From: Mathematics Editorial Office

Subject: [Mathematics] Review Request

Date: December 26, 2018 at 7:43:31 AM CST

Dear Professor Calegari,

Happy new year. We have received the following manuscript to be considered for publication in Mathematics (http://www.mdpi.com/journal/mathematics/) and kindly invite you to provide a review to evaluate its suitability for publication:

Type of manuscript: Article

Title: **Common fixed point theorems of generalized multivalued** –**contractions in complete metric spaces with application.**

If you accept this invitation we would *appreciate receiving your comments within 10 days.* Mathematics has one of the most transparent, and reliable assessments of research available. Thank you very much for your consideration and we look forward to hearing from you.

Kind regards,

[**name**]

Assistant Editor

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From: [**name**]

Subject: [Mathematics] Manuscript ID: Review Request Reminder

Date: December 27, 2018 at 9:48:02 PM CST

Dear Professor Calegari,

On 26 December 2018 we invited you to review the following paper for Mathematics:

Type of manuscript: Article

Title: **Common fixed point theorems of generalized multivalued** –**contractions in complete metric spaces with application.**

You can find the abstract at the end of this message. As we have not yet heard from you, we would like to confirm that you received our e-mail.

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From: [**name**]

Subject: [Mathematics] Manuscript ID: Review Request Reminder

Date: January 3, 2019 at 7:48:15 PM CST

On 28 December 2018 we invited you to review the following paper for Mathematics:

Type of manuscript: Article

Title: **Common fixed point theorems of generalized multivalued** –**contractions in complete metric spaces with application.**

You can find the abstract at the end of this message. As we have not yet heard from you, we would like to confirm that you received our e-mail.

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From: Frank Calegari

Subject: Re: [Mathematics] Manuscript ID: mathematics Review Request Reminder

Date: January 3, 2019 at 11:08:03 PM CST

Dear [**name**],

Thank you for agreeing to enlist my professional reviewing services. My current rate is $1000US an hour. Please send me the contract forms and payment details. I estimate somewhere between 2-5 hours will be required to review this paper.

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From: Frank Calegari

Subject: Re: [Mathematics] Manuscript ID: mathematics Review Request Reminder

Date: January 8, 2019 at 07:50:03 AM CST

Dear [**name**],

On January 3, I invited you to forward me the contract forms and payment details for my reviewing assignment.

However, if you are unable to provide payment because you are a predatory journal, please let me know quickly to avoid unnecessary reminders.

Do not hesitate to contact me if you have any questions about this request.

My previous message is included below:

…

Professor Francesco Calegari

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From: [**name**]

Subject: Re: [Mathematics] Manuscript ID: mathematics-412859 – Review Request Reminder

Date: January 8, 2019 at 9:25:25 PM CST

Dear Professor Calegari,

… Actually, the article process charge of this manuscript is only 350CHF. We need to invite at least two reviewers for each manuscript. We can’t bear the cost you proposed. So I will cancel the review invitation for you soon.

All the best to your work.

Kind regards,

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]]>However, a number of recent changes have been taken place. JNT has introduced “JNT Prime” which seeks to publish

a small number of exceptional papers of high quality (at the level of Compositio or Duke).

(I’m not sure if free two-day delivery is also included in this package.) My question is: why bother?

I have several points of confusion.

**It’s easier to start from scratch**. It is much easier (as far as developing a reputation goes) to start a new journal and set the standards from the beginning, than to steer a massive oil tanker like the Journal of Number Theory with its own firmly established brand. Consider the Journal of the Institute of Mathematics of Jussieu. This early paper (maybe in the very first issue) by Richard Taylor set the tone early on that this was a serious journal. Similarly, Algebra & Number Theory in a very few number of years became a reasonably prestigious journal and certainly more prestigious than the Journal of Number theory has ever been during my career.**The previous standards of JNT served the community well**. Not every journal can be the Annals. Not every journal can be “better than all but the best one or two journals” either, although it is pretty much a running joke at this point that every referee request nowadays comes with such a description. There is plenty of interesting research in number theory that deserves to be published in a strong reputable journal but which is better suited to a specialist journal rather than Inventiones. Journal of Number Theory: it does what it says on the tin. Before the boutique A&NT came along, it was arguably the most prestigious specialist journal in the area. It is true that it was*less*prestigious than some specialist journals in other fields, but that reflects the reality that number theory papers make up a regular proportion of papers in almost all of the top journals, which is not true of all fields. So where do those papers go if JNT becomes all fancy?**Elsevier**. Changing the Journal of Number Theory is going to take a lot of work, and that work is going to be done (more or less) by mathematicians. So why bother making all that effort on behalf of Elsevier? Yes, Elsevier continues to “make an effort” with respect to Journal of Number Theory, including, apparently, even sponsoring a conference. But (to put it mildly) Elsevier is not a charity, and nobody should expect them to start behaving like one.

