The future is now; recap from Cetraro

I’ve just returned from the second Journal of Number Theory biennial conference in Italy. It’s always nice to get a chance to see slices of number theory one wouldn’t otherwise see at the conferences I usually go to (although this was the first conference of any kind I attended in person since 2019). Here is a brief and incomplete recap.

• There were more talks that mentioned the Manin-Mumford conjecture and its various generalizations (particularly to uniform bounds in families) than I have ever previously attended in my life. There were probably equally many talks which mentioned Ax-Schanuel as well. It was nice to see these subjects and I learnt quite a lot from these talks.
• In the talk I linked to in the last post, I claimed that the modularity of elliptic curves over the Gaussian integers is “within our grasp”; well, the future is now! James Newton talked about his work in progress with Ana Caraiani where they prove modularity of all curves over imaginary quadratic fields \(F\) such that \(\# X_0(15)(F) < \infty\), which includes \(\mathbf{Q}(\sqrt{-1})\). One of the key tools in their proof is a suitable local-global compatibility statement for Galois representations coming from torsion classes in the crystalline setting where one is not in the Fontaine-Laffaille range (because of ramification in the base, for example). This was a situation where I had even been hesitant to make a precise conjecture. The problem is that the natural conjecture one might want to make is that the map of Hecke algebras factors locally through the Kisin deformation rings. But the construction of Kisin deformation rings as closures which are flat over \(W(k)\) by default might make one worried whether it is the correct integral object for torsion representations. But Caraiani and Newton show that such concerns are unfounded, and the \(W(k)\)-flat deformation rings are indeed the correct objects. One key point of their argument is showing that the (possibly torsion) representations \(\rho \oplus \rho^{\vee}\) (for suitable twists of \(\rho\) occur inside the cohomology of the Shimura variety in such a way where (using some notion of ordinary for a parabolic other than the Borel) the local representations in characteristic zero are reducible and realize the required crystalline lifts of each factor. One remaining annoyance is that one would like to find points over twists \(X(\overline{\rho}_E,\wedge)\) of the Klein quartic \(X(7)\) over \(F\) corresponding to \(E[7]\) which lie on solvable CM fields (in order to do a switch at the prime \(7\). You could (for example) start with the point \(E\) and the \(15\)-isogenous curve \(E’\) and connect them via a line. This line will go through two further points defined over a quadratic extension \(H/F\), but there is no reason to suppose a quadratic extension of an imaginary quadratic field will be CM. I had some idea related to a half-forgotten fact I learnt from John Cremona walking in the woods near Oberwolfach, but upon further consideration this half-forgotten fact was sufficiently ephemeral that it could not be reconstructed and didn’t appear to correspond to any actual facts. I did learn from Tom Fisher the nice fact that the four curves \(3\)-isogenous to \(E\) are collinear on the curve \(X(\overline{\rho}_E),3 \wedge)\) corresponding to the same mod-\(7\) representation with the other choice of symplectic form (that is, \(\wedge\) scaled by a quadratic non-residue).
• This was my first chance seeing a talk on the work of Loeffler-Zerbes on BSD for abelian surfaces. The most difficult condition to verify in their theorem is that a certain (characteristic zero) deformation problem is unobstructed. It seems very plausible to me that one could numerically verify some interesting examples and get truly unconditional results on BSD for some autochthonous abelian surfaces, or at least autochthonous elliptic curves over imaginary quadratic fields. The idea is that to prove a certain ordinary (of some flavour) deformation problem is unobstructed it suffices to prove that it is unobstructed modulo \(p\), which reduces to a computation with ray class groups in the splitting field of \(\overline{\rho}: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_4(\mathbf{F}_p)\). This seems within the realm of practicality. Moreover, once one verifies the condition for \(A\), one immediately deduces it for all the twists of \(A\) as well. It is important here to take \(p\) small, that is at most \(3\). Certainly if the mod-\(3\) image is surjective the extension will be too large, but the case of a representation induced from an imaginary quadratic field seems completely manageable. The other possibility is to work with \(p=2\). Here I think one should work with \(H^1(\mathbf{Q},\mathrm{ad}_0(\overline{\rho}))\) where \(\mathrm{ad}_0\) is the quotient of the \(11\)-dimensional adjoint representation by the diagonal (so slightly different from trace zero matrices). Here I’m imagining starting with a modular abelian surface which has good ordinary at \(2\) and whose mod-\(2\) image is \(S_5\). It might also be convenient if the local factor at \(2\) is congruent to \((1+T+T^2)T^2 \bmod 2\) so that the local deformation problem has good integral properties. Anyone interested in computing such an example?
• During the conference the second “David Goss Prize” was awarded. This prize is for work done in the past two years in number theory by someone at most 35(!) and also by someone who has not (yet!) won any other major prizes. (I joked that it might be nice to have a prize for people 50 or older who have not won any prizes but there is something nice about no longer being eligible for any prize except those one has no hope of winning.) This year’s winners were Ziyang Gao and Vesselin Dimitrov. The laudatio is here and a live action photo is here:

  

  Congratulations to both!

  Talking of prizes, I can’t quite work out whether there are far more prizes than there used to be, or whether I was simply oblivious about them when I was younger.

• Sad to say that the coffee I had in Italy was basically not that good. I’m prepared to make the seemingly heretical claim that nowadays it’s much easier to get excellent coffee in London. (Obviously it’s much easier to get excellent coffee in Melbourne.) The Pizza is still great though, and Mercato Centrale Roma ranks as the best food I have ever eaten in a train station.
• I dropped my iPhone one too many times and ended up with a broken phone with three blindingly white strips running down the right hand side of the screen. At first it did not respond to any touch at all, but after a few hours it started responding to touches on roughly the left 2/3 of the screen. There were just enough applications which were compatible with rotating the screen (so the relevant buttons came within reach) that it was barely usable for the remainder of the trip. (Other peculiarities: it gobbled battery power at an immense speed to the extend that it would last only an hour or two unused without being charged.) It did mean I started using Siri for the first time, although Siri was unsurprisingly useless for doing things I actually wanted (like “pressing” buttons on the screen that I couldn’t touch myself). The phone was also in the habit of randomly acting as if it was being pressed all on its own. The worst example of this was when trying to check into my flight on the way home. As my flight was delayed (well in advance) by several hours, the app was pushing for users to change to a different flight. To my horror, the app randomly started acting as if buttons were being pressed and changed my direct flight to Chicago (upgraded to business class) leaving in the early afternoon to an economy ticket leaving at 8:00AM in the morning and going through Dallas. When I called American they (at first) said that I couldn’t change it back because the original flight was now sold out, but some further negotiation finally got me back on my original flight. So I got to have one pleasant evening in Rome dining on the Piazza Navona after an obligatory trip to the local toy store.