At the meal I found myself sitting next to Alistair Cooke who was very charming, and absolutely fascinating to listen to. The very next Sunday when I was back in England I turned on his “Letter from America” on the radio — he started off by saying, “I went to a very boring dinner at the White House. There was no one interesting to talk to.” That amused me a lot.

So let me start off by saying that even though this post is about one or two things I learnt at Oberwolfach, it is deliberately **not** about anything I learnt in the talks, lest my choosing some talks over others leading to false inferences on what I thought interesting. For example, the title of this post alludes to David Zureick-Brown’s talk, which I will not mention again.

Let g be a non-negative integer and p a prime. Suppose one starts with a representation

with (say) cyclotomic similitude character. To avoid later circumlocutions, let me (most of the time) assume it is absolutely irreducible. One can ask whether this representation arises infinitely often from the p-torsion on an abelian variety — perhaps additionally assuming that it does arise from at least *one* such variety, or perhaps not.

This problem is very well-studied in the case g = 1, where we know that the answer is positive exactly for the primes p = 2, 3, and 5, where the corresponding moduli space X(p) has genus zero, and the associated twists X(rho) are Brauer-Severi varieties that also turn out to be rational over Q under the given hypothesis on the similitude character. When p > 5, the curves X(p) have genus at least 3, and so their twists always have at most finitely many points over any field, by Faltings. So there is certainly a satisfactory answer in this case. (Of course, there are many more subtle versions of this question — for example, replacing “infinitely” by “at least twice” — and those variations are open in general.)

If we move to genus g = 2, then the case of p = 2 is also straightforward — the 2-torsion of the Jacobian of y^2 = f(x) for a degree 6 polynomial with Galois group G just comes from the isomorphism (One needs to be a little bit careful here because the outer automorphism of S_6 means there are two non-conjugate such maps and one has to choose the right one.) Given that one can write down families of sextics where the etale Q-algebra Q[x]/f(x) is constant, it’s easy to see that the answer is positive in this case without any restrictions. For example, given an extension, there’s a six dimensional family of polynomials one can write down whose splitting field generically gives this extension, and so after accounting for the action of on the roots, this still gives a three-dimensional rational family of genus two curves whose two torsion comes from this extension.

In my paper with Boxer, Gee, and Pilloni (coming soon!), will also give a similarly conclusive answer for p = 3, although there are some unexpected surprises, as well as some complementary results recently proved by my student Shiva Chidambaram. But more on this in a post coming up soon!

When p > 3, then the corresponding 3-folds obtained by taking full level p-structure of the corresponding Siegel 3-fold are of general type. (Note that it is essentially known when is either geometrically rational of general type, see for example Theorem II.2.1 and the surrounding comments in this paper.) Of course, unlike the case of curves, varieties of dimension greater than one of general type can have many rational points. For example, it’s obvious that there are many abelian surfaces over Q whose 5-torsion has the form because one can take A to be E + E where E is an elliptic curve whose 5-torsion has the form and there are infinitely many such E because the classical modular curve of full level 5 is rational over Q. To put it a different way, the 3-fold A_2(5) corresponding to abelian surfaces with fixed 5-torsion will contain a number of rational Shimura subvarieties coming both from Hilbert modular surfaces and from modular curves, even though it itself is of general type. This can happen even if the mod-p representation rho is irreducible. For example, given an elliptic curve over a quadratic field K/Q, there will once more be a rational curve of elliptic curves with the same mod-5 representation, and so the restriction of scalars will give a rational curve on some twist A_2(rho) of A_2(5). On the other hand, one might at least start off by making the following naive minimal guess.

**Question** Suppose that is surjective for g = 2 and p >= 5. Then are there only finitely many points on

An even more extreme version of this question would be to ask if there is at *most one* such point. This seems a little unlikely even by comparison with the case of g=1. I learnt the following nice example talking to John Cremona during the hike through the Black Forest: for g=1 and p=7 and varying E, the twist X(E[7]) has genus 3 (it is a twist of the Klein quartic). This twist is still geometrically a plane quartic. By considering the tangent to the point of X(E[7]) corresponding to E, the line has two further intersections with the curve, and one obtains two further points over X(E[7]) which now (in general) lie over a quadratic extension. But one can parametrize the E for which these points are actually *rational* and this turns out to be the rational cover of the -line corresponding to asking that the invariant c_4 is a square. So there are infinitely many elliptic curves A (even with A[7] surjective) for which there exist at least a pair of non-isogenous elliptic curves A,B with A[7]=B[7] as symplectic Galois representations. So a better question is the following:

**Question** Can one find examples of non-isogenous abelian surfaces A and B with A[5]=B[5] and such that the corresponding representation has a surjective Galois representation?