So I guess my question is: who is better off if the Journal of Number Theory becomes (or heads in the direction of becoming) a “top-tier journal” besides (possibly) Elsevier?

]]>**Question:** Fix a prime p. Does there exist a non-solvable extension of unramified everywhere except for p?

There is a (very) related question of Gross, who (and I can’t track down the precise reference) was generous and allowed ramification at infinity. That makes the question easy to answer for big enough p just by taking the mod-p Galois representations associated to either the weight 12 or weight 16 cusp form of level 1. But what if you impose the condition that the extension has to be unramified at the infinite prime as well (so *totally real*) then you are completely out of luck as far as Galois representations from algebraic automorphic forms go, because for those, complex conjugation will always be non-trivial. (Things don’t get any easier if you even allow regular algebraic automorphic representations, as Caraiani and Le Hung showed). Except, that is, for the case when p = 2. There is a different paper by Lassina on this topic, which solved Gross’ question for p=2 by finding a level one Hilbert modular form over the totally real field for a 32nd root of unity with a non-solvable mod-2 representation. But (as he shows) this extension *is* ramified at infinity — in fact, the Odlyzko discriminant bounds show that to get a totally real extension (assuming GRH) one would have to take the totally real field to be at least as large as for a 128th root of unity. Is it even possible to compute Hilbert Modular Forms for a field this big?

Leaving aside the computational question, there is also a theoretical one as well, even for classical modular forms. Given a Hilbert modular form, or even a classical modular form, is there any easy way to compute the image of complex conjugation modulo 2? One reason this is subtle is that the answer depends on the lattice so it really only makes sense for a residually absolutely irreducible representation. For example:

**Question:** For every n, does there exist a (modular) surjective Galois representation

for a finite field of order divisible by which is also *totally real?* Compare this to Corollary 1.3 of this paper of Wiese.

I don’t even have a guess as to the answer for the first question, but the second one certainly should have a positive answer, at least assuming the inverse Galois problem. As usual, an Aperol Spritz is on offer to both the second question and to the first in the special case of p=2.

]]>To quote directly from the relevant blurb,

This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding five years.

Have you read any paper in either of these journals recently which you thought was really engaging, enlightening, or just neat? Perhaps something on a topic on which you were not so familiar which left you with the feeling that you gained some insight into what the key ideas or problems were? If so, please nominate them before June 30, 2019!

]]>For a smooth manifold M, de Rham’s Theorem gives an isomorphism

which can more naturally be phrased as that the natural pairing between (classes of) closed forms and (classes of) paths given by

induces a perfect pairing on the corresponding (co-)homology groups. The class of paths in homology has a very natural integral basis coming from the paths themselves. For a general M, the de Rham cohomology has no such basis. On the other hand, if M is (say) the complex points of an algebraic variety over the rational numbers, then there are more algebraic ways to normalize the various flavours of differential forms. To take an example which doesn’t quite fit into the world of compact manifolds, take X to be the projective line minus two points, so M is the complex plane minus the origin. There is a particularly nice closed form on this space which generates the holomorphic differentials. But now if one pairs the rational mutiples of this class with the rational multiples of the loop around zero, the pairing does *not* land in the rational numbers, since

In particular, to compare de Rham cohomology over the rationals with the usual Betti cohomology over the rationals, one first has to tensor with a bigger ring such as or at least with a ring big enough to see all the integrals which arise in this form. Such integrals are usually called periods, so in order to have a comparison theorem between de Rham cohomology and Betti cohomology over one first has to tensor with a ring of periods.

