## New Results in Modularity, Part II

This is part two of series on work in progress with Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne. Click here for Part I

It has been almost 25 years since Wiles first announced his proof of Taniyama-Shimura, and, truthfully, variations on his method have been pretty much the only game in town since then (this paper included). In all generalizations of this argument, one needs to have some purchase on the integral structure of the automorphic forms involved, which requires that they contribute in some way to the cohomology of an arithmetic manifold (locally symmetric space). This is because it is crucial to be able to exploit the integral structure to study congruences between modular forms. Let’s briefly recall Wiles’ strategy. One starts out with a residual representation

$\overline{\rho}: G_S \rightarrow \mathrm{GL}_2(\mathbf{F}_p)$

which one assumes to be modular, that is, is the mod-p reduction of a representation associated to a modular form which is assumed to have some local properties similar to rho. One then considers a deformation ring R which captures all deformations of the residual representation which “look modular” of the right weight and level (some aspects of Serre’s conjecture due to Ribet are empoyed here, although Skinner-Wiles came up with a base change trick to circumvent some of these difficulties). On the automorphic side, one looks at the cohomology groups M = H^1(X,Z_p)_m of modular curves (X = X_0(N)) localized at a maximal ideal m of the Hecke algebra T associated to rhobar, and proves that there is a surjective map:

$R \rightarrow \mathbf{T}_{\mathfrak{m}}.$

Already many deep theorems have been used to arrive at this point. To begin, one needs Galois representations associated to modular forms, but moreover, one needs to know that these representations satisfy all of the expected local-global compatibilities at the primes in S. In the case of modular forms, all of these facts were basically known before Wiles.

The next step, which lies at the heart of the Taylor-Wiles method, is to introduce certain auxiliary sets Q of carefully chosen primes, and consider the spaces M_Q = H^1(X_1(Q),Z_p)_m which relate to spaces of modular forms of larger level. If T_Q is the associated Hecke algebra, and R_Q is the corresponding deformation ring in which ramification is allowed not only at S but now also at Q, there are compatible maps as follows:

The key point concerning how one chooses the sets Q is to ensure that, even though R_Q may get bigger, its infinitesimal tangent space does not. Hence all the R_Q are quotients of some fixed ring R_oo = Z_p[[X_1,…,X_q]]. (Here q is chosen so that q = |Q|.) In this process, all the rings also have an auxiliary action of a ring S_oo = Z_p[[T_1,…,T_q]] of diamond operators, coming from the Galois group of X_1(Q) over X_0(Q) on the automorphic side, and the inertia groups at Q on the Galois side. The action of S_oo on these modules factors through R_Q by construction, by local global compatibility at primes dividing Q. After throwing away the Galois representations almost entirely (but keeping the diamond operators), one can patch the modules M_Q/p^n for different sets of primes Q, and arrive at a patched module M_oo for R_oo and S_oo such that:

• The module $M_{\infty}$ has positive rank as an $S_{\infty}$ module.
• If $\mathfrak{a}$ is the augmentation ideal of $S_{\infty},$ then $R_{\infty}/\mathfrak{a} = R,$ and $M_{\infty}/\mathfrak{a} = M.$

The first statement may be viewed as saying that there are “lots” of automorphic forms. On the other hand, the fact that R_oo has the same dimension of S_oo says that there are not “too many” Galois representations. Indeed, this friction is enough in this context to prove that M_oo is free over R_oo, and then to deduce the same claim for M over R, from which R = T follows. (Already included here is a innovation due to Diamond where one deduces freeness as a consequence rather than building it in as an assumption.) The argument I have very briefly sketched above is really only a proof of modularity in the minimal case. The general case requires a completely separate argument to bootstrap from minimal to non-minimal level using two further ingredients: Wiles’ numerical criterion, and a lower bound on the congruence ideal necessary to apply the numerical criterion, which ultimately follows from Ihara’s Lemma.

The “first generation” of improvements to Wiles consisted of understanding enough integral p-adic Hodge theory to make the required arugments on the Galois side. Notable papers here include the work of Conrad-Diamond-Taylor and Breuil-Conrad-Diamond-Taylor (but let us also not forget here the contribution of The Hawk). Improvements along these lines continue to today, and are very closely interwined with p-adic Langlands program and work of Breuil, Colmez, Kisin, Emerton, Paškūnas, and many others.

