As we await the latest AI-generated results (whether gurgling through the arXiv, appearing on 4chan, or coming straight from the horse’s mouth), I thought I would put on record some interesting open problems from my own field. I don’t mention these in any way to “move the goalposts”, but rather to be a little more explicit about what makes them interesting. In particular, each of these problems—which I consider significant—is only a very special case of a much larger problem, and even solving that larger problem would not be the final goal. I had wanted to say a little more about each question, but I learnt that the OpenAI problem dump is coming today, so let’s write down some problems in advance, in no particular order.
Your exam begins now. To get full marks you must answer the full question, but partial credit is available. Outside materials including the total sum of human mathematics may be used. Each question is worth 10 points. Hints are given to some of the problems. Good Luck!
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Lehmer’s Problem I: Does there exist a prime \(p\) such that \(\tau(p)=0\), where
\[q \prod_{n=1}^{\infty} (1-q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n?\]
Although this is a classical problem, it’s hard to understand how interesting it is. Famously, we know that \(\tau(p)/p^{11/2} \in [-2,2]\) by Deligne, and we now know that these values are distributed according to the Sato–Tate measure. On probabilistic grounds, \(\tau(p)=0\) should occur only finitely often; indeed, even \(\tau(p) \equiv 0 \bmod p^2\) should occur only finitely often. So after checking the first billion primes, the answer is presumably no. But the point is not so much the answer itself as what the theory of modular forms can tell us about it. We don’t even know what type of question this is! One can relate it to a question about how many ways \(p\) can be written as a sum of \(24\) squares, but that hardly seems promising. Much more promising, to me, is the fact that the associated Galois representation
\[\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{Q}_p)\]
has large global image, while \(\tau(p)=0\) would force it to be locally induced at \(p\). Now the question starts to connect to other conjectures, such as Greenberg’s conjecture, which says that, in this setting, if the Galois representation were locally split, then \(\rho\) would have to be CM (still open!). But this relationship is undercut by the fact that there are examples of modular forms \(f\) with \(a_p=0\) that are not CM, most notably when \(f=f_E\) has weight \(2\) and \(E\) is non-CM but supersingular at \(p \ge 5\). Now weight \(k=2\) is different on probabilistic grounds, but there are also examples with \(a_p=0\) in weights \(k=4\) and \(k=6\).
One can also ask about the analogue for higher-weight forms of level \(N=1\). Now the question starts to relate to Maeda’s conjecture, which predicts that, for a normalized cuspidal eigenform of weight \(k\), the field \(K\) generated by its coefficients has degree equal to the full dimension \(d\) of the space of cusp forms. Stronger versions predict not only that the Galois group of the Galois closure is \(S_d\), but also that \(K\) is generated by any single coefficient \(a_p\) of \(f\). The latter implies that \(a_p \ne 0\) as soon as the dimension of the space is greater than one. Again, it’s totally unclear whether the conjecture is false, true for probabilistic reasons, or true for some good reason.
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Lehmer’s Problem II: Is the Mahler measure of a nonzero algebraic integer that is not a root of unity bounded away from \(1\)? Progress here would be really interesting. One aspect of this problem is that I don’t think we have a convincing heuristic either way. There has been some nice progress in this general area recently (by Dimitrov, Smith, Orloski–Sardari…), although these methods do not seem to have much to say about Lehmer’s problem. A proof would probably be more interesting than a family of counterexamples, but who knows.
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Artin’s conjecture for \(2\)-dimensional Galois representations. Naturally, one can restrict to the case of even representations with projective image \(A_5\). If one can answer this, then one can also ask about the full Artin conjecture for any nontrivial irreducible Artin representation of \(G_{\mathbf{Q}}\).
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The modularity of elliptic curves over general number fields. The case of imaginary quadratic fields seems very much like something that could be done, although I don’t know how to do it. But general number fields are more interesting. Here one can also ask about—and might have to solve—the problem of constructing Galois representations for automorphic forms of algebraic type, as explained in Buzzard–Gee. One can further ask for the modularity of all compatible systems of Galois representations, which encompasses both this question and the previous one on Artin’s conjecture.
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Leopoldt’s conjecture. It used to be said that you were not a real algebraic number theorist until you had proved Leopoldt’s conjecture (usually incorrectly). More generally, for geometric Galois representations \(V\), there are conjectures about \(H^1(\mathbf{Q},V)\) and the dimension of its image in \(H^1(\mathbf{Q}_p,V)\), including Janssen’s conjecture.
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The algebraic independence of \(\zeta(2),\zeta(3),\zeta(5),\zeta(7),\ldots\). More generally, the Grothendieck period conjecture (even just for mixed Tate motives!).
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The existence of infinitely many ordinary primes for any cuspidal modular eigenform. More generally, a Newton = Hodge theorem for a density-one set of primes for any motive \(M\) over any number field \(F\) whose Sato–Tate group is connected.
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The BSD conjecture in its full form, in arbitrary rank. The BSD conjecture elliptic curves over any number field. Then Beilinson’s conjecture…
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A disproof of Vandiver’s conjecture. It’s surely false (?!), so this is more a question of finding some really clever way to compute \(K_{12}(\mathbf{Z}),K_{16}(\mathbf{Z}),K_{20}(\mathbf{Z}),\ldots\) until the order of one of these groups has an interesting prime factor. More generally, prove that there are infinitely many primes \(p\) for which it fails. At the same time, prove there are infinitely many regular primes.
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Last but not least, RH, or GRH, or GRH for \(L(s,M)\) attached to any irreducible motive \(M\).
One could go on and on, for extra credit: Schanuel’s conjecture. Given an algebraic curve \(X/\mathbf{Q}\), is there an algorithm to determine \(X(\mathbf{Q})\)? Are there infinitely many Mersenne primes and finitely many Fermat primes? What comes after the Langlands conjectures? …
I don’t know how well ChatGPT will do, but 10/100 would be extremely impressive to say the least. I would certainly guess that Vandivier’s conjecture is the easiest to disprove since it is “only” a computation. But it might be quite a difficult computation, I’m not sure. And even knowing ALL of these wouldn’t answer many questions I am still interested in.