Update on Sato-Tate for abelian surfaces

Various people have asked me for an update on the status of the Sato-Tate conjecture for abelian surfaces in light of recent advances in modularity lifting theorems. My student Noah Taylor has exactly been undertaking this task, and this post is a summary of his results.

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:

1. 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.
2. 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.
3. 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.
4. The case when the geometric endomorphism ring of A is \(\mathbf{Z}.\)

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 \(\epsilon > 0,\) there exists a positive density of primes \(\mathfrak{p}\) (with \(N(\mathfrak{p}) = p),\) one has

\(\displaystyle{\# C(\mathcal{O}/\mathfrak{p}) – p – 1 > \left(\frac{2}{3} – \epsilon \right) \sqrt{p}}.\)

Compare this to the Hasse bounds, which imply that the quantity on the LHS has absolute value at most \(4 \sqrt{p}.\)

Of course this theorem is much weaker than the Sato-Tate conjecture. But even the weaker version of this theorem which says that \(\#C(\mathbf{F}_p) > p + 1\) 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 \(y^2 = x^3 – x\) which admits CM by the Gaussian integers. One has a formula for the difference \(a_p= 1+p-\#E(\mathbf{F}_p)\) 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

\((a + b i) \equiv 1 \mod (1 + i)^3.\)

Then one has the formula \(a_p = 2a.\)

The problem then becomes: do there exist infinitely many primes p = 1 mod 4 such that one has \(a > 0?\) This seems suspiciously like something that can be proven using Cebotarev, but it is not. The problem is that the infinite places of \(F = \mathbf{Q}(\sqrt{-1})\) 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 \(\mathbf{A}^{\times}_F.\)

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 \(a_n \in [-2,2]\) such that

\(\displaystyle{\frac{1}{n} \sum_{i=1}^n a_i \rightarrow 0, \qquad \frac{1}{n} \sum_{i=1}^n a^2_i \rightarrow 1,}\)

One can draw the conclusion that \(a_n > 1/2 – \epsilon\) infinitely often, by (for example) considering the average of the quantity \((2a_n – 1)(a_n + 2).\) 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 \(A = \mathrm{Jac}(C)\) is of type (3). Then there exists a positive density of primes \(\mathfrak{p}\) (with \(N(\mathfrak{p}) = p),\) such that

\(\displaystyle{\# C(\mathcal{O}/\mathfrak{p}) – p – 1 > 2.47 \sqrt{p}}.\)

(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 \(\mathbf{C}.)\) 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 \([-2,2]\) would require knowing something close to functoriality of the map \(\mathrm{Sym}^2: \mathrm{GL}(4) \rightarrow \mathrm{GL}(10),\) 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).