The nearly ordinary deformation ring is (usually) torsion over weight space

Let F/{\mathbf{Q}} be an arbitrary number field. Let p be a prime which splits completely in F, and consider an absolutely irreducible representation:

\rho: G_{F} \rightarrow {\mathrm{GL}}_2({\overline{\mathbf{Q}}}_p)

which is unramified outside finitely many primes.

If one assumes that \rho is geometric, then the Fontaine–Mazur conjecture predicts that \rho should be motivic, and the Langlands reciprocity conjecture predicts that \rho should be automorphic. This is probably difficult, so let’s make our lives easier by adding some hypotheses. For example, let us assume that:

  • A: For all v|p, the representation \rho|G_{v} is crystalline and nearly ordinary,
  • B: The residual representation {\overline{\rho}} has suitably big image (Taylor–Wiles type condition.)

Proving the modularity of \rho under these hypotheses is still too ambitious — it still includes even icosahedral representations and Elliptic curves over arbitrary number fields. Natural further hypotheses to make include conditions on the Hodge–Tate weights and conditions on complex conjugation.

We prove the following:

Theorem I: Assume, in addition to conditions A and B+ above, that

  • C: The Hodge-Tate weights [a_v, b_v] at each v|p are sufficiently generic,
  • D: If F is totally real, then there exists at least one infinite place such that \rho is even.

Then \rho does not exist.

The condition B+ is more restrictive than the usual Taylor–Wiles condition — we shall see from the proof exactly what it entails. Condition C will also be explained — but let us note that, for any suitable method of counting, almost all choices of integers are generic, even after imposing some condition on the determinant (say a_v + b_v is constant) to rule out stupidities.

One should think of this theorem as follows. If F is totally real, then condition D should be sufficient to rule out the existence of any automorphic \rho in regular weight, because (for motivic reasons) such representations should be totally odd. On the other hand, if F is not totally real, then the weights of any motive (with coefficients) should satisfy a certain non-trivial symmetry property with respect to the action of complex conjugation. So, for example, if F has signature (1,2), then either condition C or D should be sufficient, but we will require both. In fact, even condition C is stronger than what should be necessary. In addition to assuming regularity at all primes, it amounts to (on the representation theoretic side) insisting that none of the \mathrm{GL}_2(\mathbf{C}) weights are fixed by any conjugate of complex conjugation, whereas a single such example should be enough for a contradiction.

Perhaps a useful way to think about Theorem I is to make the following comparison. Hida proves the following theorem:

Theorem [Hida]: The nearly ordinary Hida family for \mathrm{SL}(2)/F is finite over weight space and has positive rank if and only if F is totally real and the corresponding {\overline{\rho}} is odd at all infinite places.

On the other hand, a consequence of Theorem I is:

Theorem II: The fixed determinant nearly ordinary deformation ring of a residual representation {\overline{\rho}} satisfying condition B+ is finite over weight space and has positive rank if and only if F is totally real and the corresponding {\overline{\rho}} is odd at all infinite places.

In both cases, I am only considering the deformation rings up to twist — the deformation ring of the character is torsion over the corresponding weight space whenever \mathcal{O}_F has infinitely many units. Also in both cases, it is of interest to determine the exact co-dimension of the ordinary family — this is a difficult problem, because strong enough results would allow you do deduce Leopoldt by considering induced representations.

OK, so what is the argument? If you have read some of my papers, you can probably guess.

Assume that \rho exists. Let U be the representation corresponding to \rho. Now replace U by V = {\mathrm{Sym}}^2(U). Now replace V by the tensor induction:

\displaystyle{ W = \bigotimes_{G_F/G_{{\mathbf{Q}}}}  V}

of dimension 3^{[F:{\mathbf{Q}}]}. We now let C be the condition that W has distinct Hodge–Tate weights. To see that this is generic, it really suffices to show that there is at least one choice of weights for which this is true. But one can let the weights of U up to translation consist of the 2-uples [0,1], [0,3], [0,9], etc. and then the weights of W are, again up to translation, [0,1,2,\ldots,3^{[F:{\mathbf{Q}}]} - 1]. We now let B+ be the condition that the residual representation is absolutely irreducible, and that the prime p > 2 \cdot 3^{[F:{\mathbf{Q}}]} + 1. This is generically true, and amounts to saying that the conjugates of {\overline{\rho}} under G_{{\mathbf{Q}}} are sufficiently distinct. Since the dimension of W is odd, and because it is essentially self-dual (exercise), orthogonal (obvious), nearly ordinary (by assumption), has distinct Hodge–Tate weights (by construction), satisfies the required sign condition (automatic in odd dimension), we deduce that it is potentially modular by [BLGGT]. In order to win, it suffices to show, by a theorem I made Richard prove, that the action of complex conjugation on W has trace \pm 1. However, this is equivalent to condition D (see below). QED.

One can relax condition B+ slightly by only inducing down to the largest totally real subfield of F. On the other hand, there are plenty of examples to which Theorem II applies. I think one can take any elliptic curve E/F without CM and such that j_E \in F does not lie in any subfield of F, and then take p to be any sufficiently large ordinary prime which splits completely in F (caveat emptor, I didn’t check this). Of course, the condition that p splits isn’t really necessary either, I guess…

The second theorem follows along the exact same lines — the conditions are strong enough to ensure, using results of Thorne, that the nearly ordinary deformation ring of (the now residual) representation W is finite over weight space, which translates back into finiteness of deformations of U over weight space. The result is obvious if F is totally real and {\overline{\rho}} is odd. Otherwise, we choose a sufficiently generic point in weight space (in the sense of C), and then, by Theorem I, we see that the specialization of the nearly ordinary deformation ring at that point must be torsion.

It remains to compute the sign of W. This is an exercise in finite group theory, we only recall enough of the details for our purposes. Let V be a representation of H of dimension d. Consider the tensor induction:

\displaystyle{\bigotimes_{G/H} \sigma V}.

Let T denote a set of representatives of right cosets of H in G. Let t g \in T denote the corresponding choice for the coset Htg. For g \in G, let n(t) denote the size of the \langle g \rangle-orbit which contains T. If g = c has order 2, then either n(t) = 1 or n(t) = 2 Certainly

t c^{n(t)} t^{-1} \in H, \quad t \in T.

