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

]]>The Sato-Tate conjecture implies that the normalized trace of Frobenius for a non-CM elliptic curve is equidistributed with respect to the pushforward of the Haar measure of SU(2) under the trace map. This gives a perfectly good account of the behavior of the unnormalized over regions which have positive measure, namely, intervals of the form for distinct multiples of

If one tries to make global conjectures on a finer scale, however, one quickly runs into difficult conjectures of Lang-Trotter type. For example, given a non-CM elliptic curve E over if you want to count the number of primes p < X such that (say), an extremely generous interpretation of Sato-Tate would suggest that probability that would be

and hence the number of such primes < X should be something like:

except one *also* has to account for the fact that there are congruence obstructions/issues, so one should multiply this factor by a (possibly zero) constant depending one adelic image of the Galois representation. So maybe this does give something like Lang-Trotter.

But what happens at the other extreme end of the scale? Around the boundaries of the interval [-2,2], the Sato-Tate measure converges to zero with exponent one half. There is a trivial bound where is the largest square less than 4p. How often does one have an equality Again, being very rough and ready, the generous conjecture would suggest that this happens with probability very roughly equal to

and hence the number of such primes < X should be something like:

Is it at all reasonable to expect primes of this form? If one takes the elliptic curve one finds to be as big as possible for the following primes:

but no more from the first 500,000 primes. That's not completely out of line for the formula above!

Possibly a more sensible thing to do is to simply ignore the Sato-Tate measure completely, and model by simply choosing a randomly chosen elliptic curve over Now one can ask in this setting for the probability that is as large as possible. Very roughly, the number of elliptic curves modulo up to isomorphism is of order and the number with is going to be approximately the class number of where perhaps it is even exactly equal to the class number for some appropriate definition of the class number. Now the behaviour of this quantity is going to depend on how close is to a square. If is very slightly — say — more than a square, then is pretty much a constant, and the expected probability going to be around in On the other hand, for a generic value of the smallest value of will have order and then the class group will have approximate size and so one (more or less) ends up with a heuristic fairly close to the prediction above (at least in the sense of the main term being around

But why stop there? Let's push things even closer to the boundary. How small can get relative to For example, let us restrict to the set of prime numbers p such that

For such primes, the relative probability that is approximately So the expected number of primes with this property will be infinite providing that

is infinite, or, in other words, when So this leads to the following guess (don't call it a conjecture!):

** Guess:** Let be an elliptic curve without CM. Is

Of course, one can go crazy with even more outrageous guesses, but let me stop here before saying anything more stupid.

]]>Only in the same sense as Captain Renault could you possibly be shocked (*shocked!*) by what Bancroft drops into his pants.

The 90th percentile quality coffee in Melbourne (random mall coffee) is at (approximately) the level of the 10th percentile coffee in Chicago. While there’s plenty of good coffee in Chicago, you don’t want into a random cafe and expect to get something drinkable. You also don’t expect any random place to have a top of the line Marzocco machines. But if you want a few recommendations in the neighbourhood of either Lygon street or near the state library, I can suggest Market Lane/Pool House/Seven Seeds/Vincent the Dog/The League of Honest Coffee/Vertue of the Coffee Drink to get you started. Expert tip at US hipster cafes: order a magic (3/4 flat white with double ristretto), then look unimpressed when they don’t know what you are talking about.

While you’re near the state library, stop off in the reading room for some speed chess (victory is mine!)

Australia has a lot of long beaches, and I don’t mean long in the sense of fractal dimension greater than one. I mean in the sense of having several miles of pristine beach to yourself:

Fight terrorism with philosophy! (and concrete bollards):

I always assumed that A’Beckett St was named after the turbulent priest. Not So! Apparently it is named after the first chief justice of Victoria. Upon learning this, I checked out the origins of the other street names in Melbourne’s CBD. Four of the North-South (ish) streets in order include (at some point) King-William-Queen-Elizabeth, and it is “common knowledge” that these streets are so named in pairs. Also false! William is named for King William IV, and Queen for Queen Adelaide, but King is named for Philip Gidley King, the governor of NSW from 1800-1806, and Elizabeth was “possibly” named for the wife of another Governor of NSW, Richard Bourke. (I did of course know that Bourke St (named after the guv) was not named after Burke, the explorer who (with Wills) became famous for his ludicrous incompetence.