This is the type of question where it is useful to have Andrew Sutherland nearby with a laptop. Within an hour or two, he sent me the following examples:

both of conductor with surjective and isomorphic mod-5 Galois representations which are not isogenous. Nice!

Naturally, the question turned to the existence of a pair with A[7] = B[7]. That proved a tougher challenge, but not an insurmountable one, and here is such a pair (again found by Andrew the same day):

this time of conductor

Any guesses as to whether there are any such pairs for I’m not sure I have any idea.

Other news from Oberwolfach:

I do appreciate being invited to the Oberwolfach conference on computational number theory — it pushes me outside my usual range of interests. It’s also the conference I have attended most often, now 8 times since 2003, although even that is far fewer than some of the regular participants. The conference is also chance to see a bunch of people I pretty much never get to see anywhere else. Even better, they are all nice enough to still invite me after this post. On the other hand, every time I give a talk I think that *this is the time that I finally have something interesting to say to this audience*, and it never quite seems to work out that way. I was certainly convinced that this was going to be the year, but then during my talk I managed to catch three people asleep in the front row. To be fair, it was the third last talk of the conference. On the other hand, Mike Bennett talked directly after me and completely failed to rise to my level of soporificness, despite his best efforts and his own predictions he would do otherwise.

There was a lunar eclipse on the final night of our stay. Most of us took to the roof to observe it, but the tall mountains of the Schwarzwald obscured our view until the final moment. Mike Bennett took the following photo, which he describes as the “best of a bad bunch”

For comparison, here is my best photo of the same scene:

Finally, if you are travelling to Oberwolfach during the summer, you mind will naturally turn to the question of whether you can enjoy the warm evenings by sipping on an Aperol spritz or two. Certainly they have lots of sparkling water, and so if Champagne (or equivalent) is available, you will easily be able to set yourself up simply by bringing along a bottle of Aperol. You might then consider trying to determine in advance whether they actually have any sparkling wine. Let me inform you, gentle reader, that the answer to that question is a definite yes, although you may have to look at the alcohol supply near the lecture theatre rather than near the dining room:

Alas, my own intelligence was not up to snuff, as my informant on Facebook who was visiting the previous week was unable to locate this bottle.

There can be only one way to end this post (mostly only relevant to those brought up in Australia in the early 80s or perhaps in England a decade or so earlier:

**Now for a walk in the Black Forest:**

(10:40, 11:00, 18:52, 22:06 for relevant times if the timestamp link doesn’t work for you…)

]]>The AIM model of conferences encourages real time collaboration, which is unusual as far as mathematical conferences go. But the ne plus ultra of such a conference (among those I have attended) was not at AIM at all, but rather organized by the University Bristol (Although to be fair, I believe it was organized specifically by Brian Conrey). The mission was to take a group of mathematicians and have them work on a very specific problem (which we were not told about in advance). The result: we failed to solve it (c’est la vie). On the other hand, I met a bunch of interesting mathematicians for the first time. My records are spotty, but I did manage to dig out some (poorly executed) photographic evidence from the time, which I present to you below.

The conference was actually located in Clifton rather than Bristol. It didn’t look much like its namesake Clifton Hill (in Melbourne) to me.

Emmanuel Kowalski and Mark Watkins heading towards the Clifton Suspension Bridge. (Check out the snazzy red suitcase!) The conference centre was located in an old manor (Burwalls house, now apparently sold by the University of Bristol to a developer) which is visible in the photo as the orange brick building to the left.

I can’t quite tell if the red suitacase has now transformed into a red backpack or if this is a different day and my fellow blogger has a predilection for vermillion satchels.

Akshay Venkatesh (I’m not going to comment on the hair colour.)

Soundararajan (see comment above)

Elon Lindenstrauss and Erez Lapid

Ben Green and Jon Keating

Brian Conrey and David Farmer.

This collection of photos is definitely incomplete: attending but missing from the photos includes William Stein (who I’m pretty sure was there) and Andy Booker and Sally Koutsoliotas (who were both definitely there) [also Mike Rubinstein]. I think there were a few more local Bristol people as well.

Other things I learnt at this conference: the naive Ramanujan conjecture is false for GSp(4), pork pies are pretty much best avoided, and Collins’s 628.

