Chidambaram on genus two curves, II

Posted on April 24, 2020 by Persiflage

We now continue a series of posts on the work of my student Shiva Chidambaram. (Click here for part I.) Today I would like to discuss another project with Shiva that was also joint with David Roberts (no, not David Roberts).

We saw last time that the moduli spaces \(\mathcal{A}_2(\rho)\) and \(\mathcal{M}_2(\rho)\) are not in general rational over \(\mathbf{Q}\). On the other hand, the degree six cover \(\mathcal{M}^w_2(\rho)\) is always rational. So the next question is: what is an explicit parametrization? Slightly differently, start with a genus two curve with a Weierstrass point

\(y^2 = x^5 + a x^3 + b x^2 + c x + d\)

Problem: Parametrize all other genus two curves with a Weierstrass point which have the same \(3\)-torsion representation.

It might be worth briefly revisiting the argument from [BCGP] that such a parameterization exists. The key point is that there is an birational map

\(\mathcal{M}^{w}_2(3) \rightarrow \mathbf{P}^3\)

which is \(\mathrm{PSp}_4(\mathbf{F}_3)\)-equivariant. This allows one to show that the corresponding twists are Brauer-Severi varieties, and then deduce they are rational by the same group theoretic trick which appears in this paper of Shepherd-Barron and Richard Taylor. More explicitly, there are maps

\( H^1(\mathbf{Q},\mathrm{GL}_4(\overline{\mathbf{Q}}))
\rightarrow H^1(\mathbf{Q},\mathrm{PGL}_4(\overline{\mathbf{Q}})) \rightarrow H^2(\mathbf{Q},\overline{\mathbf{Q}}^{\times})\)

Here the LHS is trivial by Hilbert 90. One shows, using the fact that the Darstellungsgruppe of \(\mathrm{PSp}_4(\mathbf{F}_3)\) is \(\mathrm{Sp}_4(\mathbf{F}_3)\), that the cocycle corresponding to any Galois representation \(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_4(\mathbf{F}_3)\) with cyclotomic determinant lifts in an explicit way to a cocycle in the LHS and hence is trivial. The problem is to bridge the gap between a theoretical argument that a cocycle is trivial and a way to produce an equation of the corresponding twist. That amounts to the problem of taking a cocycle

\(Z^1(\mathbf{Q},\mathrm{GL}_4(\overline{\mathbf{Q}}))\)

and writing it as a coboundary. Before going further, it’s worth pointing out that the case of \((g,p) = (2,3)\) is very similar (but more complicated) than the case of \((g,p) = (1,3)\). In the latter case, one has that

\(\mathrm{dim} H^0(X(3),\omega) = 2,\)

and this simple equality leads to an idenfication of \(X(3)\) with \(\mathrm{Proj} H^0(X(3),\omega)\). So, let’s talk about the problem of parametrizing elliptic curves as a warm-up case. If you start with a curve

\(E: y^2 = x^3 + a x + b\)

for which (for convenience of exposition) you assume \(\rho_{E,3}\) surjective, then the splitting field \(L/\mathbf{Q}\) is a \(\mathrm{GL}_2(\mathbf{F}_3)\) extension \(L\) which contains \(F = \mathbf{Q}(\sqrt{-3})\). There is an isomorphism of \(H = \mathrm{SL}_2(\mathbf{F}_3)\)-modules

\(L = F[H]\).

The group \(H\) admits a certain specific \(2\)-dimensional representation \(V\), and the representation \(\rho\) can be interpreted as giving an explicit map

\(V \rightarrow L.\)

OTOH, the identification above means there is an inclusion \(V^2 \rightarrow L\) because \(\mathrm{dim}(V) = 2.\) The problem of solving Hilbert 90 (ignoring a certain descent from \(F\) to \(\mathbf{Q}\)) then becomes the question of finding the “other” extension. Now if you can write \(L\) down you can do this by linear algebra. But even in any specific example, \(L\) has degree 48, and computations with fields of that size can be pretty formidable and are at the limit of what one can do with explicit number fields in (say) pari/gp or magma.