The “second generation” of improvements consisted of relaxing the assumption that R_oo is smooth, by allowing instead R_oo to have multiple components (but still of the same dimension) associated to different components in the local deformation rings at primes in S (at p and away from p). This innovation was due to Kisin, who also introduced the notion of framing to handle this.

The “third generation” of improvements (somewhat orthogonal to the second) cames from replacing 2-dimensional representations with n-dimensional representations, but still under some very restrictive assumptions on the image of rho. One key consequence of these assumptions is that the spaces of modular forms M_Q = H^*(X_1(Q),Z_p)_m all occur inside a single cohomology group, which allows one to control the growth of these spaces when patching. Here one thinks of the work of Clozel-Harris-Taylor. Also pertinent is that the analog of Ihara’s Lemma is open for higher rank groups; Taylor came up with a technique to bypass it when proving modularity lifting theorems now known as “Ihara avoidance.”

(Of course there were many other developments less directly relevant to this post, including but not limited to Skinner-Wiles and Khare-Wintenberger.)

The problem with considering general representations for GL(n) for n > 2, even over Q, is that the automorphic forms are spread over a number of different cohomology groups, in fact in some range [q_0,q_0 + 1, … ,q_0 + l_0] for specific invariants q_0 and l_0.
This manifests itself in two ways:

1. There are not enough automorphic forms; the patched modules M_oo will not be free over S_oo.
2. There are not enough Galois representations: the ring R_oo does not have the same dimension as S_oo but rather dim R_oo = dim S_oo – l_0.

Of course these problems are related! My work with David Geraghty was precisely about showing how to make these problems cancel each other out. The rough idea is as follows. The cohomology groups H^*(X_1(Q),Z_p)_m which contain interesting classes in characteristic zero occur in the range [q_0,…,q_0+l_0]. Suppose one knows this to be true integrally as well, even with coefficients over F_p instead of Z_p. Then instead of patching the cohomology groups M_Q themselves, one instead patches complexes P_Q of length l_0. The result is a complex P_oo of finite free S_oo modules of length l_0, with an action of R_oo on the cohomology of this complex. But the only way the cohomology of this complex can be small enough to admit an action of R_oo is if the complex is a free resolution of the patched module M_oo of cohomology groups in the extreme final degree, and moreover it also implies that M_oo is big enough as in Wiles’ original argument to give an R=T theorem. Note that it is crucial here that one work with the torsion in integral cohomology. It is quite possible that, at all auxiliary levels Q, there are no more automorphic forms at level Q than are were at level 1. (This can only happen for l_0 > 0, and the idea that torsion should be a suitable replacement is the moral of my paper with Barry Mazur.) These argument is also compatible with the improvements to the method including Taylor’s “Ihara Avoidance” argument.

On the other hand, there is a big problem. This argument required many inputs which were completely unknown at the time we worked this out, so our results were very conditional. To be precise, our results were conditional on the following desiderata:

1. The existence of Galois representations on Hecke rings T which acted as endomorphisms of H^*(X,Z/p^nZ) for locally symmetric spaces X associated to GL(n)/F.
2. The stronger claim that the Galois representations constructed in part 1 satisfied the correct “local-global” compatibility statements for all v in S (including v dividing p).
3. The vanishing of the cohomology groups H^i(X,Z/p^nZ)_m outside the range i in [q_0,…,q_0+l_0], for a non-Eisenstein ideal m.

A different approach to some of these questions (which Matt and I discussed, see here) involves first passing to completed cohomology, where one expects (or hopes!) that all the cohomology groups except in degree q_0 should vanish after localization at a non-maximal ideal.

The first big breakthrough was the result of Scholze, who proved part 1 above, at least up to issues concerning a nilpotent ideal (this was discussed previously on this blog). Another innovation appeared in Khare-Thorne, where it was observed that one can sometimes drop the third assumption under the strong condition that there existed global automorphic forms with the exact level structure corresponding to the original representation. (Unfortunately, in the l_0 > 0 setting, there is no way to produce such forms.)