]]>One could (perhaps) equally offer commiserations — at this point, such a prize has little to no effect on your professional advancement, but may well seriously increase low-level harassment both from the press and from certain types of graduate student who will now follow you around in a ring at conferences at a respectful distance of two to six feet. (Perhaps they already do.)

Of course, echoing Emmanuel’s sentiments, in addition to not forgetting the collaborators of this year’s Fields medalists, let us *especially* not forget those people (currently estimated as a set of size one cough cough) who have collaborated with *at least two* of this year’s winners — you have ~~clearly inspired~~ [**edit:** adeptly ridden the coattails of] the next generation of mathematicians.

First, let me recall the previous status of this conjecture. An explicit form of this conjecture (detailing all the 52 possible different Sato-Tate groups which could occur for abelian surfaces over number fields — 34 of which occur over Q) was given in a paper of Fité, Kedlaya, Rotger, and Sutherland (I recommend either reading these slides or especially watching this video for the background and some fun animations). Christian Johansson gave proofs of this conjecture over totally real fields in many of the possible cases in which the abelian surface had various specific types of extra endomorphisms over the complex numbers by exploiting modularity results that had been used in the proof of the Sato-Tate conjecture for elliptic curves. Over totally real fields, this left essentially four remaining cases:

- The case when the Galois representations associated to A decomposes over a quadratic extension L/F into two representations which are Galois twists of each other, and L/F is not totally real.
- The case when the Galois representations associated to A decomposes over a quadratic

extension L/F into two representations which are not Galois twists of each other,*and L/F is CM.* - The case when the Galois representations associated to A decomposes over a quadratic

extension L/F into two representations which are not Galois twists of each other, and L/F is*neither totally real nor CM.* - The case when the geometric endomorphism ring of A is

Noah has something to say about each of these cases.

**Case 1**: Noah completed the proof of Sato-Tate in this case using only the methods from BLGGT, by exploiting the fact that the corresponding two-dimensional representations — while possibly only defined over a field L which need not be totally real or CM — in fact give rise to *projective* representations which extend to F. By a theorem of Tate, each of these representations can be extended to F after twisting by a character, and so the original 4-dimensional representation looks like the tensor product of a 2-dimensional representation over F (which is potentially modular) and an Artin representation. At this point one is in good shape.

**Case 2**: The Sato-Tate conjecture is proved in this case. This case required the least amount work, because it is pretty much an immediate consequence of the modularity results proved in the 10-author paper.

If the totally real field is Q, this implies the Sato-Tate conjecture for all abelian surfaces except those of type (4).

**Cases 3 & 4**: In these cases, one can apply the potentially modularity results proved in my (very close to being finished) paper with Boxer, Gee, and Pilloni. It is too much to expect a full proof of Sato-Tate at this point. However, knowing potential modularity allows one to obtain partial results, similar to those of Serre and Kim-Shahidi for the case of elliptic curves (after Wiles but before Clozel-Harris-Taylor). Here is a sample result:

**Theorem** (Noah Taylor). Let C be a genus two curve over a totally real field F. Then, for any there exists a positive density of primes (with one has

Compare this to the Hasse bounds, which imply that the quantity on the LHS has absolute value at most

Of course this theorem is much weaker than the Sato-Tate conjecture. But even the weaker version of this theorem which says that for infinitely many primes was *completely open* before such curves were known to be potentially modular. Similarly, I don’t think one can prove the corresponding result for elliptic curves without either using something very close to modularity (in the non-CM case) or the equidistribution theorems of Hecke in the CM case. I think the following example is instructive: take the elliptic curve which admits CM by the Gaussian integers. One has a formula for the difference as follows: for a prime which is 1 mod 4, one may write p = a^2 + b^2 uniquely in integers by imposing the additional congruence

Then one has the formula

The problem then becomes: do there exist infinitely many primes p = 1 mod 4 such that one has This seems suspiciously like something that can be proven using Cebotarev, but it is not. The problem is that the infinite places of are all complex, so there is no choice of “conductor” which differentiates between complex numbers with positive or negative real part at the infinite places in

Noah’s proof of the theorem above exploits the following idea. Potential modularity not only gives meromorphy of the L-function, but more importantly (in this case) holomorphy and non-vanishing in the (analytically normalized) halfplane Re(s) >= 1. Moreover, again using functorialities, potential automorphy, and results of Shahidi, one obtains similar results not only for the degree 4 L-function, but also the degree 5 L-function, and also crucially the Rankin-Selberg L-functions of degrees 16, 20, and 25. From this one can obtain various “prime number theorem” estimates for quantities involving the Frobenius eigenvalues, and then one has to massage these into an inequality. A simple version of this argument is as follows: given some infinite set of real numbers such that

One can draw the conclusion that infinitely often, by (for example) considering the average of the quantity Moreover, this is the best possible bound given these constraints.