Of course, one wants to do this over the field \(\mathbf{Q}(a,b)\) with general parameters in order to have the general formula. The extension \(L\), for example, is the Galois closure of the equation

\(27 y^8 + 216 b y^6 – 18 \Delta y^4 – \Delta^2 = 0,\)

but you probably don’t want to even write down a polynomial of degree 48 in these general variables which gives \(L\), let alone try to compute the Galois action. We did succeed in solving this problem by a certain amount of trickery — working in special cases and making the right ansatz for the general case. There were many intermediate formulas which involved polynomials with (say) 100 terms, but the final answer turns out to be perhaps surprisingly simple, namely, the general equation is given by

\(y^2 = x^3 + A(a,b,s,t) x + B(a,b,s,t),\)

where

\(
\begin{eqnarray*}
3A(a,b,s,t) & = & 3 a s^4 +18 b s^3 t -6 a^2 s^2 t^2 -6 a b s t^3 -(a^3+9
b^2) t^4, \\
9B(a,b,s,t) & = & 9 b s^6-12 a^2 s^5 t-45 a b s^4 t^2-90 b^2 s^3 t^3 + 15 a^2 b s^2 t^4 \\
&& \qquad -2 a
(2 a^3+9 b^2 ) s t^5 -3 b (a^3+6 b^2 ) t^6.
\end{eqnarray*}
\)

Here \([s,t]\) is the \(\mathbf{P}^1\) parameter. Curiously enough this exact formula was also found before in the literature. That reflects something a little surprising about this equation. The moduli space we are looking for is a \(\mathbf{P}^1\), and this has many automorphisms. On the other hand, we are starting with an \(E\) so we have a fixed point normalized here to be \([1,0]\). But the projective line with one fixed point still has many automorphisms! However, it turns out that there is some extra hidden structure which gives rise to a second canonical point normalized here as \([0,1]\), which is why different people would possibly end up with the same equation independently. (\(\mathbf{P}^1\) with two fixed points still has a \(\mathbb{G}_m\)’s worth of automorphisms, but an informal consideration of the integral structure can be used to pin this down further.) The map which takes one \(E\) and spits out the other point therefore ends up giving a canonical rational map on \(\mathbf{P}^1_j\) which has the property that it preserves the (projective) \(3\)-torsion representation. Explicitly it is given by:

\(\displaystyle{j \mapsto \frac{(6912 – j)^3}{27 j^2}}\)

I wonder if this has interesting dynamical properties?

The computation above was not so easy, even though the answer turned out to be simple enough. But for \((g,p) = (2,3)\) things are looking pretty bad. First of all, the extension \(L\) now has degree \(103680\), which one is not going to write down explicitly. Even the analogue of the degree \(8\)-polynomial above is a degree \(40\) polynomial in \(x^6\) with \(1673\) terms.

Despite that, we found the answer:

Theorem: [C, Shiva Chidambaram, David Roberts] There exist (and we compute) explicit polynomials \(A,B,C,D\) in \(\mathbf{Q}[a,b,c,d,s,t,u,v]\) which specialize to \(a,b,c,d\) at \([s,t,u,v]=[1,0,0,0]\) such that

\(y^2 = x^5 + A x^3 + B x^2 + C x + D\)

is the general genus two curve with a rational Weierstrass point and fixed \(3\)-torsion representation.

Even though we find the simplest form of these polynomials, they turn out to be quite big. As in, the number of monomial terms they contain are \(14671\), \(112933\), \(515454\), and \(1727921\) respectively. The text files were so big that I ran into space problems on my university account! (OK, so it’s only 200MB or so, but that’s a big text file!)

The reason such a computation is ultimately possible relates to an accidental fact that is common between the two cases, namely, that the groups \(\mathrm{SL}_2(\mathbf{F}_3)\) and \(\mathrm{Sp}_4(\mathbf{F}_3) \times \mathbf{Z}/3\mathbf{Z}\) are two of the 37 exceptional complex reflection groups as determined by Shephard and Todd. The story is explained in our paper so I won’t discuss it here, but it might be worth mentioning two further facts:

The first is that these methods can also deal (in principle) with an analogue of this problem for \(g = 3\) and \(p = 2\). Just as with \(g = 2\), the moduli space which admits an equivariant birational map to \(\mathbf{P}^6\) is not \(\mathcal{M}_3(2)\) but once more a finite cover, and this cover does not correspond to any level structure but rather some cover coming genuinely from the mapping class group. This picture relates to the isomorphism \(\mathbf{Z}/2 \mathbf{Z} \times \mathrm{Sp}_6(\mathbf{F}_2) \simeq W(E_7)\), another exceptional complex reflection group. There is even a less analogous version for \(g = 4\) and \(p = 2\) related to the fact that the largest complex reflection group \(W(E_8)\) admits a description \(W(E_8) \simeq 2.\mathrm{O}^{+}_8(\mathbf{F}_2):2\), and the projective version of this group \(\mathrm{O}^{+}_8(\mathbf{F}_2):2\) is a subgroup of \(\mathrm{Sp}_8(\mathbf{F}_2)\), although of genuine index (\(136\)) rather than as an isomorphism, which is the main reason why this is a little different to the other cases. We estimated that an explicit version of the last moduli problem would involve polynomials with approximately 100 trillion terms, so needless to say we did not try to compute it.