So this is roughly where we stood in 2016. The key new ingredient which led to this project was the new result of Caraiani and Scholze proving vanishing theorems for the cohomology of non-compact Shimura varieties in degrees above the middle dimension (localized at m) under the assumption of certain genericity hypotheses on m. Since the cohomology of the boundary (for suitably chosen Shimura varieties) is precisely related to the cohomology of arithmetic locally symmetric spaces for GL(n) over CM fields, this allowed for the first time a new construction of the Galois representations for GL(n) which directly related them to the Galois representations coming from geometry. (I say “directly related,” but perhaps I mean simply more direct than Peter’s original construction.) In particular, it was clear to Caraiani and Scholze that this result should have implications for the required local-global compatibility result above. Meanwhile, the IAS had just started a new series of workshops on emerging topics. I guess that Richard must have had conversations with Ana about her work with Peter, which led them to choosing this as the theme, namely:

Ana Caraiani and Peter Scholze are hopeful of extending the methods of their joint paper arXiv:1511.02418 to non-compact Shimura varieties. This would give a new way to attack local-global compatibility at p for some of the Galois representations Scholze attached to torsion classes in the cohomology of arithmetic locally symmetric spaces. The aim of this workshop will be to understand how much local-global compatibility can be proved and to explore the consequences of this, particularly for modularity questions.

So now (1) was available, there was an approach to (2), and a technique for avoiding (3). One issue with the Khare-Thorne trick, however, was that it involved localizing at some prime ideal of characteristic zero, and so did not interact so well with Ihara Avoidance, which was crucial for any sort of applicable theorem. Here’s the subtely, which can be described even in the case when l_0 = 0. The usual Ihara avoidance game is to compare deformation rings R and R’ at Steinberg level and ramified principal series level respectively (after making a base change to ensure that the prime v at the relevant prime q satisfies N(v) = 1 mod p). Let M and M’ be the corresponding modules. One has that M/p = M’/p and R/p = R’/p. Suppose, however, that M behaved perfectly as expected, so that M_oo was free (even of rank one say) over S_oo and free over R_oo. What could happen, if one doesn’t have vanishing of cohomology outside a single degree, is that M’_oo/p = M_oo/p is free over S_oo/p, but that M’_oo is the cohomology of a non-trivial complex S_oo —> S_oo given by multiplication by p. So M’_oo is trivial in characteristic zero, even though M’_oo/p = M_oo/p. So this is a problem. But it is exactly a problem which was resolved during the workshop. The point, very loosely speaking, is that even though the complexes “S_oo” and “S_oo –>[p]—> S_oo” have the same H^0 after reducing modulo p and taking cohomology, their intersection with S_oo/p are quite different on the derived level, so if one can formulate a version of derived Ihara avoidance, then one is in good shape.

So what remained? First, there were a number of technical issues, some of which could be dealt with individually, and one had to make sure that all the fixes were compatible. For example, it is straightforward to modify the original strategy in my paper with David to handle the issue of only having Galois representations up to nilpotence ideals of fixed nilpotence, but one had to make sure this would not interfere with the more subtle derived Ihara avoidance type arguments. Relevant here was the work of Newton and Thorne which placed some of the arguments with complexes more naturally in the derived category. Second, there was the issue of really proving local-global compatibility from the new results of Caraiani-Scholze. A particularly interesting case here was the ordinary case. The rough problem one has to deal with here is deducing that rho is ordinary from knowing that $\rho \oplus \rho^{\vee}$ is ordinary. But be careful — the latter representation is reducible and so really a pseudo-representation — so it’s not even clear what ordinary this means (though see work of Wake and Wang Erickson, as well as of my student Joel Specter). It turns out that some interesting and subtle things turn up in this case which were found by the “team” of people who wrote up this section. (Although we acheived quite a lot in a week, there were obviously a list of details to be worked out, and we divided ourselves up into certain groups to work on each part of the paper.) But I think we were fairly confident at this point that everything would work out. What was my role in the writing up process you ask? I was selected as the ENFORCER, who goes around harassing everybody else to work and write up their sections of the paper while sipping on Champagne. Presumably I was less selected for my organizational skills and more for my ablity to tell Richard Taylor what to do.

So there we have it! It was clear even during the workshop that some improvements to our arguments were possible, but since the paper is already going to be quite long, we did not try to be completely comprehensive. I expect a number of improvements will follow shortly. I would not be surprised to see in a few years a modularity result for regular weight compatible systems over CM fields which are as complete as the ones (say) in BLGGT.

Finally, I should mention that while the paper is almost completely written, the usual caveats apply about work in progress which has not been completely written up (although we are almost done…)

## New Results In Modularity, Part I

I usually refrain from talking directly about my papers, and this reticence stems from wishing to avoid any appearance of tooting my own horn. On the other hand, nobody else seems to be talking about them either. Moreover, I have been involved recently in a number of collaborations with multiple authors, thus sufficiently diluting my own contribution enough to the point where I am now happy to talk about them.