Let T_0 be a set of representatives for the \langle g \rangle orbits on T_0. Then (proof omitted)

\displaystyle{\phi^{\otimes G}(c) = \prod_{t \in T_0} \phi(t c^{n(t)} t^{-1})}.

We observe that:

  1. If n(t) = 2, then \phi(t c^{n(t)} t^{-1}) = \phi(t t^{-1}) = \phi(1) = d.
  2. If n(t) = 1, then tct^{-1} \in H and \phi(t c t^{-1}) is what it is. For example, it is 0,\pm 1 if and only if V is {\mathrm{GL}}-odd with respect to tct^{-1}.

Now suppose that G = G_{{\mathbf{Q}}} and H = G_{F}. The elements tct^{-1} are exactly the different complex conjugations of the representations of the conjugates of H. We deduce:

  1. If \dim(V) is even, then W is {\mathrm{GL}}-odd if and only if there exists at least one real place of F such that V is {\mathrm{GL}}-odd.
  2. If \dim(V) is odd, then W is {\mathrm{GL}}-odd if and only if F is totally real and V is {\mathrm{GL}}-odd at every real place.

Equivalently, a product of even integers can equal zero only if at least one of them is zero, and a product of odd integers can equal \pm 1 if and only if all of them are \pm 1.

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Tricky Fingers

How is one supposed to play this exactly?

Contrapuctus XIII

One can neither can play a 14th in the right hand (my hands are not that big) nor play legato parallel 10ths in the left; hence some sort of arpeggiation is required. But I can’t quite seem to reproduce how Glenn Gould plays this measure. Then again, that phenomenon is not unique to this passage.

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The Artin conjecture is rubbish

Let \rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_N(\mathbf{C}) be a continuous irreducible representation. Artin conjectured that the L-function L(\rho,s) is analytically continues to an entire function on \mathbf{C} (except for the trivial representation where the is a simple pole at one) and satisfies a functional equation of a precise shape. Langlands later had the profound insight to link this conjecture to functoriality in the Langlands program, which would additionally imply that \rho is automorphic which implies, inter alia, that L(\rho,s) = L(\pi,s) for a cuspidal automorphic representation \pi for \mathrm{GL}(N)(\mathbf{Q}).

This is a beautiful and fundamental conjecture. However, it does appear to be completely useless for any actual applications. The most natural application of Artin’s conjecture is to prove … the Cebotarev density theorem. This is why Cebotarev’s density theorem is so amazing! True, one can upgrade the error estimates if one knows Artin, but to do this one also has to know GRH. And if you know GRH, you are not too far away from Artin anyway, because then L(\rho,s) at worst has poles on the critical strip, and so you can (essentially) get close to optimal bounds for Cebotarev anyway.

I thought a little bit about applications of Artin’s conjecture when I wrote a paper about it, but I came up empty. Then recently, I had occasion to look at my paper again, and found to my chagrin that when Springer made the final edit they lopped off a sentence in the statement of one of the main theorems. I guess that’s why the good people at Springer get paid the big bucks. (My best ever copy editing job, by the way, was for a paper in an AMS journal.) In a different direction, I guess it also reflects the deep study of this paper by people in the field that nobody has asked me about it. However, I did notice a statement in the paper that could be improved upon, which I will mention now.

To set the context, let K^{\mathrm{gal}}/\mathbf{Q} be a Galois extension with Galois group S_5, and suppose that complex conjugation in this group is equal to (12)(34). Now suppose that \rho is a representation of \mathrm{Gal}(K^{\mathrm{gal}}/\mathbf{Q}). We already know that L(s,\rho) is meromorphic, as proved by Brauer and Artin. One thing that can be proven is that, in the particular case above, L(s,\rho) is holomorphic in a strip \mathrm{Re}(s) > 1 - c for some constant c > 0 which I described as “ineffective.” But looking at it again, I realized that it is not ineffective at all, due to a result of Stark. What one actually shows is that if L(s,\rho) has a pole in the strip \mathrm{Re}(s) > 1 - c, then there must also be another L-function for the same field which has a zero on the real line in this interval. Note that, again from by Artin, it is trivially the case that a pole of one L-function must come from the zero of another L-function, since the product of all such L-functions is the Dedekind zeta function. So the content here is that the offending pole has to be on the real line. One consequence is that, in any particular case, one can rigorously check that the L-function in question has no such zeros, and hence (combined with other results in this paper) that \rho is automorphic. With help from Andrew Booker, I was able to compute one such example (Jo Dwyer has since gone on to compute a number of other examples.) On the other hand, back to the general case, we do have effective results for zeros on the real line! The result in the paper is stated in terms of the existence of a zero of \zeta_H(s) for a certain subfield H of K^{\mathrm{gal}} of degree twelve. (The definition of H was exactly what was swallowed up by Springer, so it’s not actually defined in the paper. To define it, note that S_5 has a faithful representation on six points. There is a degree six extension E which is the fixed field of the stabilizer of a point; then H is the compositum of E and the quadratic extension inside K^{\mathrm{gal}}.) However, the actual argument produces a zero in an Artin L-factor of \zeta_H(s) which is not divisible by the Dirichlet L-function for the quadratic character of S_5. Stark shows (Some Effective Cases of the Brauer-Siegel Theorem) that such an L-function does not have Siegel zeros, and also gives an explicit estimate for the largest zero on the real line. In particular, for the L(\rho,s) of interest, one deduces that they are analytic on the strip \mathrm{Re}(s) > 1 - c where one can take

\displaystyle{1 - c = 1 - \frac{1}{4 \log |\Delta_H|}}.

The result of Stark, BTW, is why one could effectively solve the class number at most X problem for totally complex CM fields which were not imaginary quadratic fields before Goldfeld–Gross–Zagier.

