log \infty) \otimes
\mathcal{L})$

OK. So this is just a definition, it isn’t supposed to obviously be functorial: we are taking the special fibre, lifting to characteristic zero, taking a toroidal compactification, then looking at the hypercohomology of the de Rham complex with log poles at the boundary. Well I guess one can do whatever one wants, I suppose.

So what is this? The hycohomology of the de Rham complex of a smooth variety $M$ with log poles along some divisor $D$ with normal crossings should just be the Betti cohomology of the complement of $D$ in $M$. The factor $\mathcal{L}$ is the difference between the sub-canonical and canonical extensions, not entirely sure why it is there, presumably for some fundamentally important reason. So morally, I think the RHS should be computing something like the Betti cohomology of $Y^{\mathrm{ord}}$, with the proviso that these are dagger spaces, not smooth complex varieties. So one should think of the LHS is some type of algebraic Betti cohomology of the ordinary locus.

(update: the remark about the Betti cohomology of the complement of $D$ is correct, but the presence of the boundary divisor $\mathcal{L}$ is exactly what, in the classical sense, changes the answer from the cohomology of the open variety to the interior cohomology. So the cohomology is somehow compactly supported towards the boundary of $W$, but not the “other” part of the boundary (that is, the difference between $W$ and $W^{\mathrm{ord}}$.)

Let’s write down a spectral sequence:

$ H^i(W^{\mathrm{ord}}, \Omega^{\bullet}_{W^{\mathrm{ord}}}(\log \infty) \otimes
\mathcal{L}) \Rightarrow H^{i+j}_{c,\partial}(\overline{X}^{\mathrm{ord}}),$

The existence of this spectral sequence must be a formal consequence of the definition and properties of hypercohomology. Note that the $\Omega^j_{W^{\mathrm{ord}}}(\log \infty)$ are canonical automorphic sheaves of the standard type, so with the boundary piece $\mathcal{L}$ the LHS consists of terms of the form $H^i(W^{\mathrm{ord}},\xi^{\mathrm{sub}})$. To compute these terms, one can push foward via the map $\pi: W \rightarrow X$ from the toroidal compactification to the minimal one. Then one notes that:

1. The higher direct images $R^i \pi_* \xi^{\mathrm{sub}}$ vanish.
2. Since $X^{\mathrm{ord}}$ is affinoid, its higher cohomology also vanishes.

The second point seems reasonable, I have no idea why the first is true. It is probably a really key point, which I might talk about in part III (note: RLT said nothing about this and there is no pre-print, so I have no idea how to prove this at the moment). Apparently it is important that one uses the subcanonical extension here. This implies that every class which occurs in the RHS in this new cohomology actually occurs in an $H^0$ term on the LHS. Now one has Galois representations of terms of the form $H^0(W^{\mathrm{ord}},\xi^{\mathrm{sub}})$, by the first construction – here it must be OK to pass between $W$ and $X$ using the Kocher principle. So we are reduced to showing that $\Pi$ contributes to this new cohomology $H^{\bullet}_{c,\partial}(X^{\mathrm{ord}})$.

(update: here is some more about higher direct images. Let’s say a little bit about what the toroidal compactifications look like. Let’s even imagine we are working with $\mathcal{A}_2$ and are looking at a cusp where one has purely toric reduction. For the purposes of computing the higher direct images all that matters is the formal completion of $W$, which at the boundary looks something like $Z/\Gamma$ for some toric variety $Z$ which is not of finite type. One shows that $H^i(Z,\mathcal{O}_Z) = 0$ using Cech cohomology for $i > 0$, which allows one to think of $Z$ as contractible. Then one would like to say that $H^i(Z/\Gamma,\mathcal{O}_Z)$ is also zero, which comes down to understanding the action of $\Gamma$ on $H^0(Z,\mathcal{O}_Z)$. Roughly one would like to say that $\Gamma$ acts with no fixed points and and use Shapiro’s Lemma. Back to the specific example, one finds that $H^0(Z,\mathcal{O}_Z)$ corresponds to positive semi-definite $2 \times 2$ matrices, and $\Gamma$ a finite index subgroup of $\mathrm{GL}_2(\mathbf{Z})$. Here one should be reminded of the $q$-expansions of Siegel modular forms at the cusp — recall that $q$-expansions are given in terms of such matrices whose coefficients are invariant under $M \mapsto X M X^{T}$. This action is free as long as $\det(X) \ne 0$; at the level of $q$-expansions this corresponds exactly to working with cusp forms; this is why working with the sub-canonical extension allows one to restrict the positive definite forms on which the action is indeed free. In the degenerate case when $n = 1$, then $\Gamma$ is trivial, and so it even acts freely on the non-cusp form $1$, which is why it doesn’t matter in that case.)