It is too simplistic to say that p-adic Hodge theory (at least rationally) is a p-adic version of this story, but that is not the worst cartoon picture to keep in your mind. Returning to the example above, note that the period is a purely imaginary number. This is a reflection of the fact that some arithmetic information is still retained, namely, an action of complex conjugation on the complex points of a variety over the rationals is compatible (with a suitable twist) with the de Rham pairing. A fundamental point is that, in the local story, something similar occurs where now the group generated by complex conjugation is replaced by the much bigger and more interesting group Very (very) loosely, this is related to the fact that p-adic analysis behaves much better with respect to the Galois group, for example, the conjugate of an infinite (convergent) sum of p-adic numbers is the sum of the conjugates. In particular, there is a Galois action on the ring of all p-adic periods. So now there is a much richer group of symmetries acting on the entire picture. Moreover, the structure of the p-adic differentials can be related to how the variety X looks like when reduced modulo-p, because smoothness in algebraic geometry can naturally be interpreted in terms of differential forms.

So now if one wants to make a p-adic comparison conjecture between (algebraic) de Rham cohomology on the one side, and etale cohomology (the algebraic version of Betti cohomology) on the other side, one (optimally) wants the comparison theorem to respect (as much as possible) all the extra structures that exist in the p-adic world, in particular, the action of the local Galois group on etale cohomology, and the algebraic structures which exist on de Rham cohomology (the Hodge filtration and a Frobenius operator), and secondly, involve tensoring with a ring of periods B which is “as small as possible”.

Identifying the correct mechanisms to pass between de Rham cohomology and etale cohomology in a way that is compatible with all of this extra structure is very subtle, and one of the fundamental achievements of Fontaine was really to identify the correct framework in which to phrase the optimal comparison (both in this and also in many related contexts such as crystalline cohomology). (Of course, his work was also instrumental in proving many of these comparison theorems as well.) I think it is fair to say that often the most profound contributions to mathematics come from revealing the underlying structure of what is going on, even if only conjecturally. (To take another random example, take Thurston’s insight into the geometry of 3-manifolds.) Moreover, the reliance of modern arithmetic geometry on these tools can not be overestimated — studying global Galois representations without p-adic Hodge theory would be like studying abelian extensions of without using ramification groups.

Two further points I would be remiss in not mentioning: One sense in which the ring is “as small as possible” is the amazing conjecture of Fontaine-Mazur which captures which *global* Galois representations should come from motives. Secondly, Fontaine’s work on *all* local Galois representations in terms of modules which is crucial even in understanding Motivic Galois representations though p-adic deformations, the fields of norms (with his student Wintenberger, who also sadly died recently), the proof of weak admissibility implies admissibility (with Colmez, another former student, who surprisingly to me only wrote this one joint paper with Fontaine), and the Fargues-Fontaine curve. (I guess this is more than two.)

Probably the first time I talked with Fontaine was at a conference in Brittany (Roscoff) in 2009. That was the first time I ever gave a talk on my work on even Galois representations and the Fontaine-Mazur conjecture, about which Fontaine had some very kind words to say. (One of the most rewarding parts of academia is getting the respect of people you admire.) I never got to know him too well, due (in equal parts) to my ignorance of the French language and p-adic Hodge theory. But he was always a regular presence at conferences at Luminy with his distinct sense of humour. Over a long career, his work continued to be original and deep. He will be greatly missed.

]]>They are both similar and very different at the same time — Lehrer is definitely the more acerbic of the pair, as evidenced by the following pair of quotes concerning satire (themselves satirical):

When Kissinger won the Nobel peace prize, satire died.

]]>The purpose of satire, it has been rightly said, is to strip off the veneer of comforting illusion and cosy half-truth. And our job, as I see it … is to put it back again!”

What should the expectations of a new postdoc be? Many universities assign research mentors to new postdocs, but (in practice) this is essentially meaningless unless it carries with it certain expectations for mentor and mentee to interact. How much of the role should senior faculty help in suggesting problems for postdocs to work on? No doubt the answer to many of these questions is “it depends on the postdoc” but I would love to hear personal stories (positive and negative) about your postdoc experiences, especially as it relates to practical steps that an institution can make to improve the experience.

Feel free to leave your comment anonymously (well, people feel free to do that anyway). I don’t particularly trust my own experience since I feel that I was probably more independent than most as a graduate student, and was fairly happy working alone in my office (not to mention already having a number of collaborations ongoing with Kevin Buzzard and Matthew Emerton). Harvard was a welcoming and friendly place (to me), but my best interactions happened serendipitously more often than not. The initial seeds of my collaboration with Barry started by joining in conversations he was having with Romyar Sharifi and William Stein in front of their offices (all on the 5th floor I believe) discussing (early forms of) Sharifiology in the context of Barry’s paper on the Eisenstein ideal. I had a few lunches with Richard Taylor at the law school (I have a vague memory that I realized this was possible from Toby — could that be right?). Richard is definitely generous with his time, and (in this context) he was ideal for bouncing off ideas. On the other hand, I don’t think Richard’s style in mathematical conversation is to be very speculative; he certainly never suggested any particular problem to me but nor did I ask. My collaboration with Nathan surely started out by virtue of the fact that we would chat socially at tea time.