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K_2(O_F) for number fields F

Belabas and Gangl have a nice paper ( Generators and relations for K_2({\mathcal{O}}_F), which can be found here) where they compute K_2({\mathcal{O}}_E) for a large number of quadratic fields E. There main result is a method for proving upper bounds for K_2({\mathcal{O}}_E) in a rigorous and computationally efficient way. Tate had previously computed these groups for small imaginary quadratic fields by hand;  — the problem is finding an efficient way to do this in general. (Brownkin and Gangl had previously found a non-rigorous way of computing these groups using K_3({\mathcal{O}}_E) and regulator maps, but more on that later.) A good analogy to keep in mind is the problem of computing the class groups of imaginary quadratic fields. In the latter case, however, there are rigorous ways to determine whether an element in the class group is non-trivial, and this is missing from the computation of K_2({\mathcal{O}}_E).  To produce lower bounds, [BG] use theorems of Tate and Keune to relate the p-primary part of K_2({\mathcal{O}}_E) to class groups of E(\zeta_p), which they can then compute in some cases. One nice example they give is

K_2\left({\mathbf{Z}} \displaystyle{\left[\frac{1 + \sqrt{-491}}{2} \right]} \right) = {\mathbf{Z}}/13 {\mathbf{Z}}.

Akshay and I used this as one of the examples in our paper; in our context, it implies that the order of the group

H_1(\Gamma_0({\mathfrak{p}}),{\mathbf{Z}})

is always divisible by 13 where \Gamma_0({\mathfrak{p}}) is the congruence subgroup of \mathrm{PGL}_2({\mathcal{O}}_E) for E = {\mathbf{Q}}(\sqrt{-491}) and {\mathfrak{p}} is any prime — even though the group H_1(\Gamma,{\mathbf{Z}}) = ({\mathbf{Z}}/2{\mathbf{Z}})^{26} is not so divisible. (Because we are talking about \mathrm{PGL} rather than \mathrm{PSL}, the cusps are quotients of tori by involutions, so only contribute 2-torsion to H_1. This group is occasionally infinite; we use the convention that \infty is divisible by 13.) It’s always nice to see a theoretical argument come to life in an actual computation — fortunately, Aurel Page was kind enough to compute a presentation for \Gamma in order for us to do this. Now that I think about it, this and many other interesting things didn’t make it into the submitted version of the paper; you’ll have to read the “directors cut” to learn about it.

Alexander Rahm pointed out to me that the computation of K_2({\mathcal{O}}_E) we used was annotated with an asterisk in [BG], meaning that what was proved was only an upper bound. The issue is as follows. Let p = 13, and let F = E(\zeta_p), let G = {\mathrm{Gal}}(F/E) = ({\mathbf{Z}}/p {\mathbf{Z}})^{\times}, and let {\mathrm{Cl}}(F) denote the class group of F. What is required is to show, in light of Tate’s work on K_2, is that

({\mathrm{Cl}}(F)[p])^{G = \chi^{-1}} \ne 0,

where \chi: G \rightarrow {\mathbf{F}}^{\times}_p is the cyclotomic character. The problem is that F has degree 24, and it is difficult to compute class groups explicitly in such cases. Let H = {\mathrm{Gal}}(F/{\mathbf{Q}}), so there is a canonical decomposition H = G \times {\mathbf{Z}}/2{\mathbf{Z}}. There are two extensions of \chi to H, given (with some abuse of notation) by \chi and \chi \eta, where \eta is the non-trivial character of {\mathrm{Gal}}(F/{\mathbf{Q}}). The main conjecture of Iwasawa Theory (Mazur–Wiles) allows one to easily compute minus parts of class groups in terms of L-values without actually computing with explicit number fields. However, we should not expect this to help us here. Namely, it’s not hard to show that there is an isomorphism ({\mathrm{Cl}}({\mathbf{Q}}(\zeta_p))[p])^{G = \chi^{-1}} \simeq ({\mathrm{Cl}}(F)[p])^{H = \chi^{-1}}. However, the former is trivial by Herbrand’s theorem, because B_2 = 1/6 is not divisible by 13. That leaves us with the problem of proving that ({\mathrm{Cl}}(F)[p])^{H = \chi^{-1} \eta} \ne 0, which is a statement about the class group of a totally real cyclotomic extension. Since \chi \eta^{-1} is an even character, we get some savings by working in the totally real subfield F^{+} of degree 12. Now \text{\texttt{pari}} happily tells me via \text{\texttt{bnfinit}} and \text{\texttt{bnfclgp}} that the class group of this field is {\mathbf{Z}}/13 {\mathbf{Z}}, so it looks like we are in good shape. However, \text{\texttt{pari}} has the habit when computing class groups of assuming not only GRH but something stronger. What information does \text{\texttt{bnfinit}} actually contain? It certainly gives, inter alia:

  1. The Galois automorphisms of F^{+}, using \text{\texttt{gal:=nfisisom(nf,nf)}}.
  2. A finite index subgroup V of the unit group U:={\mathcal{O}}^{\times}_{F^{+}}, using \text{\texttt{bnfinit[8][5]}}.

Let me show how, just with this data, one can prove that the relevant part of the class group we are interested in is non-zero. BTW, if you tell \text{\texttt{pari}}: can you confirm this answer is really correct? (using \text{\texttt{bnfcertify}}) it complains, and says the following:

 *** bnfcertify: Warning: large Minkowski bound: certification will be VERY long.
*** bnfcertify: not enough precomputed primes, need primelimit 59644617.

A rough guess (in part) as to what it might be doing: to compute all the invariants necessary for class field theory, one needs to know the full unit group. To do this, one can take the units V found so far and saturate them in the entire unit group U. For each prime q, one can do this by taking representatives in V/q V and determining whether or not they are perfect qth powers. By taking enough primes, on either rules out the existence of such an element, or finds a candidate v \in V and then checks whether it is a qth power. On the other hand, from V, one can compute a pseudo-regulator R_V, which is related to the actual regulator R_U by the unknown index. So to make this computation finite, it suffices to have some a priori bound on the regulator (to give an upper bound on the index), which will ultimately come down to some a priori bound on an L-value at one, which GRH probably tells you something useful about.