Note: the spectral sequences above is, like the Hodge-de Rham spectral sequence, a 1-st page spectral sequence. Thus the vanishing above does *not* imply that it degenerates. Moreover, it certainly won’t degenerate, since the RHS will turn out to consist of finite dimensional vector spaces, whereas the terms on the LHS are certainly not (as they are spaces of p-adic or overconvergent forms). (Note to self: compare to work of Coleman.)

The next point is the following. Suppose one now simply replaces $X^{\mathrm{ord}}$ by $X$. Then the cohomology theory $H^{\bullet}_{c,\partial}$ is probably *literally* computing the Betti cohomology of $Y$. The Betti cohomology of $Y$ does indeed see the classes coming from the boundary that we would like to find.

Recall that $W \setminus Y$ is a normal crossings divisor. Let $\partial_0$ denote the variety, $\partial_1$ the (disjoint) union of the irreducible components of the boundary divisor, $\partial_2$ the union of the intersection of these components, and so on. One now writes down another 1st page spectral sequence as follows:

$\mathbb{H}^j(\partial_i, \Omega^{\bullet}_{W^{\mathrm{ord}}}(\log \infty)) \Rightarrow H^{i+j}_{c,\partial}(X^{\mathrm{ord}}).$

This is supposed to be an example of the following: in a nice geometric situation (normal crossings divisor) one may compute cohomology with compact supports in terms of the cohomology of the boundary strata. (I’m still a little confused why $H^*_{c,\partial}$ is cohomology with compact supports rather than the cohomology of the interior, but anyway…(update: this is explained above: the presence of $\mathcal{L}$ means it has compact supports in the direction of $W\setminus Y$, but not $W \setminus W^{\mathrm{ord}}$)). Moreover, a key point is that the LHS can be interpreted as the rigid cohomology of $\partial_i$. This allows one to use results of Berthelot and Chiarellotto to deduce that the terms of the LHS are given in terms of the rigid cohomology of (open) varieties. In particular:

1. They admit a theory of weights,
2. $H^j$ is mixed of weight at least $j$.
3. They are finite dimensional.

We deduce that RHS is also mixed of weight at least $i+j$ and finite dimensional.

We want our $\Pi$ to occur in the RHS, so it certainly suffices to show it actually occurs in $H^0$. But then by weights it suffices to show that it is coming from the $H^0$-terms in the LHS. These are simply given by component groups, and so the computation reduces to a problem concerning the combinatorics of the boundary, on which we shall say more in part III.

Posted in Mathematics | Tagged Galois Representations, Jack Thorne, Kai-Wen Lan, Michael Harris, RLT | Leave a comment

Cambridge Days

Posted on April 10, 2013 by Persiflage

Dinner at the Helmand: Zagat rating 27 (Food) and 20 (Service). After I was seated, it was fifteen minutes until I was brought a menu. Sixty seconds later, the waitress asked what I wanted to order, and then pouted when I said I needed a little extra time. References to the owner being a brother of Hamid Karzai were far less conspicuous then on my last visit. The bread is great!

Voltage coffee and art with Sug Woo Shin: pretentious in a hipster kind of way, elegant design, a payment system that involves an ipad, and a serious attitude towards coffee; in otherwords, my kind of place. I ordered a cortado, which was everything one could ask it to be. Who would have thought even five years ago that Kendall square would have a place like this!

Poonen can give a beautiful exposition of Coleman-Chabauty in 10 minutes. And he knows how to wield it too; see the recent beautiful result of Poonen-Stoll.