Second, there is an interesting story concerning the auxiliary copy of \(\mathbf{Z}/3 \mathbf{Z}\) that turns up in the \(g = 2\) setting. The formulas that we write down actually correspond not only to projective spaces \(\mathbf{P}^1\) and \(\mathbf{P}^3\) but actually to affine spaces \(\mathbf{A}^2\) and \(\mathbf{A}^4\) which represent moduli problems related to the complex reflection groups. In these affine families, not only is the representation corresponding to \(\mathrm{ker}(\rho)\) fixed, but the splitting field of \(X^3 – \Delta\) also remains unchanged. When \(g = 1\), this is not surprising, because the \(S_3\) extension comes from the map \(\mathrm{GL}_2(\mathbf{F}_3) = \widetilde{S_4} \rightarrow S_4 \rightarrow S_3\). On the other hand, that’s obviously not happening in the genus two case where the group is almost simple. This is a little peculiar! However, it related to the fact that the splitting field of \(X^3 – \Delta\) for genus two curves depends on the Weierstrass equation. If you scale the Weierstrass equation by (\(x,y) \mapsto (t^2 x,t^5 y)\), this sends \(\Delta \rightarrow t^{40} \Delta\). So the affine equation represents a moduli space for some larger group which disappears when considering the equation projectively, and you can always normalize your Weierstrass equation so that \(\Delta\) is a perfect cube.

Posted in Mathematics, Students, Work of my students | Tagged , , , , , , , , , | 2 Comments

En Passant VIII

Posted on April 23, 2020 by Persiflage

I just found out that Lucien Szpiro recently passed away. I met him only once in late 2018 when I gave the joint NY number theory at CUNY. When I arrived at my hotel (the Gregory) around 10pm, I was somewhat shocked to find that nobody had made me a reservation, and the hotel was completely booked out. I was tempted to try the Langham on the opposite side of the street, but thought that might probably bankrupt the CUNY seminar budget for the next few years, so instead I started wandering the streets of NYC trying to find a more modest hotel with a vacancy (without any success — I’m not sure if there was something going on or whether it’s always hard to find a hotel room in NYC in mid-October). With a little help from JC back in Chicago with a reliable internet connection and access to hotels.com, I eventually found the one remaining room in a Best Western a few streets away for about $450 a night. When Lucien found out about what happened the next day, he vigorously tried to persuade me to accept a personal check from him to cover my costs before the reimbursement process could be sorted out. Though I insisted on declining his generous offer, a few days later I received a personal check from him in the mail, and this time I cashed it. And it was just as well, because CUNY did deny my reimbursement request for going over the daily limit for hotels, before finally [with a little more help from locals] “generously” agreeing to a “one time exception” to cover their own incompetence. I was then finally able to pay Szpiro back via Venmo (at least through the intermediary of Alex Gamburd, friend of the blog). Although this was our only ever interaction, it certainly left a very positive impression on me.

Tim Brooke-Taylor also recently died. If you grew up in Australia in the 1980s then there’s a good chance you have a deep place in your heart for the Goodies. (Further evidence: see this week’s Australian letter in the New York Times.) In addition to the absurdist humor, there were also plenty of 70s British cultural and political references which were somewhat lost on a 10 year old kid in 80s Melbourne; I wonder what a 10 year old kid in 2020s Chicago would think?

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En Passant VII

Posted on April 19, 2020 by Persiflage

Idle question which has surely been asked (and answered!). If \(X^{+}_{\mathrm{nsp}}(p)\) is the modular curve corresponding to the normalizer of the non-split Cartan, then one reason it is hard to find all rational points is that the all factors of the Jacobian have positive rank (probably contingent on BSD). Is the same true for \(X^{+}_{\mathrm{nsp}}(pq)\)?