One can identify the automorphims of F^{+} computed by \text{\texttt{pari}} with the elements of the Galois group given by the corresponding quotient of H = G \times {\mathbf{Z}}/2{\mathbf{Z}} by (-1,1). This group is generated by the image of \sigma = (2,1) = \mathrm{Frob}_2, so it is enough to find the automorphism \sigma such that \sigma \theta - \theta^2 \equiv 0 \mod 2. View \chi \eta, a character of degree 12, as being valued in {\mathbf{F}}^{\times}_{13}. Now choose a random unit, say \text{\texttt{u:=bnf[8][5][6]}} (Warning! I have a feeling that \text{\texttt{bnfinit}} does something different each time you run it, which means you might have to tweak the choice of index 6 above if you are doing this at home. And by “you,” I really mean “me” in six months time. I guess I should also tell myself that the relevant \text{\texttt{pari}} file is \text{\texttt{\~{}fcale/Zagier/BG491}}.) We may write down a second unit as follows:

\displaystyle{{\epsilon} = \prod_{i=0}^{12} (\sigma^i(u))^{\chi \eta (\sigma^i)} \in ({\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times p}_F)^{H = \chi^{-1} \eta}}

What we have done is apply the appropriate projector in the group ring {\mathbf{F}}_{13}[H] to u. Naturally enough, we can lift {\epsilon} to an actual unit in F^{+}.

Now choose an auxiliary prime q which splits completely in F, say q = 38299. I chose this because it actually splits completely in {\mathbf{Q}}(\zeta_{13 \cdot 491}), which will make a computation below slightly easier. We reduce {\epsilon} modulo a prime {\mathfrak{q}} above q in {\mathcal{O}}_{F^+} and we find that

{\epsilon}^{(q-1)/13} \not \equiv 1 \mod {\mathfrak{q}}.

What this last computation proves is that {\epsilon} actually generates
({\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times p}_F)^{H = \chi^{-1} \eta}, which has dimension one by Dirichlet’s theorem. Note also that the inequality above does not depend on the choice of {\mathfrak{q}} — any other choice is conjugate to {\mathfrak{q}} which replaces {\epsilon} by \sigma {\epsilon} and the latter is a non-zero scalar multiple of the former modulo 13th powers by construction.

On the other hand, let \zeta be a primitive 13 \cdot 491th root of unity. Then we may consider the projection of 1 - \zeta modulo 13th powers to the \chi^{-1} \eta eigenspace (the latter is naturally also a character on F(\zeta_{491})). Remember this eigenspace is generated by {\epsilon}. Take q = 38299 again, so q - 1 = 13 \cdot 491 \cdot 6. Then 2^{6} is a primitive 13 \cdot 491th root of unity modulo q. On the other hand,

\displaystyle{\left(\prod_{({\mathbf{Z}}/13 \cdot 491 {\mathbf{Z}})^{\times}} (1 -2^{6n})^{n \left(\frac{n}{491}\right)} \right)^{(q-1)/13}  \equiv 1 \mod q}

(The exponent of (1 - 2^{6n}) is just the value of \chi \eta(n) — remember that the character gets inverted in the projection formula — and that \eta^{-1} = \eta.) This implies that the projection of (1 - \zeta) to the \chi^{-1} \eta-eigenspace of units modulo p = 13 is trivial, because the image of {\epsilon}^{(q-1)/13} computed above was not 1 \mod q. The same is trivially true for the units in {\mathbf{Q}}(\zeta_{491}) and {\mathbf{Q}}(\zeta_{13}), because the projection of any unit in a subfield of F can only be an eigenvalue for a character of the correponding quotient of the Galois group. In particular, if C denotes the group of circular units, we have shown that the map

(C/13 C)^{\chi^{-1} \eta} \rightarrow ({\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times 13}_F)^{\chi^{-1} \eta}

is the zero map. This proves that the index of the circular units in the entire units is divisible by 13. This is enough to prove that 13 divides h^{+}_F, but even better, by the Gras conjecture (also proved by Mazur–Wiles, following Greenberg) it follows that the \chi^{-1} \eta-part of the class group is non-zero, and hence, given the previous upper bound, this gives a proof that

K_2({\mathcal{O}}_E) = {\mathbf{Z}}/13 {\mathbf{Z}}.

Further Examples: Let’s now look at other examples in [BG]. Consider the following example:

\displaystyle{K_2\left( {\mathbf{Z}} \left[ \frac{1 + \sqrt{-755}}{2} \right] \right) =^{?} {\mathbf{Z}}/41 {\mathbf{Z}} \oplus {\mathbf{Z}}/2 {\mathbf{Z}}}.

Let F = E(\zeta_{41}), and let F^{+} be the totally real subfield of F of degree 40.  Well we certainly won’t be able to say so much about the class group of F^{+}. On the other hand, we can do the latter part of the computation, namely, testing that the \chi^{-1} \eta-eigenspace in the circular units looks like it has index divisible by p in the entire units. For example,  if q = 123821 = 1 + 41 \cdot 755 \cdot 4, we can compute that

\displaystyle{\left(\prod_{({\mathbf{Z}}/41 \cdot 755 {\mathbf{Z}})^{\times}} (1 -2^{n(q-1)/(41 \cdot 755)})^{n \left(\frac{n}{755}\right)} \right)^{(q-1)/13}  \equiv 1 \mod q}

For good measure, the same congruence holds for the next seven primes which split completely in F(\zeta_{755}). (One also has to check that the multiplicative order of 2 for all these primes is co-prime to 41 \cdot 755.) But, although this is compelling numerically, it doesn’t prove anything. If {\epsilon} is a generator of ({\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times p}_F)^{\chi^{-1} \eta}, it might be the case that {\epsilon}^{(q-1)/p} \equiv 1 \mod {\mathfrak{q}} for {\mathfrak{q}} above the first thousand primes of norm q \equiv 1 \mod p.  This would simply correspond to a certain ray class group being divisible by p. By Cebotarev, we know that we can find some prime q for which this congruence does not hold, but explicit Cebotarev bounds tend to be rubbish in practice.

