Today’s post is about a new paper by my student Shiva. Suppose that \(A/\mathbf{Q}\) is a principally polarized abelian variety of dimension \(g\) and \(p\) is a prime. The Galois representation on the \(p\)-torsion points \(A[p]\) gives rise to a Galois representation:
\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_{2g}(\mathbf{F}_p)\)
with the property that the similitude character coincides with the mod-\(p\) cyclotomic character. A natural question to ask is whether the converse holds. Namely, given such a representation as above with the constraint on the similtude character, does it necessarily come from an abelian variety (principally polarized or not)?
When \(g=1\), the answer is that all such representations come from elliptic curves when \(p \le 5\), but that for \(p \ge 7\) there exist representations for any \(p\) which do not. For \(p \le 5\), more is true: the twisted modular curves \(X(\rho)\) all are isomorphic to \(\mathbf{P}^1\). When \(p \ge 7\), the curves \(X(\rho)\) are of general type, so one might expect a “random” such example to have no rational points. Dieulefait was the first person to find explicit representations (for any such \(p\)) which do not come from elliptic curves (and there is a similar result in my paper here). Both of these arguments exploit the Hasse bound. Namely, if
\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_p)\)
is unramified at \(l \ne p \ge 5\) and \(\rho\) comes from \(E/\mathbf{Q}\), then \(E\) must have either good or multiplicative reduction at \(l\). But this puts a constraint on the possible trace of Frobenius at the prime \(l\). For \(l = 2\), for example, this leads to explicit examples of non-elliptic mod-\(p\) representations for \(p \ge 11\). The case \(p = 7\), however, requires a different argument. More generally, while the Hasse argument does generalize to larger \(g\), it only works when \(p\) is large compared to \(g\). On the other hand, the Siegel modular varieties \(\mathcal{A}_g(p)\) of principal level \(p\) are rational over \(\mathbf{C}\) for only very few values of \(g\) and \(p\). Indeed, they are rational only for
\((g,p) = (1,2), (1,3), (1,5), (2,2), (2,3), (3,2)\)
whereas \(\mathcal{A}_g(p)\) turns out to be of general type for all other such pairs. When \((g,p)\) is on this list, then, as discussed in these posts, the twists \(\mathcal{A}_g(\rho)\) can all be shown to be unirational over \(\mathbf{Q}\) and so any such representation \(\rho\) does indeed come from infinitely many (principally polarized) abelian varieties.
Thus one is left to consider all the remaining pairs. This is exactly the question resolved by Shiva:
Theorem [Chidambaram]: Suppose that \((g,p)\) is not one of the six pairs above such that \(\mathcal{A}_g(p)/\mathbf{C}\) is rational. Then there exists a representation:
\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_{2g}(\mathbf{F}_p)\)
with cyclotomic similitude character which does not come from an abelian variety over \(\mathbf{Q}\).
Shiva’s argument does not use the Weil bound. Instead, the starting point for his argument is based on the following idea. Start by assuming that \(\rho\) comes from an abelian variety \(A\). Suppose also that \(\rho\) is ramified at \(v \ne p\) and the image of the inertia group at \(v\) contains an element of order \(n\) for some \((n,p) = 1\). Using this, one deduces (using independence of \(p\) arguments) that
\(|\mathrm{Sp}_{2g}(\mathbf{F}_l)| = l^{g^2} \prod_{m=1}^{g} l^{2m} – 1\)
is divisible by \(n\) for all large enough primes \(l\), and hence divides the greatest common divisor \(K_g\) of all these orders. This is actually a very restrictive condition on \(n\). For example, using Dirichlet’s theorem, the number \(K_g\) is only divisible by primes at most \(2g+1\). But now if the order of the group \(\mathrm{Sp}_{2g}(\mathbf{F}_p)\) for any particular \(p\) is divisible by a prime power \(n\) with \(n\) not dividing \(K_g\), then one can hope to construct a mod-\(p\) Galois representation whose inertial image at some prime \(v\) has order divisible by this \(n\), and this representation cannot come from an abelian variety over \(\mathbf{Q}\).
