Many years ago, Khare asked me (as I think he asked many others at the time) whether I believed their existed an irreducible motive \(M\) over \(\mathbf{Z}\) (so good reduction everywhere) with Hodge-Tate weights \([0,1,2,\ldots,n-1]\) for any \(n > 1\). (Here the Motive is allowed to have coefficients.) When \(n=2\), the answer is no. Assuming all conjectures, such an \(M\) must be modular associated to a cusp form of weight \(2\) and level one, but no such cuspform exists. But the answer is also no unconditionally (for any notion of motive), and this fact is intertwined with the (inductive) proof by Khare and Wintenberger of Serre’s Conjecture. The hope might be that if no such motive existed for all \(n\), it could serve as the inductive basis for a more general form of Serre’s Conjecture.
My response at the time was that I guessed that no such motive existed for any \(n\). I generally feel that my intuition is quite good in these matters, so it was surprising to learn some time later a convincing meta-argument that such motives should really exist. This idea, which I can’t now remember whether I learned from Chenevier or Clozel, is related to trying to construct such forms which are in addition self-dual and so come from a classical group. In favourable situations, there exists a compact inner form on this group, so that computing these forms “reduces” to computing on certain finite sets. One such finite set turns out to be the set of positive definite lattices of discriminant one and dimension \(n\). As is well-known, they only occur in dimensions a multiple of 8. For \(n=8\) there is just \(E_8\), and for \(n=16\) there are two, and for \(n=24\) there turn out to be exactly \(24\), as classified by Niemeier, and which include the famous Leech lattice whose automorphism group is a central extension of the first sporadic group discovered Conway. Easier to compute is the weighted sum of such lattices by automorphisms; for \(n=24\), for example, this weighted sum is
\( \displaystyle{\frac{1027637932586061520960267}{129477933340026851560636148613120000000}}\)
which is very small, and of course is related to the fact that these lattices are quite symmetric. For \(n = 32\), however, the weighed sum is bigger than \(10^7\), and so there are lots of lattices. You might then think that the existence of these lattices (even just \(E_8\) when \(n=8\)) implies the existence of automorphic forms which then should give rise to the desired automorphic forms on \(\mathrm{GL}(n)\). But there are issues. One concerns the technical issue of trasfering forms between groups which is of course a subtle problem. But there is another. A form which is cuspidal on some group need no longer be cuspidal after transferring to \(\mathrm{GL}(n)\). So to see which forms are cuspidal you really need to do a computation. But these objects are of large complexity — already computing Hecke actions on supersingular points for \(X_0(11)\) is a non-trivial exercise; here the objects involve lattices of enormously high dimension. Chenevier and his co-authors, (including Lannes, Renard, and Taïbi) have done a remarkable job understanding what is going on here. The most basic example of the type of theorem they prove is as follows. When \(n = 16\) so there are two lattices; one can try to compute the action of a Hecke operator \(T_p\), and it turns out (see for example Theorem A here) that the answer involves Ramanujan’s function \(\tau(p)\). But this also tells you that the transfer to \(\mathrm{GL}_{16}\) will have some explicit isobaric decomposition corresponding to twists of the modular form \(\Delta\), and in particular the associated \(\pi\) will clearly not be cuspidal.
At the same time, there are some automorphic arguments which show that cusp forms of level one (and cohomologically trivial weight) cannot exist. Here the idea goes back to the (automorphic) proof of lower bounds for discriminants of number fields by Stark and Odlzyko. The idea in that case to use the explicit formula for \(\zeta_{K}(s)\) to construct an expression (and in particular the normalized version \(\Lambda_K(s)\) which satisfies the functional equation and involves \(N^s\) where \(N\) is the level which is directly related to the discriminant of \(K\)) and ultimately arrive at some expression which is provably non-negative unless the root discriminant of \(K\) is larger than some explicit constant (minus some explicit \(o(1)\) depending on the degree). Mestré generalized this argument to automorphic forms corresponding to other Motives, in particular proving that, assuming conjectures of Langlands type, that there did notexist any abelian varieties over \(\mathbf{Z}\) of dimension at least one (which was proved unconditionally by Fontaine) but also that the conductor of such an abelian variety had to be at least \(10^g\). This was then later generalized by Fermigier (a student of Mestré) and then by Stephen Miller (Rutgers!) to prove that there are no automorphic forms \(\pi\) for \(\mathrm{GL}_n/\mathbf{Q}\) of level one which are cohomological for the trivial representation when \(n < 27\). These are exactly the forms associated (conjecturally) to the motives of weight \([0,1,\ldots,n-1]\). Returning to the conference at Orsay: Chenevier gave a talk on understanding automorphic representations \(\pi\) of level one and low motivic weight, and once again raised the automorphic version of Khare’s question. Now I have known about this question for a long time, but somehow being reminded of a problem can sometimes be the spark to help one think about the question again.
Correcting what was a past failing of my own intuition, I was very happy that George Boxer, Toby Gee, and I were able to come up with a very simple argument to answer both questions; there does exist a compatible family of crystalline Galois representations with Hodge-Tate weights \([0,1,2,\ldots,n-1]\) for some \(n\); for example one can take \(n=105\). Moreover, this compatible system is even automorphic and associated to a cuspidal
\(\pi\) of level one and cohomological weight zero for \(\mathrm{GL}_n\). (With work, it is even “motivic” in the sense that the compatible system can be found inside an explicit algebraic variety over \(\mathbf{Q}\), so it is in particular also pure.) Now while the argument is very simple, it must also be said that is uses some extremely hard theorems; for a start, it uses both the full modularity lifting results of BLGGT (Barnet-Lamb, Gee, Geraghty, Taylor), following Clozel-Harris-Taylor and many others, *and* it uses the even more recent full symmetric power functoriality result for classical modular forms by Newton and Thorne. (Since the paper is only nine pages and the proof only half of that, I won’t explain it here.)