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Chidambaram on genus two curves, I

Posted on April 15, 2020 by Persiflage

Before we start, just to alert you to a minor blogpage design change: all the posts (including this one) which talk about my students work can be accessed in one place by clicking the “work of my students” tab just below the picture on the top of this page.
resume normal service.

Those who study elliptic curves certainly know that if you start with an elliptic curve \(E/\mathbf{Q}\), the \(p\)-torsion gives rise to a Galois representation:

\(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GL}_2(\mathbf{F}_p)\)

with cyclotomic determinant. Conversely, if \(p = 2,3,5\) then the converse is true, that is, any such Galois representation comes from an elliptic curve. Moreover, any such representation comes from an infinite number of curves which are parametrized by \(\mathbf{P}^1_{\mathbf{Q}}\). This is intimately related to the fact that the curves \(X(p)\) have genus zero for these \(p\).

What is also true is that, given any \(E\), one can write down explicit parametrizations of these families. This was done by Rubin and Silberberg for \(p=3,5\) around the time Fermat’s last theorem was proved. Indeed, the idea of passing between elliptic curves with the same mod-\(3\) Galois representation features prominently in Wiles’ argument.

One might ask what happens for higher genus. First of all, there is a geometric problem over the complex numbers: when is the moduli space \(\mathcal{A}_g(p)\) of PPAV of dimension \(g\) with full \(p\)-level structure a rational variety? It turns out the only possibilities when \(g > 1\) are \(p=2,3\) when \(g = 2\) and \(p=2\) when \(g = 3\). The case \((g,p) = (2,3)\) arose in my work with Boxer, Gee, and Pilloni (discussed here). In that paper, we proved a weaker version of the result above, namely the following:

Proposition: [BCGP] If \(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_4(\mathbf{F}_3)\) is any continuous representation with cyclotomic similitude character, then the corresponding twist \(\mathcal{A}_2(\rho)\) is unirational over \(\mathbf{Q}\) via a map of degree at most six. In particular, it has many rational points.

The degree six cover is not so mysterious. When \(g = 2\), PPAV are (more or less, the less being the Humbert surface) are Jacobians of genus two curves. So certainly birationally one can replace \(\mathcal{A}_2(\rho)\) by \(\mathcal{M}_2(\rho)\), the moduli of genus two curves whose Jacobian has the given \(3\)-torsion. The degree six cover is then the moduli of genus two curves with a fixed Weierstrass point, or, more prosaically, the genus two curves of the form:

\(y^2 = x^5 + a x^3 + b x^2 + c x + d \)

whose Jacobian has the given \(3\)-torsion. (Any fixed Weierstrass point can be moved to \(\infty\), and then the \(x^4\) term can be surpressed by an obvious linear transformation.) This moduli space will be discussed in more detail in the next post. But for now, this leaves open the question of whether \(\mathcal{A}_2(\rho)\) itself is rational.

Over the complex nubmers, things are well understood. The space \(\mathcal{A}_2(3)\) has a number of compactifications, including the (singular) Satake compactification, and the various smooth toroidal compactifications. When \(g = 2\), things work out extra nicely: there is a somewhat canonical compactification \(\mathcal{A}^*_2(3)\) due to Igusa. It turns out that \(\mathcal{A}_2(\rho)\) is birational to a very nice \(3\)-fold known as the Burkhardt quartic. The Burkhardt quartic is given explicitly in \(\mathbf{P}^5\) by the equations:

\(\sigma_1 = x_0 + x_1 + x_2 + x_3 + x_4 + x_5 = 0\),
\(\sigma_4 = x_0 x_1 x_2 x_3 + \ldots + x_2 x_3 x_4 x_5 = 0.\)

Eliminating any variable using the first equation leads to a quartic in \(\mathbf{P}^4\), but this is the most symmetric presentation. This variety \(\mathcal{B}\) is singular and has \(45\)-nodes — a maximal number, as it turns out. Not surprisingly, it also has an action by automorphisms of the simple group \(G = \mathrm{PSp_4}(\mathbf{F}_3)\). Blowing up \(\mathcal{B}\) at these nodes gives the smooth variety \(\mathcal{A}^*_2(3)\).