]]>Somewhat less salaciously, I received a Christmas card from an academic couple of whom I am absolutely sure I have never met. (I just checked my mailbox at work for the first time this year, which is where this card was sent.) They work in a state I have never visited, and neither of them are mathematicians (though one appears to have a math PhD). They wished me the best “*on my career*.” Apparently (according to their website) one of them is an expert on “targeted killings.” I hope that “on my career” is not a euphemism.

As for mail more directly relevant to me and this blog, I did (three or so years ago?) receive an unsolicited package in the mail from a blog reader. The sender’s name (Leslie, I think?) was suitably unisex, so I naturally assumed that it was a swooning 20-something female who had fallen for my prose and occasional deliberate grammatical and spelling errors. But the reality was better: it was (as far as I could tell from a google search) a 60 year old male with a PhD in math, who send me a CD with some Schumann Lieder, in particular an Edith Mathis CD (with Christoph Eschenbach on the piano) entitled “Frauenliebe und Leben & other Lieder”. Absolutely wonderful! Through a quirk of fate this CD has ended up in my car, and has been in heavy rotation over the past year during my commute. Because I wouldn’t want you to miss out, I’ve given a youtube link to one of the songs below (*Kennst du das Land* — not from the titular cycle, but chosen in part because the accompaniment reminds me stylistically of *Dichterliebe*, partly because it sounds good, and partly because Robert wrote it for Clara and it is Valentine’s day). But it makes me wonder: what type of fan mail does Quomodocumque get? (or, for that matter, Terry Tao and Sir Timothy Gowers, FRS)

From smoked juniper bushes to edible stones, I arrived at Alinea fearing that I would be underwhelmed but left leaving very happy. Rather than try to describe my meal in any detail, let me instead just post a few teaser pictures. I came into the experience without any preconceived notions of what to expect, which I recommend. (I also wasn’t paying, which I also recommend.)

The trick is to guess how many of these are edible.

The decorations and the food are intertwined. This was a feature of many dishes of this —and I am guessing most — meals at Alinea.

This had some Foie Gras, I believe.

This chocolate mousse hiding in one of these stones was delicious.

I am not a number! I am a free man!

Many dishes omitted, of course. One notable course involved a pomegranate cocktail which had a slight bouquet of christmas pudding; possibly the best cocktail I have ever had. Indeed, that has inspired one of my upcoming culinary choices for next two weeks, which include: the Violet Hour, L’Etoile for the number theory seminar dinner at Madison (this one is more of a suggestion than a concrete plan, but JSE, can we make it happen?), and the Victoria Market.

]]>But this post is less about the mathematics (hopefully more on that when the theorems are proved), but rather my other (mostly culinary) adventures.

My first night out, I was curious how Ethiopian food in Israel would compare to Chicago. However, my taxi driver had other ideas, and instead took me to a Kurdish restaurant (Ima) where I ended up with a pretty nice lamb dish. While walking home from dinner, I stumbled across the Chords Bridge (the “Bridge of Strings”). My particular approach presented me with a visual paradox: the bridge appeared to be straight with a central column with cables to either side. These cables appeared as lines sweeping out a ruled surface. Since the bridge was straight, these two surfaces should essentially have formed one surface, but they appear to meet at right angles at the column, which made no sense. Since my description also probably doesn’t make so much sense, I took a video:

(Admittedly my geometrical intuition is not so great, but I couldn’t work out what was going on until I saw it again from a different angle.)

Tuesday morning was my “time off” as a tourist. I think the old city would have been much better with a local guide, but I mostly just wandered around between ancient sites and an infinite number of tchotchke shops. Next stop was Machane Yehuda market, and lunch at the hippest restaurant in town. The shikshukit was delicious:

Next stop was the fanciest coffee in Jerusalem (not that good)

Akshay and I went back to the market for dinner and had the shamburak at Ishtabach along with some pretty good local beer.