The good news is that one can show that (most) symplectic groups have orders divisible by large primes using Zsigmondy’s theorem. Combined with a few extra tricks and calculations for some boundary cases, the groups \(\mathrm{Sp}_{2g}(\mathbf{F}_p)\) contain elements of “forbidden” orders exactly when one is not in the case of the six exceptional pairs \((g,p)\). Note that Zsigmondy’s theorem already arises in the literature in this context in order to understand prime factors of the (corresponding) simple groups.
So now one would be “done” if one could (for example) solve the inverse Galois problem for \(\mathrm{GSp}_{2g}(\mathbf{F}_p)\) with local conditions. The inverse Galois problem is solved for these groups, but only because there is an obvious source of such representations coming from abelian varieties. Of course, these are precisely the representations Shiva wants to avoid.
Instead Shiva looks for solvable groups inside \(\mathrm{GSp}_{2g}(\mathbf{F}_p)\) containing elements of order \(n\) for suitable large prime powers \(n\). Note that the obvious thing would simply be to take the cyclic group generated by the element of the corresponding order. The problem is that there is no way to turn the corresponding representation into a Galois representation whose similitude character is cyclotomic. The groups Shiva actually uses are constructed as follows. Start by finding prime powers \(n | p^{m} + 1\) for some \(m \le g\), then embed the non-split Cartan subgroup of \(\mathrm{SL}_2(\mathbf{F}_{p^m})\) into \(\mathrm{GSp}_{2g}(\mathbf{F}_p)\), and then consider the normalizer of this image. One finds a particularly nice metabelian subgroup whose similitude character surjects onto \(\mathbf{F}^{\times}_p\). Shiva then has to prove the existence of a number field whose Galois group is this metabelian extension with the desired ramification properties at some auxiliary prime \(v\) but also crucially satisfying the cylotomic similitude character condition. This translates into a (typically) non-split embedding problem — such problems can be quite subtle! Shiva solves it by a nice trick where he relates the obstruction to a similar one which can be shown to vanish using methods related to the proof of the Grunwald-Wang Theorem. Very nice! In retrospect, the case of \(g = 1\) and \(p = 7\) in my original paper is a special example of Shiva’s argument, except it falls into one of the “easy” cases where the relevant metabelian extension actually is a split extension over the cyclotomic field. In general, this only happens when the the maximum power of \(2\) dividing \(g\) is strictly smaller than the maximum power of \(2\) dividing \(p-1\) which is automatic when \(g\) is odd. (The case when \(p = 2\) is easier because the cyclotomic similitude character condition disappears!)
The link on the name “Shiva” in the first sentence seems to be pointed at the wrong url.
It’s very impressive that he was able to completely resolve this question!
I do, though, feel tempted to ask an evil question. It seems that in all these cases the obstruction to finding a suitable abelian variety is local, i.e., one shows that there are no Q_v points on A_g(rho) for a suitable prime v. So these constructions don’t produce counterexamples to the Hasse principle. But one certainly expects counterexamples to the Hasse principle to be very common for varieties of general type. So can one produce examples?
In genus one, one possible global obstruction is Mazur’s torsion theorem. For p>7, no elliptic curve has a p-torsion point, so any extension of the cyclotomic character mod p by a trivial mod p representation will do. The split extension, at least, can be obtained v-adically from an elliptic curve with semistable reduction at p.
Although maybe that is not so much of a Hasse principle failure as it arises from a rational point on the compactified modular curve…
URL now fixed, thanks!
Perhaps then one wants to twist the modular curve “slightly” so that the cusps are no longer rational but one can still rule out the existence of rational points. For example, if the mod-p Galois representation has the shape \(\chi^{-1} \epsilon \oplus \chi\), then \(E\) would have to have a p-isogeny which one can rule out using Mazur’s isogeny theorem (if \(p\) is big enough) but choosing \(\chi\) so that it doesn’t present any local obstructions. I have no idea how one would make this work for abelian surfaces, though.
Another postage stamp computation one might do is to ask what is the “density” of \(\mathrm{GL}_2(\mathbf{F}_7)\) representations which are locally elliptic actually come from elliptic curves. Does one have a reasonable guess here? These are more or less just twistsof \(X(7)\) which have local points everywhere. Is the proportion which actually has global points supposed to be positive??
Using the isogeny theorem instead of the torsion theorem is a good idea. We can hope that we will eventually have such strong theorems in higher genus – maybe DL will even prove the geometric analogues of all of them – but in the near future things look grim.