Things are more subtle over \(\mathbf{Q}\). It turns out that for the trivial level \(3\) structure corresponding to the representation \((\mathbf{Z}/3 \mathbf{Z})^2 \oplus (\mu_3)^2\) with the obvious symplectic structure, the corresponding variety \(\mathcal{A}^*_2(3)\) is still rational (e.g. see here). But it is no longer so obvious whether the twists we are considering should be rational over \(\mathbf{Q}\) or not. (There are actually some twists of a different flavor which don’t have points, but all the ones we are considering do.) Note there is a big difference between what happens in higher dimensions and what happens in dimension one: In dimension one the only unirational smooth projective curve with a rational point is projective space itself, but this is completely false in higher dimensions (for example, take products of projective spaces).

We left the question of the rationality of \(\mathcal{A}_2(\rho)\) open in [BCGP]. But my student Shiva Chidambaram took up the question. The first question is how can you prove a smooth projective variety \(X\) is not rational over \(\mathbf{Q}\) assuming that it is rational over \(\mathbf{C}\) and has rational points. One obstruction was found by Manin. If \(X\) is projective space, then the geometric Picard group of \(X\) is \(\mathbf{Z}\). The Picard group does not always have to be \(\mathbf{Z}\) for a smooth rational variety, but Manin showed that, still assuming that \(X\) is smooth and projective, if it is birational to projective space then its (geometric) Picard group is similar to the trivial representation in a technical sense we now explain. Here we say that two \(\mathbf{Z}[G_{\mathbf{Q}}]\)-modules \(A\) and \(B\) (which are free finitely generated abelian groups) are similar if there are integral permutation representations \(P\) and \(Q\) of \(G_{\mathbf{Q}}\) such that

\(A \oplus P \simeq B \oplus Q.\)

(I think one should imagine a sequence of birational maps where one introduces (or removes) the class of some cycle and all of its conjugates.)

Now we can hope to apply this in practice if we can compute the \(G_{\mathbf{Q}}\) action on \(M = \mathrm{Pic}_{\overline{\mathbf{Q}}}(\mathcal{A}^*_2(\rho))\).

How might one go about computing \(M\)? First of all, consider the non-twisted space \(\mathcal{A}^*_2(\rho)\). Using the explicit geometry of this space, one can hope to go about computing the Neron-Severi group completely explicitly, together with the action of \(G = \mathrm{PSp}_4(\mathbf{F}_3)\). And this was indeed done by Hoffman and Weintraub (amongst other things) in this paper. In particular, they show that the cohomology of this variety is all torsion free, trivial in odd degrees, and satisfies

\(H^2(X,\mathbf{Z}) \simeq H^4(X,\mathbf{Z}) = \mathbf{Z}^{61}.\)

Moreover, the cohomology is entirely generated by cycles, and these cycles are all defined over \(E = \mathbf{Q}(\sqrt{-3})\) and can be written down explicitly, together with the corresponding intersection pairing, and the action of the group \(G = \mathrm{PSp}_4(\mathbf{F}_3)\) on these cycles is self-evident because of their geometric nature. Clearly the Neron-Severi group of any twist will also be \(\mathbf{Z}^{61}\), because the geometric object is the same — the only thing that will change is the Galois action. For this, it is more convenient to work over \(E = \mathbf{Q}(\sqrt{-3})\). In this case, the \(G_{E}\) action will be as follows: the projective image of \(\rho\) when restricted to \(G_E\) factors through \(\mathrm{PSp}_4(\mathbf{F}_3)\) given the assumption on the similitude character. Thus \(\rho\) gives a canonical map

\( G_E \rightarrow G,\)

and the action of \(G_E\) on \(M\) is simply the restriction of the action of \(G\). Manin’s obstruction says that for the variety to be rational over \(E\), the action of \(G_E\) has to factor through a representation similar to the trivial representation. But that depends only on the image \(H \subset G\). Thus the problem (at least in terms of when we can apply Manin’s criterion) is “reduced” to group theory.

Some more caveats: It turns out to be pretty hard to tell if a representation is similar to the trivial representation. There is one obstruction coming from cohomology: using Shapiro’s lemma, if \(H\) is acting on \(M\) by a permutation representation, then

\(H^1(P,M) = H^1(P,M^{\vee}) = 0, \quad \text{all} \ P \subset H.\)

But then it follows that the same is true of \(M\) is similar to a permutation representation. This gives a way to explicitly verify in some cases that \(M\) is not similar to a permutation representation by finding a subgroup \(P\) for which the group above is non-trivial. Moreover, computing cohomology is something that magma can do! So it remains to:

  1. Explicitly translate the description of Hoffman-Weintraub into a presentation of \(\mathbf{Z}^{61}\) as a \(G = \mathrm{PSp}_4(\mathbf{F}_3)\)-representation.
  2. Determine for what subgroups \(P\) of \(G\) one has \(H^1(P,M) = H^1(P,M^{\vee}) = 0\).
  3. Deduce that \(\mathcal{A}^*_2(\rho)\) is not rational whenever the projective image contains such a \(P\) as above.