Seminars by Calegari & Calegari at MIT and Harvard. This off-broadway show will come to Chicago soon, book your tickets now!

Suppose that $X$ is a PEL Shimura variety. Can one classify all $X$ for which the image in $A_g$ lands in the Torelli locus? In Rachel Pries’ talk at MIT, a result of Moonen was mentioned which classified the $A/X$ which satisfied certain exceptional isomorphisms with families of Jacobians of curves which are cyclic covers of the projective line, but I wasn’t sure whether this was supposed to give the complete answer to this question.

Sitting next to me in Rachel’s talk: David Treumann, who has a nice paper with Venkatesh on the way (I’ll update the link when it becomes available.)

Unconfirmed Rumor: Mark Kisin’s student Thanos Papaïoannou will be the next president of Cyprus.

Posted in Waffle | Tagged Bjorn Poonen, Calegari, David Treumann, Helmand, Shimura Varities, Voltage | Leave a comment

Galois Representations for non self-dual forms, Part I

Posted on March 26, 2013 by Persiflage

This is the first of a series of posts discussing the recent work of Harris, Lan, Taylor, and Thorne on constructing Galois representations associated to regular algebraic automorphic forms for $\mathrm{GL}(n)$ over a CM field $F/F^{+}$. I will dispense with any niceties about why one should care, and try simply to decipher the scribbles I made during a talk RLT gave at the Drinfeld seminar. I should warn the reader of two difficulties: this paper does not exist as a public manuscript, and it also involves technical details which I generally prefer not to avoid thinking about. So caveat emptor.

First, some simplifying assumptions. Let’s assume that:

$\pi_{\infty}$ has trivial infinitesimal character.

$ \pi_p$ is unramified.

$F$ is an imaginary quadratic field in which $p$ splits.

For examples, I will generally consider the case $n = 1$ and $n = 2$.
The goal will be to construct a Galois representation

$R_p(\pi) = r_p(\pi) \oplus \epsilon^{1-2n} r_p(\pi^{c,\vee})$

If one can do this for $\pi$ and for $\pi \otimes \chi$ for enough characters $\chi$, then one can recover $r_p(\pi)$. Naturally enough, $R_p(\pi)$ will be associated to an automorphic form $\Pi$ for a bigger group. Now $\pi \boxplus \epsilon^{1-2n} \pi^{c,\vee}$ is automorphic for $\mathrm{GL}(2n)/F$; it is, moreover, an essentially conjugate self-dual (RAESD) although no longer cuspidal. It does, however, come from a smaller group, namely, the unitary similitude group $G$ which is ubiquitous in the papers of of Harris and Taylor. Over the complex numbers, $G$ looks like $\mathrm{GL}(2n) \times \mathrm{GL}(1)$, but over the real numbers I think it must look like $\mathrm{GU}(n,n)$. Although it’s true that the natural — i.e. occurring in cohomology of $X(G$) — Galois representations associated to RAESDC forms $\varpi$ for $G$ will actually be nth exterior powers, I don’t think that matters so much, since once one has congruences between $\varpi$ and $\Pi$ one gets Galois representations of the right degree for $\Pi$.

OK. Now associated to $G$ and an open compact $U$ of $G(\mathbf{A}^f)$ one has three natural objects: a smooth quasi-projective Shimura variety $Y = Y_U$, a (typically non-smooth) normal minimal compactification $X = X_U$, and a (family of) smooth toroidal compactifications $W = W_U$. The complement of $Y$ in $W$ is SNCD (smooth normal crossing divisor). I’m using somewhat non-standard terminology as far as the letters go because I don’t want too many subscripts. If $n = 1$, then $Y$ is an open modular curve, $X = W$ is a smooth compactification, and the complement of $Y$ in $W$ is a finite number of points (cusps). If $n = 2$, then $Y$ has complex dimension $4$. More on that example later.