I can’t quite distill from my own experiences either any recommendations for new postdocs or specific recommendations for institutions (particularly the University of Chicago) to put things in place to improve the lives of postdocs. But perhaps you can help!

]]>For example:

** Lemma:** Let F be an imaginary quadratic field in which p > 3 splits, and suppose that is a congruence subgroup of of level N prime to p. Let

be a semi-simple Galois representation associated to a Hecke eigenclass in

Assume that the image of this representation contains SL_2(F_p). Then is finite flat at primes dividing p.

The point is as follows. One wants to apply Theorem 4.5.1 of the 10-author paper, but not all the conditions are satisfied. First consider the decomposed generic condition. This is guaranteed (a tedious lemma) by the big image assumption. (In fact, this hypothesis is no doubt much too strong, and possibly — in this setting where F is an imaginary quadratic field — something close to irreducibility should be enough, but I don’t really want to bother checking that now.) The more serious hypothesis in 4.5.1 is that a certain inequality holds for the degrees of various local extensions at primes dividing p in F. This inequality **never** holds unless there are at least three primes above p, not something that usually happens for imaginary quadratic fields. But it *is* possible to achieve this via a cyclic extension. For characteristic zero forms, we can appeal to cyclic base change, but this doesn’t apply for torsion classes. On the other hand, we see that we *can* achieve a transfer of Galois representations in the case of a cyclic extension of *degree p*, by the main result of this paper (I checked with at least one of the authors this preserves the property of having level prime to p). We still have to assume that p splits in F because another condition of 4.5.1 is that F contains an imaginary field in which p splits, and one can’t force this to happen after a cyclic extension H/F of (odd) degree p unless it was true to begin with. So this hypothesis will always be required if one wants to use the results of Venkatesh-Truemann in this way.

It’s an intriguing question to ask to what extent this argument could also be applied to valued representations, where is the Hecke algebra acting on mod-p classes and I is some nilpotent ideal with nilpotence of some fixed (absolute) order. This boils down to the corresponding question of how much of one sees after the cyclic degree p extension through the Venkatesh-Truemann argument. I don’t know the answer to this, but possibly a reader will. (Having done that, there are further tricks available in which one might hope to access the ring corresponding to all of rather than just the p-torsion.)

]]>I feel that I should preface this post with the following psychological remark. Occasionally you have the germ of an idea at the back of you mind that you sense is in conflict with your world view. Perhaps you try subconsciously to banish it from your mind, or perhaps you are drawn towards it. But inevitably, the idea breaks through your consciousness and demands to be addressed. The game is now winner-takes-all — either you can defeat the challenge to your world view, or you will be swallowed up by this new idea an emerge a new person. This is how I came face to face with the non-trivial multiplicities in cohomology for non-split forms of GL(2) over an imaginary quadratic field. Part of me somehow, unconsciously, worried about the conflict between extra multiplicities on the one hand and, on the other hand, the “numerical” equality between the space of “newforms” on the split side with the corresponding space on the non-split side (this equality is not known for each maximal ideal of the Hecke algebra, but rather the “averaged” version over all maximal ideals is the topic of my paper with Akshay). Then, earlier this week, I turned my face directly towards the problem and admitted its existence, which lead to the previous post. But now… there may be a way to defeat the beast after all!

Here is the issue. I talked last time about two types of local framed Steinberg deformation rings at l=1 mod p. The first was defined by imposing conditions on characteristic polynomials, but the second was a more restrictive quotient which demanded the existence of an eigenvalue which was genuinely equal to 1. This modification seemed to pass some consistency checks, and more importantly resolved the compatibility issue between having both the equality |M| = |M’| but also having M be cyclic whilst M’ was not. Then I went away for a few days and was distracted by other math, until I flew back to Chicago this evening. While on the plane, I tried to flesh out the argument a little more by writing down more carefully what these two deformation rings R (and its smaller quotient R’) were like. And here’s the problem. It started to seem as though this quotient R’ didn’t really exist — after all, demanding the existence of an eigenvector without pinning it down in the residual representation is a dangerous business, and runs into exactly the same issues one sees when trying to give an integral definition of the ordinary deformation ring for l=p. Then I thought a little more about the ring R, and it turns out that, for all the natural integral framed deformation rings one writes down, the ring R is a Cohen-Macaulay normal integral domain! In particular, since R’ has to be of the same dimension of R, this pretty much forces R to equal R’. So it seems that my last post is completely bogus.