If we re-think our original computation, what we really want is a “generic” unit of F^{+} in order to project. Since F is abelian, we actually know how to compute a finite index subgroup of the unit group, namely, by projecting (via the norm map) the group of circular units from some cyclotomic overfield. Of course, this exactly won’t be good enough to find a candidate unit {\epsilon}. One approach is to take our lattice V \subseteq U = {\mathcal{O}}^{\times}_{F^+} and saturate it. Now we only have to saturate it at p = 41. In fact, we only need to saturate the \chi^{-1} \eta eigenspace, which is one dimensional. That is, it suffices to show that

e_{\chi^{-1} \eta} N_{F(\zeta_{41})/F^{+}} (1 - \zeta)

is a pth power in F^{+}. (Before taking the norm, the element is already in F^{+} up to pth powers, and [F(\zeta_{41}):F^{+}] has order prime to 41.) But if I ask \text{\texttt{pari}} to compute the following:

\displaystyle{\left(\prod_{({\mathbf{Z}}/41 \cdot 755 {\mathbf{Z}})^{\times}} (1 -\zeta^n)^{n \left(\frac{n}{755}\right)} \right)^{(q-1)/13}  \mod \Phi_{41 \cdot 755}(\zeta)}

it complains and conks out. Well, probably René Schoof could do this computation, but let’s think about these things a little differently.

Higher Regulators: So far, we’ve been relying on the fact that the fields E we are considering are abelian, in order to be able to explicitly write down some finite index subgroup of the full unit group using circular units. But what if we want to compute K_2({\mathcal{O}}_E) for non-abelian fields E?  For this, I want to talk about an earlier paper of Gangl with Brownkin ( Tame and wild kernels of quadratic imaginary number fields…  oh bugger, this should also be cited as [BG].) Their approach is through the study of higher regulators. Borel constructs a higher regulator map for odd K-groups (the even ones are trivial after tensoring with {\mathbf{Q}}). For imaginary quadratic fields and K_3, this amounts to a map

K_3(\mathcal{O}_E) \rightarrow (2 \pi i)^2 \cdot \mathbf{R},

where the co-volume of the image is a rational multiple of \zeta_E(2). The Quillen–Lichtenbaum conjecture predicts that the covolume differs exactly from \zeta_E(2) by a factor coming from the torsion in K_3({\mathcal{O}}_E), which has order dividing 24, some slightly mysterious powers of 2 which I will ignore, and — the most relevant term for us — the order of K_2({\mathcal{O}}_E). Now the Quillen–Lichtenbaum conjecture is true. So how does this help to compute anything? Well, first one has to ask how to compute K_3({\mathcal{O}}_E).  As an abelian group, it is easy to compute, but this is not enough to compute the regulator map. One could give explicit classes in \pi_3(\mathrm{BGL}({\mathcal{O}}_E)^{+}), of course, but that may not be the most practical approach. It turns out that the group K_3 is computable in a natural way because of its relation to the Bloch group B(E), due to theorems of Bloch and Suslin. (That is, via the Hurewicz map we get classes in H_3(\mathrm{GL}_N({\mathcal{O}}_E),\mathbf{Z}) which turn out to be seen by \mathrm{GL}_2.) To recall, the Bloch group is defined as the quotient of the pre-Bloch group:

\displaystyle{ \sum n_i [x_i], x_i \in E^{\times}, \ \text{such that} \ \sum n_i (x_i \wedge (1 - x_i)) = 0 \in \bigwedge^2 E^{\times}}

by the 5-term relation

\displaystyle{[x] - [y] + \left[\frac{y}{x} \right] - \left[ \frac{1-y}{1-x} \right] + \left[ \frac{1 - y^{-1}}{1 - x^{-1}} \right] = 0, x,y \in E^{\times} \setminus 1}.

Now the Bloch group admits a very natural regulator map

B(E) \rightarrow {\mathbf{R}}^{r_2}

(where E has signature (r_1,r_2)) given by (under the various complex embeddings) the Bloch–Wigner dilogarithm

D(z) = \mathrm{Im} (\mathrm{Li}_2(z)) + \arg(1-z) \log |z| \in \mathbf{R}.

Now all of this is (almost) very computable. Namely, one can replace E^{\times} by the S-units of {\mathcal{O}}_E for some (as large as you can) set S, compute the pre-Bloch group, then do linear algebra to find the quotient. Since (roughly) K_3({\mathcal{O}}_E) = {\mathbf{Z}}^{r_2} \oplus T for an easy to understand finite group T which has order dividing 24, as soon as one has a enough indepdenent elements in the Bloch group (which can be detected by computing D(z)) you can compute a group B_S(E) which has finite index in B(E). Moreover, the dilogarithm is also easy to compute numerically, and so one can compute a regulator D_S(E) coming from the Bloch group. Now this regulator map is known to be rationally the same as Bloch’s regulator map (by Suslin and Bloch). Assuming this is also true integrally, we expect there to be a formula:

\displaystyle{\frac{3 |d_E|^{3/2}}{\pi^2 D(E)} \cdot \zeta_E(2) =^{?} K_2({\mathcal{O}}_F)},

at least for primes p > 3.  (The 3 is coming from the torsion of K_3, and this formula is probably only true up to powers of 2 — this formulation above comes from Brownkin–Gangl.) For S big enough, D_S(E) should stabilize to D(E), which gives a method of computing the order of K_2({\mathcal{O}}_E). This is what Brownkin and Gangl do. There are two issues which naturally one has to worry about. The first is that it’s not known that the regulator map coming from dilogarithms is the same on the nose as Bloch’s map. However, even granting this (and it should be true), this algorithm will not certifiably end, because one can never be sure that D_S(E) = D(E). If you compare this to the computation that \text{\texttt{pari}} is doing with the class group, the problem is that there is no a priori bounds on the size of the corresponding regulators. Well, I guess this algorithm can sometimes end, namely, when one can be sure if the indicated upper bound for K_2({\mathcal{O}}_E) matches with a known lower bound. However, we are exactly in a situation in which we are trying to prove a lower bound. For example, when E = {\mathbf{Q}}(\sqrt{-755}), Brownkin and Gangl predict that |K_2({\mathcal{O}}_E)| = 2 \cdot 41 because, for a set of larger and larger primes S, the index formula above stabilizes. So, beyond the issue of relating two different higher regulator maps, we have the problem of determining whether a class in K_3({\mathcal{O}}_E) is divisible by a prime p or not. This seems harder than our previous problem of determining whether a unit was divisible or not! (To be fair, however, it seems impossible to find units in E(\zeta_p) once E is non-abelian and p is in any sense large.)