The first such paper I want to discuss has 9(!) co-authors, namely Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne. The reason for such a large collaboration is a story of itself which I will explain at the end of the second post. But for now, you can think of it as a polymath project, except done in a style more suited to algebraic number theorists (by invitation only).

In this first post, I will start by giving a brief introduction to the problem. Then I will state one of the main theorems and give some (I hope) interesting consequences. In the next post, I will be a little bit more precise about the details, and explain more precisely what the new ingredients are.

Like all talks in the arithmetic of the Langlands program, we start with:

The Triangle:

Let F be a number field, let p be a prime, and let S be a finite set of places containing all the infinite places and all the primes above p. Let G_S denote the absolute Galois group of the maximal extension of F unramified outside S. In many talks in the Langlands program, one encounters the triangle, which is a conjectural correspondence between the following three objects:

• A: Irreducible pure motives M/F (with coefficients) of dimension n.
• B: Continuous irreducible n-dimensional p-adic representations of G_S (for some S) which are de Rham at the places above p.
• C: Cuspidal algebraic automorphic representations $\pi$ of $\mathrm{GL}(n)/F.$

In general, one would like to construct a map between any two of these objects, leading to six possible (conjectural) maps, which we can describe as follows:

• A->B: This is really the only map we understand, namely, etale cohomology. (I’m being deliberately vague here about what a motive actually is, but whatever.)
• B->A: This is the Fontaine-Mazur conjecture, and maybe some parts of the standard conjectures as well, depending on exactly what a motive is.
• B->C: This is “modularity.”
• C->B: This is the existence of Galois representations associated to automorphic forms.
• A->C: We really think of this as A->B->C and also call this modularity.
• C->A: Again, this is a souped up version of C->B. But note, we still don’t understand how to do this even in cases where C->B is very well understood. For example, suppose that $\pi$ comes from a Hilbert modular form with integer coefficients of trivial level over a totally real field F of even degree. We certainly have an associated compatible family of Galois representations, and we even know that its symmetric square is geometric. But it should come from an elliptic curve, and we don’t know how to prove this. The general problem is still completely open (think Maass forms). On the other hand, often by looking in the cohomology of Shimura varieties, one proves C->A and uses this to deduce that C->B.

This triangle is also sometimes known as “reciprocity.” The other central tenet of the Langlands program, namely functoriality, also has implications for this diagram. Namely, there are natural operations which one can easily do in case B which should then have analogs in C which are very mysterious.

Weight Zero: For all future discussions, I want to specialize to the case of “weight zero.” On the motivic/Galois side, this corresponds to asking that the representations are regular and which Hodge-Tate weights which are distinct and consecutive, namely, [0,1,2,…,n-1]. The hypotheses that the weights are distinct is a restrictive but crucial one — already the case when F = Q and the Hodge-Tate weights are [0,0] is still very much open (specifically, the case of even icosahedral representations). On the automorphic side, the weight zero assumption corresponds to demanding that the $\pi$ in question contribute to the cohomology of the associated locally symmetric space with constant coefficients.

For example, if n=2, then we are precisely looking at abelian varieties of GL(2) type over F (e.g. elliptic curves). This is an interesting case! We know they are modular if F is Q, or even a quadratic extension of Q. More generally, we know that if F is totally real, then such representations are at least potentially modular, that is, their restriction to some finite extension $F'/F$ is modular. This is often good enough for many purposes. For example, it is enough to prove many cases of (some version of) B->A. In this case, we have quite complete results, although still short of the optimal conjectures, especially in the case when the residual representation is reducible.

There are many other modularity lifting results generalizing those for n=2, but they really involve Galois representations whose images have extra symmetry properties. In particular, they are either restricted to representations which preserve (up to scalar) some orthogonal or symplectic form, or they remain unchanged if one conjugates the representation by an outer automorphism of G_F (for example when $F/F^+$ is CM and one conjugates by complex conjugation). There were basically no unconditional results which applied either in the situation that n > 2 or that F was not completely real, and the representation did not otherwise have some restrictive condition on the global image. Our first main theorem is to prove such an unconditional result. Here is such a theorem (specialized to weight zero):

Theorem [ACCGHLNSTT]: Let F be either a CM or totally real number field, and p a prime which is unramified in F. Let

$\rho: G_S \rightarrow \mathrm{GL}_n(\overline{\mathbf{Q}_p})$

be a continuous irreducible representation which is crystalline at v|p with Hodge-Tate weights [0,1,..,n-1]. Suppose that

