How to reject a paper

I just had the paper discussed in this post very quickly rejected. Since it was such a short paper, I thought it not too unreasonable to submit it to a strong journal, so I am not terribly disappointed. I can always say I chose such a journal for the benefit of my junior collaborator, of course. Rejections are almost always a bit of a kick in the gut, but I have to say that these were the nicest rejection letters I have ever received. It’s honestly more positive feedback than I receive on most of my papers which are accepted. Because of this, I thought I would share them (both) with you, in the hope that they will inspire (as a referee) how you phrase your future rejections. Remember that most authors get precious little positive feedback on their work. You of course have to evaluate the paper as you see it, but there’s almost always something positive and encouraging you can say. Maybe imagine that the author is your student… and you are giving them your opinion in person. The reports below have not been redacted in any way by me. (It may be possible that the editor stripped out less generous paragraphs, in which case kudos to the editor as well)

Report I: There is an old question about how often the Fourier coefficients of a modular form can vanish. I don’t know how important this question is, but it has intrigued people on and off for some time. The authors consider a variant of this question. they show that if you fix a prime p and a level N coprime to p then there are only finitely many non-CM Hecke eigenforms of level N and even weight for which the p’th Fourier coefficient vanishes. One might reasonably ask why one would care. The authors only answer seems to be the analogy to things like Lehmer’s conjecture. On the other hand the proof is a beautiful application of ideas from the deformation theory of Galois representations. I took me only a few minutes to understand the argument, which is very elegant and short. I guess is there is a case for publication it is that the proof is so nice. I’m not really sure what to recommend. The paper is very short, which I suppose is a plus, and it is a pleasure to read. As I say the downside is that it is not clear why one should care about the main theorem.

Report II: On the positive side, the result is simple to state and appealing, and the proof is indeed relatively short and elegant (although it is obtained more through an application of existing techniques in deformation theory of modular forms and Galois representations, than through the development of new techniques in this area.) The result is presented as being in the spirit of Lehmer’s conjecture, which asks how often a_p(f) can be zero for a fixed eigenform f of weight >2; in this paper one is fixing p and varying f instead over all forms of a fixed level but varying weights. This leads to a very different kind of problem, which can be tackled by different methods, involving the p-adic deformation theory of modular forms. To my knowledge, this “vertical Lehmer conjecture” had never been considered before, and the introduction, which is very brief, is somewhat lacking in motivation. And it seems very unlikely that the Vertical Lehmer conjecture which is proved will tell us anything about the original, horizontal version! I guess that, although the result is quite nice, its importance is perhaps not made clear enough, in the paper, to justify the appearance in a very top-flight journal like redacted.

I would see it well in a journal of a caliber right below redacted. Perhaps the authors could be asked make a somewhat better case in their introduction for the importance and eventual impact of this result . Assuming such a case can be made: they shouldn’t of course be asked to say more than they believe; if they just think the result is of interest in its own right without further applications or ramifications, that is perfectly fine as well, but then perhaps redacted is aiming a bit high.

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First of all, thank you to both reviewers for those kind words; it’s hard to find fault with anything you say. To answer the implicit question of the second reviewer, I personally am happy for the result to stand on its own, and didn’t particularly want to make a case for the importance of the result. I think of it more as an amuse-bouche. But good journals can publish amuse-bouches too! I guess one way to try to sell the paper would be to draw analogies where one replaces counting modular forms with \(a_p = 0\) by a condition at the infinite prime. One analogue (more fundamental of course) is counting Maass forms with \(\lambda = 1/4\). The narrative would be that these problems are fundamentally hard — for similar reasons — to study from an analytical point of view, and it is only when one can relate the problem to very arithmetic questions can one hope to make progress. Chacun à son goût as they say, at least they do in Austrian operetta, I have no sense if actual Francophone people say this or not (but please tell me in the comments).