Chern Class Maps: We want to understand whether a class in the Bloch group B(E) or in K_3({\mathcal{O}}_E) is divisible by p or not. Instead of working over {\mathbf{R}}, another approach is to work modulo a prime q. (It may seem a little strange to work modulo q to detect divisibility by p, but bear with me.) Soulé constructed certain Chern class maps, which include a map:

c_2: K_3({\mathcal{O}}_E) = K_3(E) \rightarrow H^1(E,{\mathbf{Z}}_p(2)).

These maps are the boundary map in the Atiyah–Hirzebruch spectral sequence for étale K-theory. Now compose this maps with the reduction modulo p map. Then, after restricting to F = E(\zeta_p), we may identify {\mathbf{Z}}_p(2)/p with \mu_p, and so, by Kummer and Hilbert 90, we get a map:

c_2: K_3(\mathcal{O}_E)/p \rightarrow H^1(F,\mathbf{Z}_p(2)/p) \simeq  H^1(F,\mathbf{Z}_p(1)/p) = F^{\times}/F^{\times p}.

Keeping track of the various identifications, the image lands in the \chi^{-1} invariant subspace, where \chi is the cyclotomic character of G = {\mathrm{Gal}}(F/E).

Lemma Let p > 3 be a prime which is totally ramified in E(\zeta_p)/E and suppose that p does not divide the order of K_2({\mathcal{O}}_E). Then the Chern class map induces an isomorphism

({\mathbf{Z}}/p {\mathbf{Z}})^{r_2} = K_3({\mathcal{O}}_E)/p K_3({\mathcal{O}}_E) \rightarrow ({\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times p}_F)^{\chi^{-1}}.

That is, the image of c_2 in F^{\times}/F^{\times p} may be taken to land in the unit group, and the ranks of all the groups are the same and equal to r_2, the number of complex places of the field E.

This lemma follows from Quillen–Lichtenbaum, but it can also be proved directly from the surjectivity of the Chern class map as proved by Soulé, the known rank of K_3 \otimes {\mathbf{Q}} by Borel, and some knowledge of the torsion of K_3 proved by Merkuriev and Suslin.  It turns out that the hypothesis on K_2({\mathcal{O}}_E) is necessary not only for the proof but for the lemma to be true.

To detect whether a class in K_3 is divisible by p, it suffices to “compute” the Chern class map above and see whether it is zero. If one ever wants to compute anything, it makes sense to work with the Bloch group B(E) instead. On the other hand, it seems hopeless to give a “concrete” map:

B(E) \rightarrow F^{\times}/F^{\times p}.

Even though one can write down elements in the first group somewhat explicitly, it’s hard to imagine a recipe that would produce explicit elements in F^{\times} with the correct Galois action.

Instead, what we do is reduce modulo q for some prime q \equiv -1 \mod p. That is, we pass from the Bloch group over E (which will be generated by S units for some S) to the Bloch group of the field {\mathbf{F}}_q. The construction over {\mathbf{F}}_q is just the same. By a theorem Hutchinson, this group will have order q+1. The numerology here is intimately related to Quillen’s result that K_3({\mathbf{F}}_q) = {\mathbf{Z}}/(q^2 - 1){\mathbf{Z}}. Now there are some commutative diagrams one has to check commute here; I think the key point to keep in mind is that Quillen’s computation of K_3({\mathbf{F}}_q) can already be realized in the cohomology group H^3({\mathrm{SL}}_2({\mathbf{F}}_q),{\mathbf{Z}}), and so the map of Bloch groups will be the same as the map on K-groups via comparison with the Hurewicz map.

Let’s choose a prime q \equiv -1 \mod p which splits completely in E(\zeta_p + \zeta^{-1}_p) \subset F = E(\zeta_p). So we have a map B(E) \rightarrow B({\mathcal{O}}_E/{\mathfrak{q}}) \otimes {\mathbf{F}}_p = B({\mathbf{F}}_q) \otimes {\mathbf{F}}_p = {\mathbf{F}}_p. The Bloch group can be thought of in terms of (a quotient of a subgroup of) the free abelian group of elements of \mathbf{P}^1(E), so there’s no issue about this reduction map. Moreover, given an element of the Bloch group, we can explicitly compute its image in the latter group. If this image is non-zero, that gives a certificate that the original element is not divisible by p. This will be enough to compute K_2({\mathcal{O}}_E) as long as the Bloch regulator map agrees with the dilogarithm map.

This argument is still yoked to real regular maps. Let’s try to work entirely with c_2 and finite auxiliary primes q \equiv - 1 \mod p. Another manifestation of the map B(E) \rightarrow B({\mathbf{F}}_q) \otimes {\mathbf{F}}_p is the map:

c_2: K_3({\mathcal{O}}_E) \rightarrow {\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times p}_F  \rightarrow ({\mathcal{O}}_F/{\mathfrak{Q}})^{\times} \otimes {\mathbf{F}}_p = {\mathbf{F}}_p,

where {\mathfrak{Q}} is a prime above {\mathfrak{q}} in F.  Let’s go back to considering the case when E is an imaginary quadratic field. The image of a generator of K_3({\mathcal{O}}_E) will map exactly to a non-zero multiple of the non-trivial element unit {\epsilon} \in F^{\times}/F^{\times p}. If K_2({\mathcal{O}}_E) is prime to p, it will even land in ({\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times p}_F)^{\chi^{-1}}. The latter map is exactly computing (up to a non-zero scalar) {\epsilon}^{(q-1)/p} \mod {\mathfrak{q}}, and so, purely using the Bloch group, we can check whether this is trivial or not. In particular, given an element of the Bloch group B(E) which (we think) is a generator, or at least not divisible by p, we can find a prime q \equiv -1 \mod p such that the reduction to B({\mathbf{F}}_q) \otimes {\mathbf{F}}_p is non-zero, which will imply that the image of c_2 is non-zero, which will imply that

{\epsilon}^{(q-1)/p} \not\equiv 1 \mod p.