1. The residual representation $\overline{\rho}$ has suitably big image.
2. The residual representation is “modular” in the sense that there exists an automorphic form $\pi_0$ for $\mathrm{GL}(n)/F$ of weight zero and level prime to p such that $\overline{r}(\pi_0) = \overline{\rho}.$

Then $\rho$ is modular, that is, there exists an automorphic representation $\pi$ of weight zero for $\mathrm{GL}(n)/F$ which is associated to $\rho.$

One could be more precise about what it means to have big image. In fact, I can do this by saying that it has enormous image after restriction to the composite of the Galois closure of F with the pth roots of unity. Here enormous is a technical term, of course. There is also a version of this theorem with an ordinary (rather than Fontaine-Laffaille) hypothesis (more on this next time).

Let me now give a few nice theorems which can be deduced from the theorem above:

Theorem [ACCGHLNSTT]: Let E be an elliptic curve over a CM field F. Then E is potentially modular.

When I had a job interview at MIT in 2006, I was asked by Michael Sipser, the chair at the time, to come up with a theorem which (in a best case scenario) I would hope to prove in 10 years. I said that I wanted to prove that elliptic curves over imaginary quadratic fields were modular. (Reader, I got the job … then went to Northwestern.) It is very gratifying indeed that, roughly 10 years later, this result has actually been proved and that I have made some contribution towards its eventual resolution. (OK, we have potential modularity rather than modularity, but that is splitting hairs…). It is also amusing to note that a number of co-authors were still in high school at this time! (Fact Check: OK, just one…)

In fact, one can improve on the theorem above:

Theorem [ACCGHLNSTT]: Let E be an elliptic curve over a CM field F. Then Sym^n(E) is potentially modular for every n. In particular, the Sato-Tate conjecture holds for E.

Finally, for an application of a different type, suppose that $\pi$ is a weight zero cuspidal algebraic automorphic representation for $\mathrm{GL}(2)/F.$ For each prime v of good reduction, one can associate to $\pi_v$ a pair of Satake parameters $\{\alpha_v,\beta_v\}$ satisfying $|\alpha_v \beta_v| = N(v).$ The Ramanujan conjecture says that one has

$|\alpha_v| = |\beta_v| = N(v)^{1/2}.$

An equivalent formulation is that the sum $a_v$ of these two eigenvalues satisfies $|a_v| \le 2 N(v)^{1/2}.$ We prove the following:

Theorem [ACCGHLNSTT]: Let F be a CM field, and let $\pi$ be a weight zero cuspidal algebraic automorphic representation for $\mathrm{GL}(2)/F.$ Then the Ramanujan conjecture holds for $\pi.$

If F is totally real, then the Ramanujan conjecture follows from Deligne’s theorem. One can associate to $\pi$ a motive, whose Galois representation is either $\rho = \rho(\pi)$ or $\rho^{\otimes 2}.$ Then, by applying purity to these geometric representations, one deduces the result. (Of course, this was famously proved by Deligne himself in the case when F = Q. The case of a totally real field, especially in cases where one has to go via a motive assoicated to $\rho^{\otimes 2},$ is due (I think) to Blasius.) This is decidedly not the way we prove this theorem. In fact, we do not know how to prove the Fontaine-Mazur conjecture for the representation $\rho$ associated to $\pi,$ even in the weak sense of showing that $\rho$ or even $\rho^{\otimes 2}$ appears inside the cohomology of some projective variety. Instead, we prove that $\mathrm{Sym}^n \rho$ is potentially modular, then use the weaker convexity bound to prove the inequality:

$|\alpha_v|^n \le N(v)^{n/2 + 1/2}.$

Taking n sufficiently large, we deduce that $|\alpha_v| \le N(v)^{1/2},$ which (by symmetry) proves the result. Experts will recognize this as precisely Langlands’ original strategy for proving Ramanujan using functoriality! In a certain sense, this is the first time that Ramanujan has been proved without a direct recourse to purity. I say “in some sense”, because there is also the ambiguous case of weight one modular forms. Here the Ramanujan conjecture (which is $|a_p| \le 2$ in this case) was deduced by Deligne and Serre as a consequence of showing that $\rho$ has finite image so that alpha_v and beta_v are roots of unity. On the other hand, that argument does also simultaneously imply that the representations are motivic. So our theorem produces, I believe, the only cuspidal automorphic representations for $\mathrm{GL}(n)/F$ for which we know to be tempered everywhere and yet for which we do not know are directly associated in any way to geometry.