One would still like to prove, of course, that there are a huge number of self-dual forms for all sufficiently large \(n\). And one can naturally ask what is the smallest such \(n\), which we now know satisfies \(27 \le n \le 105\). The expectation is certainly that \(n\) is probably close to around \(32\). It would be nice to know!
Of course, there is an endless list of other tricky problems one can pose of this form. For example, does there exist a regular motive (with coefficients) over \(\mathbf{Z}\) with Hodge-Tate weights \([0,1,\ldots,n-1]\) for some \(n\) which is *not* essentially self-dual?
Discussing, I think, exactly this question, I believe I heard a long time ago the argument that if one counts cuspidal self-dual automorphic representations of GL_n with Hodge-Tate weights consecutive integers via the trace formula for the corresponding classical group, the main term coming from the adelic volume should grow so fast that all the other contributions, including the counts of Eisenstein series on GL_n arising from cusp forms on the classical group, should be dominated, and hence such representations should exist for all large enough n. Of course actually doing this kind of analysis with the trace formula in the large n limit seems essentially impossible, though the work of Deligne and Flicker does give a function field analogue.
I’m confused about one point in the Chenevier research report you link – why do the Galois representations considered there have a repeated Hodge-Tate weight, while the ones you are interested in do not?
In that direction I was tempted to try to, rather than find an explicit eigenvalue of the set of rank 32 even unimodular lattices and check it gives a cusp form on GL_32, bound the number of possible Eisenstein series on GL_32 that contribute to eigenforms on the set of rank 32 even unimodular lattices, but now I see why this is too hard: The worst case is perhaps the direct sum of Delta with a cusp form on GL_30 that has exactly two gaps in its sequence of Hodge-Tate weights matching the weights of Delta, so we can sum them with Delta to get a problematic form on GL_32. Such forms on GL_30 surely can’t be ruled out by L-function methods as their weights are similarly spaced to forms that can’t be ruled out, and counting them with the trace formula is clearly out of reach.
On the lower bounds side, I wonder if the results of Miller can be improved at all if one (a) assumes GRH and/or (b) restricts attention to self-dual representations. Given a lower bound of 27 and a potential upper bound of 32, even a tiny improvement would do a lot to close the gap.
With regards to the existence of modular forms of weight k and level 1 with reducible mod p reductions, I guess by Serre’s conjecture this is equivalent to the existence of certain Galois representations into GL_2(F_p). By a natural extension of Bhargava’s heuristics for counting number fields to counting GL_2(F_p) extensions with fixed GL_1(F_p)-part I get that the probability one should exist for each k is 1/(p-1) so the probability none exists for any k is (1-1/(p-1))^{phi(p-1)} which agrees closely with your heuristic. (The local factor at each non-archimedean prime is one so one only needs the local factor at infinity, which is the proportion of elements of GL_2(F_p) with determinant -1 that have order 2 and could be complex conjugation, which is 1/(p-1) for centralizer reasons.)
Chenevier considers a far broader range of cuspidal representations \(\pi\) which need not be of cohomological type; the extreme case being the \(\pi\) which should be associated to Artin representations. Here of course it’s harder to imagine how to construct forms when they exist (whose transfer is cuspidal), but the analytic methods used to rule out their existence don’t mind!
Concerning GRH: In Chenevier, the use of GRH does make a difference but only a very slight one in this range (a more general result for \(m \le 23\) can be improved to \(m \le 24\) but no more). So my guess that for this specific problem GRH probably wouldn’t move the bar much, quite possibly even not at all.
I’m considering not the finiteness theorem of Chenevier but rather the theorem relating Hecke eigenforms on the space of even unimodular rank n lattices to automorphic forms on GL_n. What I observe is that, in the research report you link and also a longer paper, this transfer seems to produce automorphic forms that don’t have cohomological weight 0, since the associated Galois representations have a repeated Hodge-Tate weight:
“Fact: There are exactly 24 representations of dimension 24 which are direct sums of representations in the list (2)–(5) above and whose Hodge-Tate numbers are 0, 1, . . . , 22 with 11 occuring twice.”
The “occuring twice” there is my point of concern.
I considered the possibility that there may be two different ways to transfer from an inner form of SO_n to GL_n but I don’t see how the L-groups can allow this.
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Let me see if I see what is going on here. All forms on SO_{2n} arising from functions on even unimodular lattice have the same infinity type. Since the trivial representation is obviously among them, they all have the infinity-type of the trivial representation. Transferring them along any sort of Langlands functoriality will preserve these infinity-types, so you will get forms which have the infinity-type of the transfer of the trivial representation. However, the Arthur parameter of the trivial representation of SO_{2n} is the representation Sym^{2n-2} + trivial of SL_2 into the dual SO_{2n}, and transferring to GL_{2n} gets you a representation with Arthur parameter Sym^{2n-2} + trivial, i.e. not the trivial representation which would be Sym^{2n-1}.
I think this is correct. I may be misreading, but I think this case is discussed in Prop. 8.2.10 of the Chenevier–Lannes book.
Also concerning Bhargava’s heuristics – yes, I learnt from Akshay some precise formulation of how to make these predictions in terms of centralizer computations, and I knew enough about those in my bones to know that the most stupid \(1/p\) heuristic should (more or less) agree with this kind of computatuon (which I didn’t do). And indeed the sentence “Minor variations of the heuristic may change the relevant constants, but not the overall prediction…” is supposed to be (in part) a reference to this.
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