As usual, one has the Hodge bundle $\mathbb{E} = \pi_* \Omega^{1}_{A/Y}$, from which one may build automorphic bundles $\xi_{\rho}$ in the usual way for suitable algebraic representations $\rho$ of what I guess amounts to the levi of $G(\mathbf{C})$. In my notes I have written:

$\xi_{st} = \mathrm{st}_{\tau} \oplus \mathrm{st’}_{\tau’}$

Here $\mathrm{st}$ means the standard $n$-dimensional representation of $\mathrm{GL}_n$, and $\mathrm{st’}$ denotes the complex conjugate representation. One must have has $\mathbb{E} = \xi_{st}$, where the decomposition into a direct sum of two rank $n$-modules comes from the action of the auxiliary ring on the tangent space to the universal abelian variety (built into the definition of $G$ which I have omitted). I also have written:

$ \mathrm{KS} = \mathrm{st}_{\tau} \otimes \mathrm{st’}_{\tau’}$

This presumably relates to the Kodaira–Spencer isomorphism. It’s certainly consistent with a surjection:

$\bigwedge^2 \pi_* \Omega^{1}_{A/Y} \rightarrow \Omega^1_{Y/k}$

Now it turns out that $\xi_{\rho}$ extends to $ W$ in two natural ways, there is the canonical extension $\xi^{\mathrm{can}}_{\rho}$ and the sub-canonical extension $\xi^{\mathrm{sub}}_{\rho}$; they differ by the divisor corresponding to the boundary. Just as in the case $n = 1$, the bundle $\xi^{\mathrm{can}}$ should be though of as having log-poles at the boundary. Last but not least, for the one dimensional representation $\wedge^{2n}(\mathrm{st}_{\tau} \oplus \mathrm{st’}_{\tau’})$, one has the line bundle $\omega$ on $Y$. Denote the canonical extension of $\omega$ to $W$ by $\omega$. Then it turns out that $\omega$ is the pull-back of an ample line bundle $\omega$ on $X$. Of course, if $n =1$, then $\omega$ is what you think it is — well, almost, since we are using $GU(1,1)$ Shimura varieties rather than $\mathrm{GL}(2)$. However, for general $n$, things are a little trickier. For example, $\omega$ is ample on $X$, but not (in general) on $W$.

If $U$ is maximal at $p$, then the previous constructions also work over a finite field $k$ of characteristic $p$ and the appropriate smoothness claims are still true. One has the Hasse invariant $H$, which is a section of $\omega^{p-1}$ over $X/k$. Since $\omega$ is ample on $X$, the complement of the zero divisor of $H$ is affine, it is of course the ordinary locus. In particular, one has Galois representations of the correct flavor associated to forms in the infinite dimensional space

$H^0(X^{\mathrm{ord}}, \xi_{\rho})$

This follows in the “usual” way; RLT sketched an argument, it goes as expected, although I think the Kocher principle must have slipped in at some point.

So far, I haven’t really said anything related to the actual argument, but I think I will stop here for now. The next step is to connect $\Pi$ in any way to classes in the p-adic modular forms arising in the cohomology group above.

Posted in Mathematics | Tagged Drinfeld Seminar, Galois Representations, RAESDC, RLT | Leave a comment

Office Toys

Posted on March 22, 2013 by Persiflage

I think I need one of these for my office, possibly to point at undergraduates when they ask for a higher grade: $quantum computer$ .

For the full story, see here.

Posted in Waffle | Tagged Death Ray, NYT | Leave a comment

Straight from the top to the bottom

Posted on March 4, 2013 by Persiflage

Possibly the worst programme on radio is NPR’s “from the top.” Ostensibly the show is about showcasing young performers in classical music. Yet, in reality, it is all about the incredibly annoying host, Christopher O’Riley, for whom condescension has become an art form in itself. Whilst I wouldn’t call a 15 year old an adult, I can’t imagine why you would talk to a 15 year old you didn’t know as if you were asking a 5 year old what their favourite colour was. My impression – both now and when I was younger – is that talking to kids you have just met as if they are adults is almost always the right thing do to.