Question: Suppose I’m sitting in my club, and Tim Gowers asks me to say what is really new about this paper. What should I say?

Answer: The distinction (say) between elliptic curves over imaginary quadratic fields and real quadratic fields, while vast, is quite subtle to explain to someone who hasn’t thought about these questions. You could explain it, but the club is hardly a place to do so. Instead, go with this narrative: We generalize Wiles’ modularity results for 2-dimensional representations of Q to n-dimensional representations of Q. If you are pressed on previous generalizations, (especially those due to Clozel-Harris-Taylor), say that Wiles is the case GL(2), Clozel-Harris-Taylor is the case GSp(2n), and our result is the case GL(n).

If you had slightly more time, and the port has not yet arrived, you might also try to explain how the underlying geometric objects involved for GSp(2n) are all algebraic varieties (Shimura varieties), but for GL(n) they involve Riemannian manifolds which have no direct connection to algebraic geometry. Here is a good opportunity to name drop Peter Scholze, and explain how this is the first time that the methods of modularity have been combined with the new world of perfectoid spaces.

## In defense of Theresa May

Nothing gives me more sympathy for Theresa May^* (or less respect for the Economist) than the following line from this article:

Mrs May made do with a dowdier college and relaxed by watching “The Goodies”, a particularly dire comedy.

Sacrilege! Everyone knows that the Goodies is the cultural touchstone to which all other cinematography should be compared!

^*: Well, not really.

## Elementary Class Groups Updated

In a previous post, I gave a short argument showing that, for odd primes p and N such that $N \equiv -1 \mod p,$ the p-class group of $\mathbf{Q}(N^{1/p})$ is non-trivial. This post is just to remark that the same argument works under weaker hypotheses, namely:

Proposition: Assume that N is p-power free and contains a prime factor of the form $q \equiv -1 \mod p,$ and that p is at least 5. Then the p-class group of $K = \mathbf{Q}(N^{1/p})$ is non-trivial.

The proof is pretty much the same. If N has a prime factor of the form $1 \mod p,$ then the genus field is non-trivial. Hence we may assume there are no such primes, from which it follows that $H^1_S(\mathbf{F}_p)$ has dimension one and $H^2_S(\mathbf{F}_p)$ is trivial, where S denotes the set of primes dividing Np. The prime q gives rise to a non-trivial class $b \in H^1_S(\mathbf{F}_p(-1))$ which is totally split at p (this requires that p be at least 5), and the field K itself gives rise to a class $a \in H^1_S(\mathbf{F}_p(1)).$ But now the vanishing of H^2 implies that $a \cup b = 0$ and hence there exists a representation of G_S of the form:

$\rho: G_S \rightarrow \left( \begin{matrix} 1 & a & c \\ 0 & \chi^{-1} & b \\0 & 0 & 1 \end{matrix} \right),$

where $\chi$ is the mod-p cyclotomic character. The class c gives the requisite extension (after possibly adjusting by a class in the one-dimensional space $H^1_S(\mathbf{F}_p)).$ The main point is that the image of inertia at primes away from p is tame and so cyclic, but any unipotent element of $\mathrm{GL}_3(\mathbf{F}_p)$ has order p if p is at least three. This ensures c is unramified over K away from the primes above p. On the other hand, the class $b$ is totally split at p. This implies that the class c is locally a homomorphism of the Galois group of $\mathbf{Q}_p,$ and so after modification by a multiple of the cyclotomic class in $H^1_S(\mathbf{F}_p)$ may also be assumed to be unmarried at p. The fact that $b \ne 0$ ensures that $c \ne 0,$ and moreover the fact that p is at least 5 implies that the kernel of c is distinct from that of a, completing the proof. (This result was conjectured in the paper Class numbers of pure quintic fields by Hirotomo Kobayashi, which proves the claim for $p = 5.)$

Posted in Mathematics | 2 Comments

## Thoughts on Paris

Nothing is more pretentious or annoying than when an American offers, uninvited, their opinions of Paris. Here, then, are some of mine.