On Wednesday, I was contemplating going straight back to the hotel and going to sleep after an undergraduate lecture by Akshay (full jetlag mode at this point, the talk itself was great). But then I ran into Alex Gamburd, who suggested going out to dinner and said he knew of a place which sold food from “biblical times.” At that point, my spirits were instantly lifted, and there was no choice about what I was going to do. So we jumped into a taxi and off we went to Eucalyptus, to have (amongst other things) chicken stuffed in figs (yes, I thought that was just a poor english translation for figs stuffed in chicken, but no, chicken stuffed in figs). The owner came out to chat with us, and claimed that this dish had won a special prize in Melbourne and had also appeared in Vogue (I couldn’t verify these claims, but they were tasty).

A few more things *en passant:*

A “reception” at Hebrew University apparently does not include Champagne, much for the worse for anyone who had to suffer though my subsequent basic notions seminar. (Hat tip to Michael Schein for telling me this in advance.)

Here’s Alex Gamburd and Andre Reznikov arguing over a point concerning Stalin:

Near the old city:

The campus appears to be overrun by cats. Well, overrun is an exaggeration, but then saying the campus is “run by cats” conveys a somewhat different image (which may or may not be accurate).

]]>Brumer and Kramer made a conjecture positing a bijection between isogeny classes of abelian surfaces over the rationals of conductor N with and paramodular Siegel newforms of level N with rational eigenvalues (up to scalar) that are not Gritsenko lifts (Gritsenko lifts are those of Saito-Kurokawa type). This conjecture is closely related to more general conjectures of Langlands, Clozel, etc., but its formulation was made more specifically with a view towards computability and falsifiability (particularly in relation to the striking computations of Poor and Yuen).

The recognition that the “optimal level” of the corresponding automorphic forms is paramodular is one that has proved very useful both computationally and theoretically. Moreover, it is almost certain that something very close to this conjecture is true. However, as literally stated, it turns out that the conjecture is false (though easily modifiable). There are a few possible ways in which things could go wrong. The first is that there are a zoo of cuspidal Siegel forms for GSp(4); it so happens that the forms of Yoshida, Soudry, and Howe–Piatetski-Shapiro type never have paramodular eigenforms (as follows from a result of Schmidt), although this depends on the accident that the field has odd degree and no unramified quadratic extensions (and so the conjecture would need to be modified for general totally real fields). Instead, something else goes wrong. The point is to understand the relationship between motives with -coefficients and motives with -coefficients which are invariant under the Galois group (i.e. Brauer obstructions and the motivic Galois group.)

It might be worth recalling the (proven) Shimura-Taniyama conjecture which says there is a bijection between cuspidal eigenforms of weight two with rational eigenvalues and elliptic curves over the rationals. Why might one expect this to be true from general principles? Let us imagine we are in a world in which the Fontaine-Mazur conjecture, the Hodge conjecture, and the standard conjectures are all true. Now start with a modular eigenform with rational coefficients and level Certainly, one can attach to this a compatible family of Galois representations:

with the property that the characteristic polynomials of Frobenius at any prime have *integer* coefficients, and the representations are all de Rham with Hodge-Tate weights [0,1]. But what next? Using the available conjectures, one can show that there must exist a corresponding simple abelian variety which gives rise to The key to pinning down this abelian variety is to consider its endomorphism algebra over the rationals. Because it is simple, it follows that the endomorphism algebra is a central simple algebra for some number field F. From the fact that the coefficients of the characteristic polynomial are rational, one can then show that the number field F must be the rationals. But the Albert classification puts some strong restrictions on endomorphism rings of abelian varieties, and the conclusion is the following:

Either:

- is an elliptic curve.
- is a fake elliptic curve; that is, an abelian surface with endomorphisms over by a quaternion algebra