Posted in Music, Uncategorized | Tagged children, From the top, NPR, radio | Leave a comment

Stephen Fry

Posted on February 14, 2013 by Persiflage

Never let it be said that this blog shies away from confronting sacred cows. Today’s target: Stephen Fry. In the video below, Fry tries to have his cake and eat it too — calling out grammar pedants for “showing off their superior knowledge” whilst … showing off his own superior knowledge. Perhaps the qualities we most despise in others are the ones which we most fear in ourselves. Plus, “Stephen Fry’s America” is absolutely terrible. Full disclosure, my own grammar is a mess; I cannot return a positive answer to Lawren Smithline’s immortal question “do you write like a Harvard man?” (Case in point, I’m not sure if I should have added a period to that last sentence or not; my guess is that a full stop would be ok but a period is not.)

Posted in Uncategorized | Leave a comment

Torsion in the cohomology of co-compact arithmetic lattices

Posted on February 6, 2013 by Persiflage

Various authors (including Bergeron and Venkatesh) have shown that the cohomology of certain arithmetic groups have a lot of torsion. For example, if $\Gamma$ is a co-compact arithmetic lattice in $\mathrm{SL}_2(\mathbf{C})$, and $\mathcal{L}$ is an acyclic local system, then

$\log |H^*(\Gamma(N),\mathcal{L}) | \gg [\Gamma:\Gamma(N)].$

The proof relies on the fact that the difference $l_0$ in ranks of $\mathrm{SL}_2(\mathbf{C})$ and $\mathrm{SU}_2(\mathbf{C})$ is one. As the invariant $l_0$ grows, one expects there to be less torsion. How much torsion should one expect in general? I’m not sure I have an answer, but the point of this post is that Poincare duality gives a non-trivial bound, at least if one restricts to covers up a $p$-adic tower. Let $\mathbb{G}$ be a semi-simple group over $\mathbf{Q}$, Let $G = \mathbf{G}(\mathbf{R})$, let $K$ be a maximal compact, let $H^* = \bigoplus H^m$, let $\Gamma$ be a co-compact lattice, and let $\mathcal{L}$ be an acyclic local system. Suppose that $n = \dim(G)$ and $d = \dim(G/K)$. Then, for a fixed prime $p$ (for which $\mathbb{G}(\mathbf{Q}_p)$ is split) and varying $m$, I claim that one has the inequality

$\log |H^*(\Gamma(p^m),\mathcal{L}) | \gg [\Gamma:\Gamma(p^m)]^{1 – \frac{d}{n}}.$

An elementary exercise shows that $\mathcal{L}/p \mathcal{L}$ is trivial as a local system for $\Gamma(p^m)$ and large enough $m$. The inequality above can then be reduced to the following claim: there is an inequality:

$\dim H_*(\Gamma(p^m),\mathbf{F}_p) \gg p^{m(n-d)}.$

Assume otherwise. The main point is as follows: taking the inverse limit over all $m$, we obtain modules $\widetilde{H}_j$ over the Iwasawa algebra $\Lambda$. This algebra, by results of Lazard and Venjakob, is essentially a regular local ring, in particular, it makes sense to talk about the dimension of modules over that ring. If the inequality above does not hold, then these modules will have small dimension, explicitly, co-dimension greater than $d$. This is so small that Poincare duality will, Ouroboros like — swallow itself completely and collapse into nothingness. However, the only way that could happen is if there was nothing to start with, which is nonsense.

More mathematically, consider the completed homology groups

$\widetilde{H}_* = \displaystyle{\lim_{\leftarrow} } \ H_*(\Gamma(p^m),\mathbf{F}_p)$

The homology groups may be computed by a complex of free $\Lambda$-modules obtain by lifting an initial triangulation on the base. (Here one thinks of group cohomology as the cohomology of the associated arithmetic quotients, of course.) Poincare duality then explains what happens when one takes the dual of this sequence and considers the corresponding homology groups, namely, there is a spectral sequence:

$\mathrm{Ext}^i(\widetilde{H}_j,\Lambda) \Rightarrow \widetilde{H}_{d-i-j}.$

This spectral sequence might be more familiar to some readers if one imagines $\Lambda$ to be a field, in which case the zeroth Ext group is a Hom and the higher Exts vanish, and one obtains the duality isomorphisms between homology and cohomology over a field. Or, if $\Lambda$ was the integers, then then zeroth Ext group is a Hom, the first Ext group is torsion, the higher Ext groups vanish, and one obtains the usual short exact sequence comparing the dual of homology to cohomology up to a torsion error term.) The dimension assumption we made implies that the limits are small as $\Lambda$-modules, in particular that $\mathrm{Ext}^i(\widetilde{H}_j,\Lambda) = 0$ for all $i \le d$. The key here is a Theorem of Ardakov and Brown relating the size of the cohomology growth under towers to the codimension of the module. Yet putting this assumption into the spectral sequence shows that all terms with $i + j \le d$ vanish, and hence that $\widetilde{H}_{0} = \widetilde{H}_{d-d} = 0$. Yet it is easy to see that
$\widetilde{H}_{0} = \mathbf{F}_p$, and thus we have a contradiction.

In fact, this is the same argument that ME and I used to give lower bounds on torsion for $p$-adic analytic covers of $3$-manifolds. There is some slack where the argument can be improved – since one only needs vanishing for a triangular portion of the spectral sequence, you are in good shape if you have extra information about the lower rows. Of course, the real answer to the amount of mod p torsion in these towers (which is a different question to the original one of torsion over the integers) should be:

$\dim H_*(\Gamma(p^m),\mathbf{F}_p) \sim p ^{m(n-l_0)},$

where $l_0$ was defined above.

Edit: In a previous version of this post, I confused the roles of $\mathrm{dim}(K)$ and $d = \mathrm{dim}(G/K)$. For complex groups one has $n = 2d$, and this is asymptotically the correct estimate for simple real groups. In general, one has $n \ge (3/2)d$, with the worse case, ironically, corresponding to (any number of copies of) $\mathrm{SL}_2(\mathbf{R})$. So you get a bound of the form:

$\log |H^*(\Gamma(N),\mathcal{L}) | \gg [\Gamma:\Gamma(N)]^{1/3}.$

Posted in Mathematics | Tagged Akshay Venkatesh, Iwasawa Theory, Matthew Emerton, Nicolas Bergergon, Ouroboros, Poincare Duality, torsion | Leave a comment

A scandal in Romania

Posted on January 28, 2013 by Persiflage

I was invited to review some research proposals for the CNCS. They offered a modest remuneration for my time (something like €168, I believe). For privacy reasons I won’t comment on the proposals I read, suffice to say that they did actually exist (and I was impressed with the quality). However, in order to process my payment, they requested a surprisingly large amount of information, including a copy of my passport and bank account numbers. The process is long over (almost two months), but I have still have not been paid, and several emails to various people have gone unanswered. Perhaps I should check my credit report…

Posted in Mathematics, Waffle | Tagged Benedict Cumberbatch, Romania | Leave a comment

Small Cyclotomic Integers

Posted on January 26, 2013 by Persiflage

Julia Robinson is a famous mathematician responsible for fundamental work in logic and in particular on Hilbert’s Tenth problem. Less well known nowadays is that her husband, Raphael Robinson, was a number theorist at Berkeley. One question R.Robinson asked concerned small cyclotomic integers. Namely, let $\alpha$ be a cyclotomic integer, and suppose that every conjugate of $\alpha$ has absolute value at most $R$. Then what can one say about $\alpha$? If $R \le 1$, then Kronecker’s theorem says that $\alpha$ is a root of unity (this statement only requires that $\alpha$ is an algebraic integer). Robinson studied the problem of what happens when $R \le 2$ and also $R \le \sqrt{5}$. He made five conjectures concerning these questions, four of which were solved in the 60’s by Jones, Cassels, and Schinzel. Five decades later, Frederick Robinson (no relation!) and Michael Wurtz proved the last of these conjectures (while working with me as summer students), and their paper has just been accepted by Acta Arithmetica. In particular, they answer the following problem: if $\alpha$ is an algebraic integer the largest of whose absolute values is $R \le \sqrt{5}$, then what are the possible values of $R$? Two such families of such numbers are those of the form

$\zeta + \zeta^{-1}, \qquad i + \zeta + \zeta^{-1}$

for a root of unity $\zeta$. These give all $R$ of the form

$2 \cos(\pi/N), \qquad \sqrt{1 + 4 \cos^2(\pi/N)}.$

Note that these sets have limit points at $\sqrt{4}$ and $\sqrt{5}$ respectively. It turns out that there exactly two further exceptions, as follows:

$\displaystyle{\frac{\sqrt{3} + \sqrt{7}}{2}, \qquad \sqrt{\frac{5 + \sqrt{13}}{2}}}$

The first element is totally real and cyclotomic, and so manifestly occurs as such an $R$. The second turns out to be the absolute value of $1 + \zeta_{13} + \zeta^4_{13}$. The proof by Robinson and Wurtz actually applies to slightly larger values of $R$, and after the limit point $\sqrt{5}$ there is another gap, and the next smallest possible $R$ is

$|1 + \zeta_{70} + \zeta^{10}_{70} + \zeta^{29}_{70}| \sim \sqrt{5.017655 \ldots}$

The first two exceptional numbers turn up in relation to subfactors. How about the last example?

Posted in Mathematics | Tagged Cassels, cyclotomic integers, REU, subfactors | 2 Comments

Random p-adic Matrices

Posted on January 23, 2013 by Persiflage

Does anyone know if the problem of random matrices over (say) $\mathbf{Z}_p$ have been studied?
Here I mean something quite specific. One could do the following, namely, since $\mathbf{Z}_p$ is compact with a natural measure, look at random elements in $M_N(\mathbf{Z}_p)$ and then ask about the distribution of several obvious quantities as $N$ goes to $\infty$. For example, one can consider the rank of $M \mod p$, which translates into an elementary counting problem over $\mathbf{F}_p$. However, I don’t mean this, that would just be rubbish for my purposes. What I am looking for is something that models a random compact operator, and then I want to understand the behavior of the normalized eigenvectors as the eigenvalue
$\lambda \rightarrow 0$. To be concrete, let $B = \mathbf{Q}_p \langle T \rangle$ be the Tate algebra corresponding to the open unit ball. Then consider a “random” compact operator $U$ acting on $B$. What does random mean? This is a good question, to which I do not know the answer. But let me give several properties that it should satisfy. Because the ball $B$ is a disk, it is “dimension 2 as a real manifold”, and so — imagining that our compact operator is a $p$-adic avatar of $e^{-\nabla}$ for the Laplacian $\nabla$ — the eigenvalues of $U$ should satisfy Weyl’s Law:

$N(T):=\{ \# \lambda \ \| \ -v(\lambda) < T \} \sim \displaystyle{ \frac{\mathrm{Vol}(B)}{4 \pi}} \cdot T.$

Here $v(\lambda)$ denotes the valuation of $\lambda \in \overline{\mathbf{Q}}_p$. Ignoring the volume factor, this just means that the Fredholm determinant $\det(1 – U T)$ has a Newton Polygon with certain quadratic growth. I’m not sure exactly what ensembles one can come up with to define such operators, which is one of my questions. Let us also assume, although this may not be necessary, that $U$ is semi-simple and admits nice convergent spectral expansions. We can’t quite insist that $U$ is a self-adjoint operator, because one doesn’t have p-adic Hilbert spaces. For such an operator, what behavior should one expect of the normalized eigenvalues $\phi_j$ of $U$? For example, suppose one knows that the number of zeros of $\phi_j$ goes to infinity. What limit distribution should the zeros of $\phi_j$ satisfy when $\lambda \rightarrow 0$? (Somewhat troubling here is that the eigenvalues will lie in $\overline{\mathbf{Q}}_p$ in general and $\overline{\mathbf{C}}_p$ has compactness issues…)

As you might guess, this is related to p-adic arithmetic quantum CHAOS, a group of subjects which gets sexier every time an extra adjective is added, and will form part of my student project at the Arizona Winter School.

Posted in Mathematics | Tagged Arizona Winter School, Avatar, PAQUE, QUE, Random Matrices, Weyl's Law | Leave a comment

Galois Representations for non self-dual forms, Part II

Cambridge Days

Galois Representations for non self-dual forms, Part I

Office Toys

Straight from the top to the bottom

Stephen Fry

Torsion in the cohomology of co-compact arithmetic lattices

A scandal in Romania

Small Cyclotomic Integers

Random p-adic Matrices

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