This gives an explicit value of q for which this is true without ever having to compute {\epsilon}. For such a prime q, we can then check that the circular units project to the identity in this space, which will prove unconditionally that K_2({\mathcal{O}}_E) is divisible by p. (Part of this computation assumed that p did not divide K_2({\mathcal{O}}_E), but that’s OK, because to prove that p does divide this group we are allowed make that assumption anyway). Back to our example. We now want a prime q \equiv -1 \mod 41, which is also a square modulo 755. We take q = 163. Now this is not the most attractive computation in the world, because the root of unity \zeta of order 37 \cdot 755 cuts out the extension {\mathbf{F}}_{q^{300}}, as we can see by computing the multiplicative order of q = 163 modulo 41 \cdot 5 \cdot 151. Let’s do it in baby steps. By choosing a suitable prime {\mathfrak{Q}} in E(\zeta_{755}), we can ensure that

\zeta^{755} + \zeta^{-755} = \zeta_{41} + \zeta^{-1}_{41}  \equiv 4 \mod {\mathfrak{Q}}.

We write

\zeta^{1510} - 4 \zeta^{755} + 1 = F(\zeta) G(\zeta) \mod 163,

where F(\zeta) is any of the four factors of degree 300 (there are also two factors of degree 150, and factors of degrees 2, 4, and 4.) Now we want to compute, with p = 41, q = 163, and r = 755,

\displaystyle{\eta:= \left(\prod_{({\mathbf{Z}}/p r {\mathbf{Z}})^{\times}} (1 -\zeta^n)^{n \left(\frac{n}{r}\right)} \right)^{(q^{300}-1)/p}  \mod \mathfrak{Q} = (163,F(\zeta))}

Of course, one should first reduce the exponents \chi^{-1} \eta(n) = n (n/r) modulo p = 41 before taking the powers. (Actually, it’s probably kind of stupid to take a product over \varphi(pr) = 24000 different terms, and one can surely set this up much more effeciently, but whatever.) We find (drum roll) that:

\eta \equiv 1 \mod 163.

To finish, we have to take an element in the Bloch group B({\mathcal{O}}_E) and show that it doesn’t vanish in B({\mathcal{O}}_E/{\mathfrak{q}}) \otimes {\mathbf{F}}_p = B({\mathbf{F}}_{163}) \otimes {\mathbf{F}}_{41}. At this point, I email Herbert (Gangl), and he sends me an email with the following beautiful element of B(E), where \alpha^2 = - 755:

\displaystyle{-8 \left[\frac{3 - \alpha}{10}\right] - 10 \left[\frac{7-\alpha}{10} \right] - 8 \left[\frac{3 - \alpha}{100} \right] +  \ldots + 6 \left[\frac{7 \alpha + 221}{972} \right]}.

(There are 114 terms in all! This should be a generator of B(E).) Into my \text{\texttt{magma}} programme it goes, which cheerily reports that the image of this element is non-zero in B({\mathbf{F}}_{163}) \otimes \mathbf{F}_{41}! So K_2 is really divisible by 41. (You might question the veracity of my programme’s output, but more on that below.)

Stark and Beyond: Here are some more general remarks. Let’s still suppose that E is imaginary quadratic. Take the image of a generator [M] of B(E), which is defined up to torsion and up to sign. The image of the Chern class map for some p > 3 and p not dividing K_2({\mathcal{O}}_E) gives a canonical unit in {\mathcal{O}}^{\times}_F/{\mathcal{O}}^{\times p}_F, where F = E(\zeta_p). Let me a be a bit more careful here: by writing F as F = E(\zeta_p), we are choosing a root of unity (this unit depends on this choice). There’s also an automorphism of {\mathrm{Gal}}(E/{\mathbf{Q}}) which acts, but this changes the sign of [M], so that is the same ambiguity we had before. What is this canonical unit? It is not just a circular unit, but a canonical one (modulo pth powers). What is it? More generally, when r_2 = 1, both K_3 and (\mathcal{O}^{\times}_F/\mathcal{O}^{\times p}_F)^{\chi^{-1}} have rank r_2 = 1, so if p is prime to K_2(\mathcal{O}_E) we are generating canonical units. It’s tempting here to conjecture some relation to Stark units here, and in particular to the special value of L(1,E,\chi^{-1}), but let me say no more about this. When r_2 > 1, one is no longer in the Stark world, but there is still a canonical map from the Bloch group to the unit group (the group \mathbf{Z}^{r_2} has no canonical generator when r_2 > 1 — but in the manifestation of this group as a Bloch group, one does have explicit elements.)

Actually, I haven’t even explained how to compute c_2. So far, I have only explained how to compute whether it is zero or not modulo p. To evaluate it exactly requires a further threading of the needle through the previous maps (on the Bloch group), and ultimately uses a test element coming from torsion in B({\mathbf{Q}}(\zeta_p + \zeta^{-1}_p)). Although this is somewhat delicate, and I have not yet proved all of the appropriate diagrams commute (blech), one can work with it in practice and it gives many consistency checks on all the computations. (So, for example, once one has the image of c_2, one can compute the reduction of the corresponding element in the Bloch group in B({\mathbf{F}}_{q}) \otimes {\mathbf{F}}_p for one prime q \equiv -1 \mod p knowing its image in the corresponding group for another such prime. Generating the same element of \mathbf{F}_p for p = 13 and twenty different primes q is pretty convincing.)

In fact, computing this map exactly is exactly the problem that I was thinking about in the first place. I did compute it explicitly for K = {\mathbf{Q}}(\sqrt{-491}) and p = 13 (and also K = {\mathbf{Q}}(\sqrt{-571}) for p = 5), and the image of a generator of the Bloch group is not a unit. Instead, it gives a generator of \frak{a}^{13} for a non-trivial ideal in the class group {\mathrm{Cl}}(F) of F = E(\zeta_p), indeed, an element of {\mathrm{Cl}}(F)[p]^{\chi^{-1} \eta}. (In particular, it gives, having fixed a root of unity, a canonical element of this class group, which is also somewhat mysterious.) Let me also mention the Coates–Sinnott conjecture, higher Stickelberger elements, and work of Banaszak and Popescu which are closely related to the topics in this post (in particular, using Chern class maps to construct Euler systems generalizing the circular unit Euler system, although not so much question of identifying these elements in some explicit way — especially because much less is known about higher analogues of the Bloch group). But perhaps this is enough for now.

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Intermezzo

Scholze is giving a course at Berkeley! Follow Jared Weinstein’s lecture notes from the course here.

I believe that the lectures are also being recorded and will be available approximately one week after each lecture; I will add a link when it becomes available. Added: Chandan Dalawat in the comments gives the following link.

