The boundaries of Sato-Tate, part I

A caveat: the following questions are so obvious that they have surely been asked elsewhere, and possibly given much more convincing answers. References welcome!

The Sato-Tate conjecture implies that the normalized trace of Frobenius b_p \in [-2,2] 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 a_p \in [-2 \sqrt{p},2 \sqrt{p}] over regions which have positive measure, namely, intervals of the form [r \sqrt{p},s \sqrt{p}] for distinct multiples of \sqrt{p}.

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 \mathbf{Q}, if you want to count the number of primes p < X such that a_p = 1 (say), an extremely generous interpretation of Sato-Tate would suggest that probability that a_p = 1 would be

\displaystyle{\frac{1}{4 \pi \sqrt{p}}},

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

\displaystyle{\frac{X^{1/2}}{2 \pi \log(X)}},

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 a_p  \le   t where t^2 is the largest square less than 4p. How often does one have an equality a^2_p = t^2? Again, being very rough and ready, the generous conjecture would suggest that this happens with probability very roughly equal to

\displaystyle{\frac{1}{6  \pi p^{3/4}}},

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

\displaystyle{\frac{2 X^{1/4}}{3  \pi \log(X)}}.

Is it at all reasonable to expect X^{1/4 \pm \epsilon} primes of this form? If one takes the elliptic curve X_0(11), one finds a^2_p to be as big as possible for the following primes:

a_{2} = -2 \ge  -2 \sqrt{2} = -2.828\ldots,

a_{239} = -30 > -2 \sqrt{239} =  -30.919\ldots,

a_{6127019} = 4950   \le  2 \sqrt{p} = 4950.563\ldots,

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 E/\mathbf{F}_p by simply choosing a randomly chosen elliptic curve over \mathbf{F}_p. Now one can ask in this setting for the probability that a_p is as large as possible. Very roughly, the number of elliptic curves modulo p up to isomorphism is of order p, and the number with a_p = t is going to be approximately the class number of \mathbf{Q}(\sqrt{-D}) where -D = t^2 - 4p; perhaps it is even exactly equal to the class number H(t^2 - 4p) for some appropriate definition of the class number. Now the behaviour of this quantity is going to depend on how close 4p is to a square. If 4p is very slightly — say O(1) — more than a square, then H(t^2 - 4p) is pretty much a constant, and the expected probability going to be around 1 in p. On the other hand, for a generic value of p, the smallest value of t^2 - 4p will have order p^{1/2}, and then the class group will have approximate size p^{1/4 \pm \epsilon}, 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 X^{1/4 \pm \epsilon}).

But why stop there? Let's push things even closer to the boundary. How small can a^2_p - 4p get relative to p? For example, let us restrict to the set S(\eta) of prime numbers p such that

\displaystyle{S(\eta):= \left\{p \ \left| \ p \in (n^2,n^2 + n^{2 \eta}) \ \text{for some} \ n \in \mathbf{Z} \right.\right\}}.

For such primes, the relative probability that a_p = \lfloor \sqrt{4p} \rfloor = 2n is approximately n^{\eta}/p \sim n^{2 \eta - 1}. So the expected number of primes with this property will be infinite providing that

\displaystyle{\sum \frac{n^{3 \eta}}{n^2 \log(n)}}

is infinite, or, in other words, when \eta \ge 1/3. So this leads to the following guess (don't call it a conjecture!):

Guess: Let E/\mathbf{Q} be an elliptic curve without CM. Is

\displaystyle{\liminf \frac{\log(a^2_p - 4p)}{\log(p)} = \frac{1}{3}?}

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

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Australiana

Some short observations from my recent trip:

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.

More Coffee!More Coffee!Even More Coffee!

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:

Beach

Fight terrorism with philosophy! (and concrete bollards):

Chicago Philosophy

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.

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Fan Mail

Edit: A previous version of this post has been edited upon request.

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)

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Alinea

Fresh off my dining experiences in Jerusalem, I returned home for some more local dining.

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.)

Stones

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

Juniper

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

Berries

This had some Foie Gras, I believe.

Dessert Stones

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

Balloon

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

Rover

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.

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This year in Jerusalem

I just returned from spending almost a week in Jerusalem (my first ever visit to Israel). The main reason for my visit was to talk with Alex Lubotzky (and Shai Evra) about mathematics, but there was plenty of time for other mathematical activities — I gave a four hour talk on cohomology to computer scientists, chatted with Kazhdan (and also Akshay), caught up with Alex Gamburd, Ehud de Shalit, and went to the presentation of the Ostrowski medal to Akshay Venkatesh (with a virtual laudation by Peter Sarnak).

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:

Lunch

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

Coffee

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:

Stalin

Near the old city:

Magnum

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).

Cats

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The paramodular conjecture is false for trivial reasons

(This is part of a series of occasional posts discussing results and observations in my joint paper with Boxer, Gee, and Pilloni mentioned here.)

Brumer and Kramer made a conjecture positing a bijection between isogeny classes of abelian surfaces A/\mathbf{Q} over the rationals of conductor N with \mathrm{End}_{\mathbf{Q}}(A) = \mathbf{Z} 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 \mathbf{Q} 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 \mathbf{Q}-coefficients and motives with \overline{\mathbf{Q}}-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 \Gamma_0(N). Certainly, one can attach to this a compatible family of Galois representations:

\displaystyle{\mathcal{R} = \{\rho_p\}, \qquad \rho_p: \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) \rightarrow \mathrm{GL}_2(\overline{\mathbf{Q}}_p).}

with the property that the characteristic polynomials P_q(T) = 1 - a_q T + q of Frobenius at any prime q \nmid Np 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 E/\mathbf{Q} which gives rise to \mathcal{R}. 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 D/F 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:

  1. E/\mathbf{Q} is an elliptic curve.
  2. E/\mathbf{Q} is a fake elliptic curve; that is, an abelian surface with endomorphisms over \mathbf{Q} by a quaternion algebra D/\mathbf{Q}.

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 \mathcal{R} 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:

  1. A/\mathbf{Q} is an abelian surface.
  2. A/\mathbf{Q} is a fake abelian surface; that is, an abelian fourfold with endomorphisms over \mathbf{Q} by a quaternion algebra D/\mathbf{Q}.

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 \mathbf{Q}, and one wants to end up with fourfolds which are simple over \mathbf{Q}. Hence one can do the following:

Example: Let B/\mathbf{Q} 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 J_0(243) with coefficient field \mathbf{Q}(\sqrt{6}) with K = \mathbf{Q}(\sqrt{-3}). Take the restriction to K, twist by a sufficiently generic quadratic character \chi, and then induce back to \mathbf{Q}. 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 \mathbf{Q}(\sqrt{-6}), 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 \mathbf{Q} (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 L/\mathbf{Q} be any Galois extension with Galois group Q, the quaternion group of order 8. The group Q has an irreducible representation V of dimension 4 over the rationals, which preserves a lattice \Lambda. If you take

A = E^4 = E \otimes_{\mathbf{Z}} \Lambda,

then A is a simple abelian fourfold with an action of an order in D, (now the definite Hamilton quaternions) and so gives rise to compatible families \mathcal{R} 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 \mathrm{Gal}(L/\mathbf{Q}), 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 \mathcal{R} with cyclotomic similitude character should be indefinite.

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Hiring Season

Lizard 1: Wait, explain again why we bury our young in the sand and thereby place them into mortal peril?

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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