• Starting the day with a two hour lecture on elliptic integals:

OUI: Who does not get a slight frisson upon seeing the identity

$\displaystyle{\frac{j}{16} = (\sqrt{k})^4 + \left(\frac{1}{\sqrt{k}}\right)^4 + \left(\frac{1 - \sqrt{k}}{1 + \sqrt{k}}\right)^4 + \left(\frac{1 + \sqrt{k}}{1 - \sqrt{k}}\right)^4 + \left(\frac{1 - \sqrt{-k}}{1 + \sqrt{-k}}\right)^4 + \left(\frac{1 + \sqrt{-k}}{1 - \sqrt{-k}}\right)^4 + 42}.$

where j is the modular invariant and k is the usual parameter of elliptic integrals, given in terms of theta functions as $\theta^2_2/\theta^2_3$ where $\theta_2 = \sum q^{(n+1/2)^2}$ and $\theta_3 = \sum q^{n^2}.$

• Starting the morning with a croissant:

NON: There are decent enough croissants available, but in the general spectrum of correctly proportioning one’s caloric intake, there are better choices.

• Starting the morning with a Kouign Amann:

OUI…ET NON: Yes, I did wake up at 6:45 so I could bike to Blé Sucré and have a Kouign Amann before they were sold out. It was indeed good. But it still didn’t live up to the buttery sugary indulgences I had in Brittany. Calling on Jacques Tillouine to organize another conference in Roscoff!

• Using Vélibs (the Paris bikeshare program):

OUI: Travelling by bike, especially from my location at Paris 7, was extremely convenient, not to mention very pleasant in the clear 70 degree days with a light breeze that were pretty much a constant throughout my stay. The bike paths were excellent, and rarely required having to get too close to cars. But even on-the-road traffic (for example, cycling around the place de la Bastille) was less stressful than it can sometimes be in Chicago or London. The Velib stations themselves were not perfect: there were a number of times the internet connection was down, or the machine inexplicably returned to the initial screen or gave some other error (the “you already have a bike rented” being the most disturbing one), or the closest stations were either all full or empty depending on whether you were trying to return or rent a bike, but this type of thing seems to happen for many such programs. Extra points for the baskets on the front of the bikes which were extremely useful. Also points for being so much cheaper than Divvy: I had about three weeks of use for 24E, wheras in Chicago the cheapest option would have been to get a \$100 yearly membership.

• Going anywhere by car:

NON: Traffic was terrible. Fortunately, I mostly avoided having to be in a car. We did go by bus to the Paris Mosque. We ended up being stuck in one stretch of road for about 10 minutes, at a point where the alternative would have been a very pleasant (and less than 10 minute) walk through the jardin de plantes.

• The Gardens at Giverny:

OUI: I had to choose a day exursion for my young charges, and I was very happy with this choice. Admittedly, a Parisian local described my choice as “American,” so make of that what you will.

• Lunch with Clozel:

OUI: I didn’t have much time for socializing on this trip, but I did get to have a very pleasant lunch with Laurent. If you leave this off your itenerary, you haven’t seen Paris!

• Orange SIM cards:

NON: My phone would randomly claim that I had used up all my data, and I would hae to turn it off and start it again before it would work. It was truly the worst SIM card I have ever used in Europe. I strongly recommend using anyone but Orange.

• Third Wave Coffee:

OUI…ET NON: It is well known that the French have mastered all aspects of cafe culture except making drinkable coffee. But I was very interested to see how much of the third wave had infiltrated into Paris. Here’s a breakdown of the third wave places I visited in order of preference: Telescope, Boot (Right Bank and Left Bank — the Right Bank store is much smaller and has wifi, the Left Bank is bigger and does not), Coffee Cuillier, Fragments, Strada (two locations), Le Peleton Cafe, Ten Belles, and Passager, although the gap between almost all of these was close to non-existent and I would revisit any of them if I was in the neighbourhood. (I had a very pleasant stay at Passager working on my laptop outside. I stayed there for so long I very nearly forgot to pay for my coffee when I left.) Given the weather and general ambience, the general experience of biking to these cafes and then sitting down for a flat white (or equivalent) or espresso was overall very pleasant. On the other hand, I would rate the coffee at these places as generally fine but not great. Many of these places seem to be run (or staffed) by Australians, which is no surprise. (As mentioned previously, Australians have also done wonders for coffee in London.)

• Background music in cafes:

NON: There seems to be some sort of cultural time warp, with Paris 7 students consisting of skateboarding dudes smoking and wearing ’80s fashion. The music in the cafes is similarly pretty bad. Of course, YMMV.