With regards to the density, first one should ask how many locally elliptic GL_2(F_7)-representations are there. One reasonable way to count them is by conductor. For squarefree conductors a conjectural count was given by Lipnowski using the Bhargava-Shankar heuristics (https://doi.org/10.1080/10586458.2018.1537864) – he conjectures that the number of such representations with conductor < X is asymptotic to X log (X).
Second one should ask how many of them come for elliptic curves. Equivalently, modulo some Frey-Mazur difficulties, we can ask how many elliptic curves have "mod 7 conductor" < X. One heuristic is to start from the fact that there are H^{5/6} elliptic curves of naive height at most H^{5/6}. For most elliptic curves, the conductor is not far from the naive height, and the mod 7 conductor is not far from the conductor, so you could guess that the number of elliptic curves of "mod 7 conductor" < X is asymptotic to X^{5/6}. I think you can even write down a product formula for the expected constant (maybe there's a power of log here too) but it's a lot to ask from a weak heuristic with next to no evidence behind it. One might feel safer predicting an asymptotic like X^{5/6+epsilon}. I don't know at all what's in the literature about this.
I think the upshot of this is that the simplest heuristics you can try suggest that the density should be zero. But whether elliptic curves choose to conform to our simplest heuristics is mysterious enough that I think people normally try to be cautious about this.
I’m a little unconvinced and/or confused by this — surely this argument would also apply to \(p = 5\) as well — or even \(p = 2\)? (Sure many elliptic curves can give the same representation, but that just makes the count worse). I think the issue is comparing the conductors of \(E\) and \(E[p]\). If you order elliptic curves by the conductor of \(E\) they they may well typically will be similar, but if you count in terms of \(E[p]\) they may not be, and certainly won’t be when \(p = 2\) even if you take the curve of smallest conductor. Possibly there is some mileage to be gained for \(p > 6\) assuming Szpiro’s conjecture, but not enough I think(?)
Both being skeptical and confused are reasonable reactions. I have a heuristic here, which I didn’t explain very well, and which also has not so much evidence behind it. But it does have a way of dealing with the behavior of p=2,3,5.
Maybe the right way to describe the heuristic is that we can hope that the reductions of a random elliptic curve (with coefficients of a given size) at different places behave like independent variables, subject to the product formula for the discriminant. Probably one can define a Cramer-like random model where we create “fake elliptic curves” by choosing p-adic and Archimedean elliptic curves at random. I think in such a model one could check that the number of fake elliptic curves with conductor of the mod p representation at most X is unbounded with probability 1 if p5.
But I didn’t do the calculations in this model exactly, I just do the calculation and assume the local factors are independent so we can separate the variables in a sum at the right point. (I didn’t do this calculation at all until you raised your objection, so I’m glad it worked out consistent with the known p 5/6.
We can write [conductor of the mod p representation of E] as [conductor of the mod p representation of E]/[discriminant of E] * [discriminant of E]/ [naive height of E] * naive height of E. Now the sum over elliptic curves E of [naive height of E]^{-s} is bounded for s> 5/6, and the other terms are products of local terms – [conductor of the mod p representation of E]/[discriminant of E] is a product of local terms at non-archimedean places, and [discriminant of E]/ [naive height of E] is a local term at the archimedean place.
Because we expect these local terms to behave independently and randomly, we may remove the local terms for each place from the sum over E at the cost of introducing the average value of that term over elliptic curves defined over that local field. In the end we get the known naive height sum times an Euler product whose local factor, at non-archimedean places q, is the average of elliptic curves over Q_p of ([ discriminant of E]/[conductor of mod p representation of E])^s.
The main type of elliptic curve where the discriminant is large and the conductor of the mod p representation is small is those with Neron model of type I_p. This has probability roughly 1/q^p of happening at a prime q . For these elliptic curves, the discriminant is q^p and the conductor of the mod p representation is 1, so this gives a term of q^{ p (s-1 ) }. If s is at least 1 -1/p then this is at least 1/q and so when we add 1 from smooth curves and take the product over all p we get a divergent Euler product. But if s is less than 1 -1/p this term, at least, doesn’t make the Euler product unbounded.
Since we take s> 5/6, we can see that the heuristic predicts the sum is infinite for p 6 (unless there is another reduction type that causes trouble), fitting the known cases.