So what then is going on? When you have eliminated the impossible, whatever remains, however improbable, must be the truth. It is impossible that R does not equal so I can only conclude the improbable — that even when the representation rhobar is unramified at l and the image of Frobenius at l under rhobar is scalar, the multiplicity on the quaternionic side ramified at l will **still have multiplicity one**. In other words, the local multiplicity behavior will be sensitive to the archimedean places. This is not what I would (or did) guess, but I cannot see another way around it. So, at the very least, we should investigate this assumption more closely.

Let’s talk about two situations where multiplicity two occurs. The first is in the Jacobian J_1(Np) for mod-p representations which are ramified *at p*. In this case, the source of multiplicities is coming from the fact that the local deformation ring R is Cohen-Macaulay but not Gorenstein. On the other hand, the stucture of the Tate module is well understood to be of the form and so the multiplicity can (ultimately) be read off from the dualizing module of R. This is what happens in my paper with David Geraghty. The second, which is something I should have paid more attention to last time, is in the work of Jeff Manning (I can’t find a working link to either the paper or to Jeff!). The setting of Manning’s work is precisely as above: one has l=1 mod p and one is looking at the cohomology of an inner form of GL(2)/F. The only difference is that F is totally real and the geometric object is a Shimura curve. The corresponding local deformation ring R — which is basically the corresponding ring R above — is Cohen-Macaulay but not Gorenstein. On the other hand, one doesn’t now know what the structure of the Jacobian is as a module over the Hecke ring. Manning’s idea is to exploit the fact that, in his setting, the module M is reflexive (and generically of rank one), and then by studying the class group of R, pin down M exactly. But here is the thing. The reflexivity of M is coming, ultimately, from the fact that the cohomology group H^1 for Shimura curves is **self-dual**. And this is fundamentally **not** the case for these inner forms for GL(2) over an imaginary quadratic field, where the cohomology is spread between H^1 and H^2. So this is where the archimedean information can change the structure. At this point, I am pretty much obligated to make the following conjecture.

**Conjecture:** For inner forms of GL(2) over an imaginary quadratic field, and for a minimal rhobar which is irreducible and finite flat at primes dividing p > 2, the multiplicity of rhobar in cohomology is one. Moreover, the correpsonding module M’ of this cohomology group localized at this maximal ideal is isomorphic (as R-modules and so as Z_p-modules) to the space of newforms on the split side, as defined in the last post.

To put it another way, in *Example 2* of the previous post, I am now forced to say that rather than

To reiterate from last time — perhaps this conjecture is worth a computation!

I guess we shall have to wait a few days to see whether there will be a part 3!

]]>In the case of imaginary quadratic fields, Akshay and I observed a number of new pathologies that don’t occur in the classical case. One of the confusing aspects was how to define a “space of newforms” which might match (in some vague sense) the cohomology of some inner form. I want to discuss here a new conjecture which is very speculative and for which I have absolutely no computational evidence. It started off as a troubling example in my mind where things seemed to go wrong in the setting of my work with Akshay, and this is the result of me trying to put down those concerns in written form. My guiding principle is that R=T in every situation, so if this doesn’t seem to work, you have to find the right definition of R (or T).

Let F be a fixed imaginary quadratic field, say of class number 1, and let P and Q be primes (of residue characteristic different from p). Suppose that

where localization is done with respect to a non-Eisenstein maximal ideal of the Hecke algebra (assume all Hecke algebras are anemic for now). It can (and does) totally happen that one might have