Note that since the Sato-Tate conjecture is known in all other cases, one only has to consider cases (3) and (4), which behave slightly differently in this argument. In fact, in case (3), one can do much better:

**Theorem** (Noah Taylor). Let C be a curve over a totally real field F such that is of type (3). Then there exists a positive density of primes (with such that

(Note that once this result is known in case (3) it is known for all curves whose Jacobian is not of type (4), that is, those whose Jacobians admit a non-trivial endomorphism over The point is that, in this case, one knows not just the potential automorphy of A, but also the potential automorphy of the corresponding two-dimensional representations over the quadratic extension L, and so one can also exploit the automorphy of symmetric powers of the corresponding GL(2)-automorphic representations (and further analyticity results for higher symmetric powers) as well as a zoo of Rankin-Selberg L-functions coming from pairs of low symmetric powers. (As for the constants involved in both of these theorems, they are essentially optimal given the automorphic input.)

These results tie in to problems raised in various talks of Nick Katz (see for example this talk). Noah’s result above implies that, given an curve C over a totally real field, one *can tell* that it doesn’t have genus one from the distribution of the traces of Frobenius *except* possibly in the case when its Jacobian has no non-trivial geometric endomorphism (the “typical” case, of course). It’s a little sad that the modularity results are not sufficient to handle that last case as well — showing that the support of the normalized trace of Frobenius extends beyond would require knowing something close to functoriality of the map and this is currently out of reach, unfortunately. Oh well, that’s a shame: wow I dearly would have loved to give a talk entitled *Simple things that Nick Katz doesn’t know (but I do)*.

**added:** The final word from Chicago ITS:

Hello,

We have looked into other reasons as to why this functionality is the way it is and it seems to be a part of DUO authentication that cannot be changed. The best suggestion we [have is] using VPN when accessing university resources if you want it to remember you for 30 days. Otherwise, 3rd party cookies will have to be enabled to have this work without using VPN. Using the University VPN does have advantages if you are working in a public network.

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====================

]]>Current obsession:

This plaintive choral work, possibly an incomplete fragment of an unfinished anthem, is essentially one long phrase building to a final resolution with just the right amount of dissonance. There are a number of fine recordings on youtube, but this performance is my current favourite. (It appears to be tuned in something close to a baroque pitch with A = 415, but I’m neither sure how accurate that estimate is nor how deliberate that specific choice was — those of you who know how to take Fourier transforms in practice can tell me how close I am.)

**Edit:** OK, thanks to an old answer here by user Lenoil (I love it how the Stack Exchange websites are stocked with competent experts able to answer so many questions; a pity there is no equivalent in mathematics), I could at least clip out the very first note in the file and let Mathematica compute the corresponding Fourier transform. There’s still a little vibrato which makes for inaccuracies, but here was the result from the opening “hear”:

Assuming (very generously to myself) that the second peak is the result of vibrato, I think this suggests an opening note very close to 245Hz. The piece itself is in C (minor) and opens on a middle C. Assuming (equal temperament) an A = 415 tuning, the resulting middle C should in fact be approximately

Make of that what you will (probably not so much, I imagine).

]]>Jordan was very careful about notation in his talk. He had previously used as a symbol to denote an integer, so he carefully used to denote an object which admits a map to :

Answered here: What is BG? Not answered here: why do two corners of this pullback square look suspiciously like the same symbol?

As for my own talk, having previously tried to give some technical talks on math related to the CG-method which went horribly wrong, this time I gave a slide talk on my work with Boxer, Gee, and Pilloni which was all candy and no vegetables. (Summary of talk: first Riemann, then Wiles, and now us). Kai-Wen legitimately expressed disappointment at the lack of details (fair enough, you can’t fault that guy for skipping details). Otherwise, it elicited the reminder from Dick Gross that — although I could get by and make a living doing this sort of thing — it was time for number theorists to escape the “ghetto of holomorphic forms” (a phrase I think he attributed to someone else, I should say). Hey, Dick, don’t I at least get points from escaping the even worse “swamplands of discrete series”?