On a different matter, it was reported to me that it was “common knowledge” amongst graduate students at a certain bay area university that “Sowa the Owl” (Сова = Owl) was in fact Andrei Suslin. I can safely confirm that is almost definitively not the case; Suslin, for one, doesn’t use email. In fact, the only way to talk to him at all is to ambush him outside his algebra class at 1:58. On this matter, I will otherwise confine my remarks to the following:

Consider a “Gowers Colloquium” which consisted of Gowers giving a talk once a week on a topic chosen by the colloquium organizer. Then I wager (and the bar is unfortunately low here) that it would be a more informative and interesting lecture series than the math colloquiums at most universities. (Although that $2000 dinner I had at Harvest restaurant after a Harvard colloquium was pretty nice; we drank a few bottles of decent Chablis, if I remember.) More generally, I think mathematicians talking about work that is not their own (in a seminar/colloquium setting) is not done enough in mathematics. Can anyone tell me how succesfull the Bourbaki seminar is as an actual seminar talk? I worry that speakers might try too hard to impress Serre, I’m not sure.

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100 Posts

Meaningless numerical milestones are a good a reason as any for an indulgent post. Today, I will discuss some facts from this blog which you might not otherwise know about. It will be in the form of an (mercifully short) interview with myself.

When did you start this blog?

I originally started it when I went to the IAS for a special year in 2010-2011, but I never ended up making the blog public at that time. The irreverence has been toned down for the current version. Sample post from the IAS: “Who wears short shorts? Deligne wears short shorts!”

What topics would you like to blog about in the future?

No promises, but here are some thoughts:

  • How does an NSF panel work?
  • What are letters of recommendation really like?
  • Who wore it better: piano v. orchestral arrangements.
  • Langlands versus the world.
  • Book reviews: Frenkel, Ellenberg, Harris.
  • India’s greatest mathematician: Harish-Chandra.
  • The top 1%: class and privilege in academia.

Does that mean you are planning to have less math in the future?

No, the math posts are not really planned in advance, they are just what I happen to be thinking about at the time. The math posts are really the main (if not exclusive) focus of this blog. Although, as one of my graduate students once remarked: “I though your post on swans was your best post ever.” Yeah, thanks for that, I’m working hard to bring you occasional insights into the vast edifice of algebraic number theory, and you like the guy who can wobble around to Saint-Saëns.

What can your readers do for you?

More audience participation! A lot of what I write is speculative, so please don’t refrain from giving your partially formed thoughts in the comments. As the readership of this blog went up, the number of comments has gone down. I think I understand this phenomenon, especially when it comes to math posts. The worst thing that can happen, however, is that you say something completely ridiculous in front of a bunch of senior number theorists. But, if you are not occasionally saying stupid things in front of smart people, then you are doing it wrong.

Does the audience have any questions?

I’m going to take audience questions in the form of random search terms which led to this blog. Perhaps those who came here were disappointed with their search results at the time, but perhaps if they search again this post will provide some answers.

  • 아리조나 윈터스쿨: I recommend going here.
  • review my paper: No thanks!
  • how can i tell how many pages my paper is: Form a bijection with one of the sets defined in Part II of this volume.
  • paskunas conference 2013: Sounds good! Alas, I was not invited.
  • maximal unramified abelian extension of a local field: It’s procyclic, and is generated by roots of unity of order prime to the residue characteristic. Assuming, of course, your local field is of mixed characteristic, which all the interesting ones are.
  • bush is the messiah: This seems to me to be an unverifiable claim.
  • galois reprsentations matrix: Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.
  • honorarium + editor + elsevier: $60 for any processed paper. It is taxable income, however.
  • galoisrepresentations+blog+who?: It’s me!
  • bach mit pedal schiff: Bach without pedal, surely?
  • how do i find out how my paper is being reviewed: Oh, I can tell you that. If you are lucky, the reviewer has completely forgotten about it. Otherwise, the reviewer is currently cursing you for generally ruining his or her life.
  • compute the average rate of change from x = -10 to x = 10. enter your answer as a fraction in simplest terms using a slash ( / ). do not include spaces in your answer: (f(10)-f(-10))/20.
  • local even galois representations: I presume you are asking about representations of a local Galois group which are even. For this to make sense, you should probably talk about Galois representations at the infinite prime, that is, representations of \mathrm{Gal}(\mathbf{C}/\mathbf{R}) on which complex conjugation acts trivially. Let me classify those for you: they are all trivial!
  • peter scholze gowers: I don’t think they wrote any joint papers.
  • ila varma grothendieck: Same answer as above.
  • ila varma galois: Same answer as above.
  • danny calegari brilliant; jacob lurie genius: self-googling, I imagine.
  • joel specters math thesis: I shall link to it on this blog after it has been written.
  • representation galois change of characteristic: I assume you are asking about the p-q-switch? It is now ubiquitous, but there are plently of good expositions available online.
  • affirmative motives: I guess these are motives which just need a little support. For only a $5 donation to this blog, I will help turn a poor motive into a bold and effective one, simply by twisting.
  • kevin buzzard chess: I’d be surprised if he had the time. I fancy my chances.
  • xxx agol in school: Personally, I rate way Agol schooled 3-manifolds as [T18+], suitable for topologists of ages 18 and above.
  • the math behind a waffle: Aah, sorry about that. I’m more an expert in the waffle behind the math. On the other hand, you can learn about the chemistry of waffles here.
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The decline and fall of Publications Mathématiques de l’lHÉS

I want to discuss the decline of a once great journal.

How did IHES go from this:

and this:

to this:

It is a sad state of affairs. To be clear, I am talking about the typesetting here. The old IHES typeface was a delicious rich dark Baskerville (according to this source). The math displays were always a little wonky, but that was more than compensated by the charm of the text. Now? Still the same wonky formulas, and an inferior electronic typeface that is the equivalent of replacing some find fine hand-made Belgian chocolate with a Cadbury flake bar that’s been left out in the sun too long. This is very sad. How good was the typeface? It was so good, some people submitted their papers to IHES precisely for the beauty of the typesetting, even though it doesn’t look like that anymore. I think I even prefer \usepackage{baskervald} to the SMF version.

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