• Restaurants:My restaurant list is somewhat longer than my cafe list, and I have a detailed set of notes, but I would say the best overall meal was at La Bourse et La Vie. For those on a budget looking for a cheap place to have a light lunch, I strongly recommend Canard & Champagne. Other notable courses: a rendition of vitello tonnato at Paul Bert, a light egg tapas dish whose name I don’t remember at Sourire tapas françaises, a fluffy squid dish which tasted like liquid quiche at Semilla, seared Foie Gras at Domaine De Lintellac, and a few more.
• The weather in May:

OUI: It poured the first day or so, and threatened in the forecast to rain quite frequently. But future forecasts faded, and for almost the entire three weeks, it was pretty close to a blissful 70 degrees, clear, with a slight breeze. Perfect!

## Who proved it first?

During Joel Specter’s thesis defense, he started out by remarking that the $q$-expansion:

$\displaystyle{f = q \prod_{n=1}^{\infty} (1 - q^n)(1 - q^{23 n}) = \sum a_n q^n}$

is a weight one modular forms of level $\Gamma_1(23),$ and moreover, for $p$ prime, $a_p$ is equal to the number of roots of

$x^3 - x + 1$

modulo $p$ minus one. He attributed this result to Hecke. But is it really due to Hecke, or is this more classical? Let’s consider the following claims:

1. The form $f$ is a modular form of the given weight and level.
2. If $p$ is not a square modulo 23, then $a_p = 0$.
3. If $p$ is a square modulo 23, and $x^3 - x + 1$ has three roots modulo $p,$ then $a_p= 2.$
4. If $p$ is a square modulo 23, and $x^3 - x + 1$ is irreducible modulo $p,$ then $a_p = -1.$

At when point in history could these results be proved?

$\displaystyle{ \prod_{n=1}^{\infty} (1 - q^n) = \sum_{-\infty}^{\infty} q^{(3n^2+n)/2} (-1)^{n}}$
Using this, one immediately sees that

$\displaystyle{f = \sum \sum q^{\frac{1}{24} \left((6n+1)^2 + 23 (6m+1)^2 \right)} (-1)^{n+m}}$

This exhibits $f$ as a sum of theta series. With a little care, one can moreover show that

$\displaystyle{2f = \sum \sum q^{x^2 + x y + 6 y^2} - \sum \sum q^{2 x^2 + x y + 3 y^2}}.$

This is not entirely tautological, but nothing that Gauss couldn’t prove using facts about the class group of binary quadratic forms of discriminant $-23.$ The fact that $f$ is a modular form of the appropriate weight and level surely follows from known results about Dedekind’s $\eta$ function, which covers (1). From the description in terms of theta functions, the claim (2) is also transparent. So what remains? Using elementary number theory, we are reduced to showing that a prime $p$ with $(p/23) = +1$ is principal in the ring of integers of $\mathbf{Q}(\sqrt{-23})$ if and only if $p$ splits completely in the Galois closure $H$ of $x^3 - x + 1.$

Suppose that $K = \mathbf{Q}(\sqrt{-23}) \subset H.$ What is clear enough is that primes $p$ with $(p/23) = + 1$ split in $K,$ and those which split principally can be represented by the form $x^2 + xy + 6y^2$ in essentially a unique way up to the obvious automorphisms. Moreover, the class group of $\mathrm{SL}_2(\mathbf{Z})$ equivalent forms has order $3,$ and the other $\mathrm{GL}_2(\mathbf{Z})$ equivalence class is given by $2x^2 + xy + 3y^2.$ In particular, the primes which split non-principally in $K$ are represented by the binary quadratic form $2 x^2 + xy + 3y^2$ essentially uniquely. From Minkowski’s bound, one can see that $H$ has trivial class group. In particular, if $x^3 - x + 1$ has three roots modulo $p,$ then the norm of the corresponding ideal to $K$ is also principal and has norm $p = x^2 + xy + 6y^2.$ This is enough to prove (3).

So the only fact which would not obviously be easy to prove in the 19th century is (4), namely, that if $p = x^2 + xy + 6y^2,$ then $p$ splits completely in $H$. The most general statement along these lines was proved by Furtwängler (a student of Hilbert) in 1911 — note that this is a different (and easier?) statement than the triviality of the transfer map, which was not proved until 1930 (also by Furtwängler), after other foundational results in class field theory had been dispensed with by Tagaki (another student of Hilbert!). Yet we are not dealing with a general field, but the much more specific case of an imaginary quadratic field, which had been previously studied by Kronecker and Weber in connection with the Jugendtraum. I don’t know how much Kronecker could actually prove about (for example) the splitting of primes in the extension of an imaginary quadratic field given by the singular value $j(\tau).$ Some of my readers surely have a better understanding of history than I do. Does this result follow from theorems known before 1911? Who proved it first?