For those playing Barry Mazur bingo (sample squares: Gorenstein, “but…that’s beautiful”, there were plenty of opportunities to see the influence of Barry’s mathematics. There was, however, a novel aspect of the conference which was an interdisciplinary day consisting of three conversation sessions of Literature/Poetry, History of Science, and Philosophy/Law/Physics respectively. By all accounts this was a wildly successful enterprise (hat-tip to the organizers). I did have one question I would have liked to ask one of the historians of mathematics, but the theme of the conversation meandered elsewhere. Instead I shall ask it here into the void (I’m not accusing you, dear reader, of being a void, merely that there are probably not any actual *historians* reading this blog):

*A working mathematician usually has a very interpretative (and somewhat anachronistic) view of the history of mathematics: Galois “knew” which groups acted on p points, Gauss “knew” XYZ about class groups, etc. Mathematicians feel confident in these interpretations even if they are not explicitly written in this form in the original texts. What are the dangers in this (Whiggish?) view of the history of mathematics?*

**Cambridge Culinary Roundup:**

With conference banquets (with some touching and amusing speeches by those who knew Barry well) and receptions going on, there was only a limited time for dining, not to mention the problem of trying to book restaurants at the last moment. Still, there was some opportunity to revisit some familiar and some new places:

**Burger versus Burger:** When it comes to Cambridge burgers, there is only one possible choice…or is there? My general impression was that the only way you could like Mr Bartley’s was if you were first exposed to it before your culinary tastes had a chance to develop (i.e. as a drunk undergraduate). On the other hand, a Cragie on Main burger (circa 2012) was as close in my mind to burger perfection as you could get. But did either of these opinions hold up today? Thus was the origin of the burger versus burger challenge. The participants for round one (Tuesday lunch) included myself, Quomodocumque,

The Hawk, Akshay, Joel, and Bisi. (Although Bisi was participating in a slightly different show, namely the latest episode of “mathematicians trick Bisi into going to a grungy restaurant.”) Round two was Tuesday dinner. Bisi and Joel dropped out on the reasonable basis that they had already consumed enough saturated fat for one week, but the rest of us continued on.

The conclusion? Cragie on Main clearly serves the superior burger (as noted by the Hawk, the fact that the request for a “medium rare” burger came out medium rare at Cragie on Main versus medium well at Bartley’s meant there could be no other conclusion). But perhaps inevitably, my opinions were forced to be somewhat softened in both directions. Bartley’s really did a decent job as far as the overall taste was concerned, and Cragie on Main’s burger — while better — stopped well short of being transformative. I suspect that they’ve been coasting for too long and haven’t maintained the level of excellence they started with (maybe that’s true of Bartley’s too, although I didn’t get a chance to eat there in 1960). In fact, there’s a generally sound principle to be a little wary about restaurants which have been around for too long. (Having said that, I would still find it hard to skip going to Rivoli restaurant on my next trip to Berkeley.)

**Chess:** Au Bon Pain has disappeared! The entire Holyoke center building is under some sort of reconstruction. The chess players are still around, however, having moved to (literally) the triangle that is Harvard square. I played a few games, and was prettily solidly crushed by a 2300 player in some lightning games. I also declined to play a $10 lightning game against an IM with generous odds, not because I thought I didn’t have a 50% chance of winning, but because I didn’t think I had a 90% chance of winning, and losing would have been at least 10 times more annoying than winning would have been pleasant.

**Coffee:** 5 (or so) years ago, Crema was a revolution in Harvard Square (i.e. drinkable coffee, reasonable hipster attitude). While their coffee was never at the level of something like Voltage Coffee (near MIT, and sadly now gone), it made staying at the car park known as the Harvard Square Hotel a more palatable option than at the “quaint” Irving street B&B. Times have changed! Crema is a victim of its own success — in a busy place which requires a frequent changeover of staff, the emphasis on coffee no longer seems to be paramount, and the quality control has dropped precipitously. The result was high inconsistency. Out of four coffees I got there, two were OK, one was pretty bad, and one was send directly into the bin. (I would like to have said “tossed in an elegant arc directly into the rubbish bin,” but if I had really attempted that, it would have been more like “unceremoniously spilt all over my shirt.”) As of today, there are definitely better options even in Harvard Square (further afield, one trip was made to Broadsheet which showed promise, even though my own flat white there was merely acceptable). Namely: Tatte Bakery & Cafe, which I really quite liked as far as the pastries and the sandwiches went, and the coffee itself rose to acceptable if not excellent standards.