The point is now that the second case can never arise; the usual argument is to note that there will be an induced action of the quaternion algebra on the homology of the real points of A, which is impossible since the latter space has dimension two. (This is related to the non-existence of a general cohomology theory with rational coefficients.) In particular, we do expect that such modular forms will give elliptic curves, and the converse is also true by standard modularity conjectures (theorems in this case!). A similar argument also works for all totally real fields. On the other hand, this argument does *not* work over an imaginary quadratic field (more on this later). In the same way, starting with a Siegel modular form with rational eigenvalues whose transfer to GL(4) is cuspidal, one should obtain a compatible family of irreducible 4-dimensional symplectic representations with cyclotomic similitude character. And now one deduces (modulo the standard conjectures and Fontaine-Mazur conjecture and the Hodge conjecture) the existence of an abelian variety A such that:

Either:

- is an abelian surface.
- is a fake abelian surface; that is, an abelian fourfold with endomorphisms over by a quaternion algebra

There is now no reason to suspect that fake abelian surfaces cannot exist. Taking D to be indefinite, the corresponding Shimura varieties have dimension three, and they have an abundance of points — at least over totally real fields. But it turns out there is a very easy construction: take a fake elliptic curve over an imaginary quadratic field, and then take the restriction of scalars!

You have to be slightly careful here: one natural source of fake elliptic curves comes from the restriction of certain abelian surfaces of GL(2)-type over and one wants to end up with fourfolds which are simple over Hence one can do the following:

**Example:** Let be an abelian surface of GL(2)-type which acquires quaternion multiplication over an imaginary quadratic field K, but is not potentially CM. For example, the quotient of with coefficient field with Take the restriction to K, twist by a sufficiently generic quadratic character and then induce back to Then the result will be a (provably) modular fake abelian surface whose corresponding Siegel modular form has rational eigenvalues. Hence the paramodular conjecture is false.

Cremona (in his papers) has discussed a related conjectural correspondence between Bianchi modular forms with rational eigenvalues and elliptic curves over K. His original formulation of the conjecture predicted the existence of a corresponding elliptic curve over K, but one also has to allow for fake elliptic curves as well (as I think was pointed out in this context by Gross). The original modification of Cremona’s conjecture was to only include (twists of) base changes of abelian surfaces of GL(2)-type from Q which became fake elliptic curves over K, but there is no reason to suppose that there do not exist fake elliptic curves which are autochthonous to K, that is, do not arise after twist by base change. Indeed, autochthonous fake elliptic curves do exist! We wrote down a family of such surfaces over for example. (We hear through Cremona that Ciaran Schembri, a student of Haluk Sengun, has also found such curves.) On the other hand, the examples coming from base change forms from Q have been known in relation to this circle of problems for 30+ years, and already give (by twisting and base change) immediate counter-examples to the paramodular conjecture, thus the title.

It would still be nice to find fake abelian surfaces over (rather than totally real fields) which are geometrically simple. I’m guessing that (for D/Q ramified only at 2 and 3 and a nice choice of auxiliary structure) the corresponding 3-fold may be rational (one could plausibly prove this via an automorphic form computation), although that still leaves issues of fields of rationality versus fields of definition. But let me leave this problem as a challenge for computational number theorists! (The first place to look would be Jacobians of genus four curves [one might be lucky] even though the Torelli map is far from surjective in this case.)

Let me finish with one fake counter example. Take any elliptic curve (say of conductor 11). Let be any Galois extension with Galois group the quaternion group of order 8. The group has an irreducible representation of dimension 4 over the rationals, which preserves a lattice If you take

then is a simple abelian fourfold with an action of an order in (now the definite Hamilton quaternions) and so gives rise to compatible families of 4-dimensional representations which are self-dual up to twisting by the cyclotomic character. However, the four dimensional representations are only symplectic with respect to a similitude character which is the product of the cyclotomic character and a non-trivial quadratic character of and instead they are orthogonal with cyclotomic similitude character. So these do not give rise to counterexamples to the paramodular conjecture. A cursory analysis suggests that the quaternion algebra associated to a fake abelian surface which gives rise to a symplectic with cyclotomic similitude character should be indefinite.

]]>Lizard 2: So they develop character! If it was good enough for me, it’s good enough for them.

(Feel free to choose your own metaphors.)

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