That is, at level PQ there are two old forms but nothing new either in characteristic zero *or* at the torsion level. In this setting, there are apparently no “newforms” of level PQ, and so one might predict that, on the quaternionic side ramified at PQ, there is no cohomology at this maximal ideal. This is certainly true in characteristic zero by classical Jacquet-Langlands. But it is false integrally! In particular, suppose that the corresponding residual representation

has the property that the image of Frobenius at Q has eigenvalues with ratio N(Q). Then one indeed expects a contribution on the non-split side. Akshay and I managed to find an interpretation of this result by giving a “better” definition of the space of newforms as the cokernel of a transfer map:

and this can have interesting torsion even in the context above. In fact, by a version of Ihara’s Lemma, one can (and we did) compute that the order of the cokernel in this case will be exactly the order of

and (again in this precise setting) Akshay and I predicted that this should have the same order as the corresponding localization at the same maximal ideal on the non-split side. (In the Eisenstein case, this is not true, and one sees contributions from various K_2 groups). We even prove a few theorems which prove results of this form taking a product over all maximal ideals of the Hecke algebra.

But even in this example, something a little strange can happen. In particular, I want to argue in this post that **there are two natural definitions of the appropriate global deformation ring, and in order to have a consistent theory, one should consider both of them**. To remind ourselves, we now have two modules, one, defined in terms of the cokernel above, call it M, and then the cohomology localized at the appropriate maximal ideal on the non-split side, which we call M’.

What should we predict about M? The first prediction is that the image of the Hecke algebra should be precisely the universal deformation ring R_Q which records deformations that are Steinberg at Q (and what they should be at the other places). But what does Steinberg at Q even mean for torsion representations? There are basically two types of guesses for the corresponding local deformation ring, and correspondingly two guesses for the associated global deformation ring.

- A deformation ring defined in terms of characteristic polynomials. In particular, the maximal quotient of R_Q which corresponds to classes unramified at Q is the unramified deformation ring where the characteristic polynomial of Frob_Q is (X-1)(X-N(Q)).
- A more restrictive ring in which (on this same unramified quotient) the image of Frob_Q must actually fix a line.

These certainly will have the same points in characteristic zero, but they need not *a priori* coincide integrally. And this will save us below.

Returning to the corresponding global deformation rings (which should be framed, but now ignore the framing), call the corresponding rings R_Q and R’_Q. There is a surjection from R_Q to R’_Q.

Now we make the following conjecture on the smell of an oily rag:

**Conjecture**: The Hecke action on M has image R_Q while the Hecke action on M’ has image R’_Q.

I base this conjecture entirely on the following thought experiment.

Let’s suppose for convenience that N(Q) is not -1 mod p. This implies that a_Q is congruent to precisely one of 1+N(Q) or its negative — assume the former. Then the “space of newforms” M as we define it (under all the hypotheses above) will be actually be isomorphic to

because one of the factors will be a direct summand. (The case when N(Q) = -1 mod p is no problem but one has to break things up more using the Hecke operator at U_Q which I am ignoring.) So the claim in this case is that R_Q is isomorphic to this ring. What about R’_Q? Let us consider two possibilities.

(**added:** Note that if N(Q) =/= 1 mod p then R_Q=R’_Q, so we are assuming that N(Q)=1 mod p in the examples below.)

**Example 1:** Suppose that a_Q – 1 – N(Q) is exactly divisible by p^2, and that

In this case, the non-split property implies that the corresponding matrix modulo p^2 will **always** have 1 as an eigenvalue, so the prediction is that R_Q = R’_Q.

**Example 2:** Suppose that a_Q – 1 – N(Q) is exactly divisible by p^2, and that

In this case, the split condition and the assumption that a_Q – 1 – N(Q) is *exactly divisible by p^2* force the lift to be of the form

where a and c are non-zero. In particular, 1 will **never** be an eigenvalue. So in this case, one predicts that R_Q = Z/p^2Z but R’_Q = Z/pZ.

So how do we see this in terms of R=T and Jacquet-Langlands and our Conjecture above? First of all, my paper with Akshay suggests indeed that |M’|=|M|= p^2, and certainly M’ should be an R_Q-module. But now the following should happen:

- In Example 1, we should have multiplicity one, and so M’ should be free of rank 1 over R_Q = R’_Q.
- In Example 2, we should have multiplicity
**two**, following Ribet (Helm, Cheng, Manning…), since multiplicities should be determined by local conditions, and in particular multiplicities should arise exactly when primes which ramify in the quaternion algebra are split and such that the image of the corresponding Frobenius is scalar. Hence M’ should be free of rank 2 over R’_Q in this case.

In particular, the Hecke action on M’ should factor through R’_Q in both cases, and R_Q does not act faithfully. Perhaps this conjecture is worth a computation!

**Update:** Read Part 2 of this series.