Darwin’s is still Darwin’s (I prefer Tatte), Night Market (inspired by asian street food) was pretty interesting (some tasty eggplant) if a little idiosyncratic, and Parsnip did a perfectly good job of replacing “Upstairs at the Square,” a restaurant at which I had many a dinner when I used to live in Cambridge. (I had my eye on a few other restaurants, but none of them could take at short notice a reservation for 4 on a Thursday, so Parsnip was especially good given the constraint of not being so popular.

If you have suggestions of better coffee that I may have missed, please make suggestions since I will be returning in November. I also hope the Cambridge weather in November is more like June weather, given that the weather this week was more like November weather:

]]>Slightly more into the future and for a slightly different audience, it has now bene confirmed that there will be a special semester on “The arithmetic of the Langlands program” at the Hausdorff Institute in Bonn during May 4 — August 21 in 2020 (organized by Ana Caraiani, Laurent Fargues, Peter Scholze, and me).

]]>**Anthony Várilly-Alvarado:** Honestly, I’ve never quite forgiven this guy for his behavior as an undergraduate. He was my TA when I taught Complex Analysis at Harvard, and he had the bad manners to do an absolutely wonderful job and be beloved by all the students. Nothing makes a (first time) professor look worse than a good TA. (It means I can’t even take any credit for the students in the class who became research mathematicians). Anyway, Tony gave a talk on his joint work with Dan Abramovich about the relation between Vojta’s conjecture and the problem of uniform bounds on torsion for abelian varieties. (Spoiler: one implies the other.) More specifically, assuming Vojta’s conjecture, there a universal bound on (depending only on and beyond which no abelian variety of dimension over can have full level structure.

If one wanted to prove this (say) for elliptic curves, and one was allowed to use any conjecture you pleased, you could do the following. Assume that for some large integer m. One first observes (by Neron-Ogg-Shafarevich plus epsilon) that E has to have semi-stable reduction at primes dividing N_E. Then the discriminant must be an th power, and then Szpiro’s Conjecture (which is the same as the ABC conjecture) implies the desired result.

If you try to do the same thing in higher dimensions, you similarly deduce that A must have semi-stable reduction at primes dividing N_E. **edit: some nonsense removed.** One then gets implications on the structure of the Neron model at these bad primes, which one can hope to parlay in order to get information about local quantities associated to A analogous to the discriminant being a perfect power. But I’m not sure what generalizations of Szpiro’s conjecture there are to abelian varieties. A quick search found one formulation attributed to Hindry in terms of Faltings height, but it was not immediately apparent if one could directly deduce the desired result from this conjecture, nor what the relationship was with these generalizations to either ABC or to Vojta’s conjecture.

**Ilya Khayutin:** Ilya mentioned Linnik’s theorem that, if one ranges over imaginary quadratic fields in which a fixed small prime is split, the CM j-invariants become equidistributed. The role of the one fixed prime is to allow one to use ergodic methods relative to this prime. My naive question during the talk: given p is split, let be a prime above p. Now one can take the subgroup of the class group corresponding to the powers of Do these equidistribute? The speaker’s response was along the lines that it would probably be quite easy to see this is false, but I didn’t have time after the talk to follow up. It’s certainly the case that, most of the time, the prime will itself generate a subgroup of small index in the class group (the quotient will look like the random class group of a real quadratic field), but sometimes it will be quite large. For example, I guess one can take

and the subgroup generated by this prime has order compared to So I decided (well, after writing this line in the blog I decided) to draw a picture for some choice of Mersenne prime. And then, after thinking a little how to draw the picture, realized it was unnecessary. The powers of in this case are given explicitly by

It is transparent that for the first half of these classes, the first factor is much smaller than the second, but since the second term also has small real part, the ratio already lies inside the (standard) fundamental domain. Hence the corresponding points will lie far into the cusp. Similarly, the second half of the classes are just the inverses in the class group of the first half, and so will consist of the reflections of those points in and so also be far into the cusp. So I guess the answer to my question is, indeed, a trivial no. So here is a second challenge: suppose that 2 AND 3 both split. Then do the CM points generated by for primes above 2 AND 3 equidistribute? Actually, in this case, it’s not clear off the top of my head that one can easily write down discriminants for which the index of this group is large. But even if you can, sometimes subgroups get you much closer to equidistribution than

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