The conclusion:

Theorem (C-Shiva Chidambaram): For all but \(27\) of the \(116\) conjugacy classes of \(G\), the corresponding twist \(\mathcal{A}^*_2(\rho)\) is not rational over \(E = \mathbf{Q}(\sqrt{-3})\) and hence not rational over \(\mathbf{Q}\) either. In particular, if the projective image over \(E\) has order greater than \(20\), the twist is not rational.

You actually get something stronger from Manin’s criterion — if the variety becomes rational over some map of degree \(d\), then the cohomology of the modules must be annihilated by \(d\). From our computations we find, for example:

Theorem (C-Shiva Chidambaram): Suppose that \(\rho: G_{\mathbf{Q}} \rightarrow \mathrm{GSp}_4(\mathbf{F}_3)\) is surjective with cyclotomic similitude character. Then the minimal degree of any dominant rational map \(\mathbf{P}^3_{\mathbf{Q}} \rightarrow \mathcal{A}^*_2(\rho)\) is six.

Note that from the construction of [BCGP] we know that there is such a cover of degree six, so the six in this theorem is best possible! (It was good that the computation was consistent with the existence of this cover!)

It turns out that (in the surjective case) one can give a softer argument that only depends on the rational representation. The point is that there can still be an obstruction to a rational representation to being a difference of permutation representations. This is easy enough to compute using the character table; you take the group of all virtual representations over \(\mathbf{Z}\) and compute the subgroup of all induced representations. For \(G = \mathrm{PSp}_4(\mathbf{F}_3)\), this quotient, sometimes called the Burnside cokernel (at least this is what it is called in the magma documentation), turns out to be \(\mathbf{Z}/2\mathbf{Z}\) (magma computes it). It’s also not so hard to see that there exist subgroups \(G_{40}\) and \(G_{45}\) of the obvious index such that

\([H^2(X,\mathbf{Q})] = [G/G_{40}] + [G/G_{45}] – [\chi_{24}],\)

where \(\chi_{24}\) is the unique representation of \(G\) of dimension \(24\) which also happens to be defined over \(\mathbf{Q}\) and also generates the Burnside cokernel. On the other hand, this method this gives weaker results for subgroups of \(G = \mathrm{PSp}_4(\mathbf{F}_3)\) and even in the surjective case only shows the minimal cover has degree two rather than six.

A word on the actual computation: Shiva went off and did the task of converting the description in Hoffman–Weintraub into a form which could be used by magma. I also went off and tried to do this independently. We then both produced codes (mine much messier) which computed the cohomology of all the subgroups and arrived at completely different answers, which was a bit troubling. But then Shiva pointed out to me that magma automatically does something with matrices that converts right actions to left actions or something like that [could it really be that Magma treats matrices as acting from the right? that sounds crazy], and so his computation of \(H^1(P,M)\) was correct, but I was computing \(H^1(P,M^{\vee})\). But fortunately both are useful! (Of course, one could easily also extract that data from Shiva’s code which was much cleaner than mine.)

Posted in Mathematics, Students | Tagged , , , , , , , , , , , , | 8 Comments

How to reject a paper

Posted on March 31, 2020 by Persiflage

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.

################################################

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).

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You are welcome, Northwestern junior faculty

Posted on March 27, 2020 by Persiflage

Nobody has even accused Persiflage of being afraid to speak truth to power. After Chicago made their announcement that they were extending tenure by one year for all (qualified) faculty, I thought that, as a good local citizen of Evanston, I should put the pressure on Northwestern to do this same. Naturally, I first considered tweeting the official Northwestern account, or even better, the twitter feed of Morty Schapiro (Northwestern’s President). Alas, he didn’t seem to have a twitter feed, so I looked up the email address instead. The best I could find was nu-president@northwestern.edu, which I presumed would just be read by a staff member in the president’s office. Still, it was the best I could find, so I forwarded the email announcement from the University of Chicago’s provost to this address, adding a few choice words: Your turn, Northwestern! Surely you’re not going to let the *University of Chicago* (and Harvard, and etc.) outdo you here? There is no better way to persuade a university to do something than to point out the bold actions of “peer institutions.” Then, less than 10 minutes later, I got a response:

And now, the very same week, Northwestern has gone ahead an taken up my suggestion and extended the tenure clock. Naturally, I am too modest to take 100% of the credit; I will give Morty Schapiro at least 10% for listening and then another 10% for being gracious enough to reply immediately to an email which was written in a manner somewhat more casual than (I imagine) the usual letter to a university president. Of course, Morty is very well known amongst the students for being far more approachable than the average university president. I think that if I had written and sent a similar email to Bob Zimmer’s office, I would have phrased it slightly more politely.

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More on Lehmer’s Conjecture

Posted on March 21, 2020 by Persiflage

Lehmer said it was a “natural question” whether there existed an integer such that \(\tau(n)=0\) or not. I’ve wondered a little bit recently about how reasonable this is. (See this post.) The historical context is presumably related to the fact that, by the multiplicativity of coefficients, the vanishing of \(\tau(p)\) for one prime guarantees that a positive proportion of other coefficients vanish. From the perspective of Galois representations, however, I’m a little confused as to whether we expect any sort of “automorphic” Lehmer’s conjecture to hold. To recall, we have

\(\Delta = q \prod_{n=1}^{\infty} (1 – q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n.\)

Deligne’s bound says that \(|\tau(p)| \le p^{11/2}\), so a probabilistic argument suggests that there should only be finitely many primes for which \(\tau(p)=0\). Since there aren’t any such primes in the first few thousand primes, that’s a fairly convincing heuristic for why it might be true. But it’s basically impossible to prove anything this way, so one might hope to formulate a more general conjecture which is true for a more systematic reason.

A first attempt might be to ask that \(a_p(f) \ne 0\) for any eigenform \(f\) of weight \(k \ge 3\) and level prime to \(p\) which is not CM. (When \(k = 2\), of course, there are plenty of modular elliptic curves without CM, and (thanks to Noam) there are plenty of primes \(p\) with \(a_p(f) =0\)). At first thought this seems a little strong; after all, if we just work in weight \(12\) (say) then we know that \(|a_2(f)| < 2^{11/2} < 46\), so surely if you take enough such forms you should find one with \(a_2(f) = 0\). However, this secretly assumes that there are many weight \(12\) forms with coefficients in \(\mathbf{Z}\), and it seems that there are only finitely many such forms. So, for most forms, the coefficients would lie in (presumably) larger and larger number fields, and there would be more possibilities for \(a_2(f)\). For those who did the computation and might be worried, note that the probabilistic heuristic above only applies when the weight \(k \ge 4\). On the other hand, an easy exercise shows that when the weight is odd and the coefficients are integral then the form has CM. The conjecture that there only finitely many non-CM forms with rational coefficients in large even weight is certainly made in this paper, although Dave seems to be hesitant on numerical grounds to make the conjecture for \(k = 4\). There seem to be enough forms of weight \(k = 4\) and integer coefficients that perhaps there exists a form of odd level with \(a_2(f)=0\). In fact, it should be easy to search for such forms if you can search the LMFDB with a fixed Hecke eigenvalue, which I remember John Voight demonstrating at the Simons Institute general meeting, but I couldn’t work out just know when writing this post. Ah, but I guess one can just search for forms with coefficients in \(\mathbf{Q}\) and just look at them by hand. It appears that there is a form

\(f = q + 4 q^3 – 8 q^4 – 5 q^5 – 22 q^7 – 11 q^9 \ldots \in S_4(\Gamma_0(95),\mathbf{Q})\)

with \(a_2(f) = 0\). Are there any examples in higher \(>4\) weight?

All of this becomes similarly confusing on the level of Galois representations. The modular forms with \(a_p(f) = 0\) have the special property that the local \(p\)-adic Galois representation \(\rho_f\) is induced from the unique unramified quadratic extension of \(\mathbf{Q}_p\). From this perspective, the Lehmer conjecture looks a little bit like Greenberg’s conjecture that an ordinary modular form is split if and only if it has complex multiplication. But whereas that conjecture (or at least a stronger version where one assumes such a splitting at all primes of the coefficient field) does follow from plausible conjectures about motives as explained by Matt. I wonder if Matt has any more opinions on what happens if one makes the assumption for only one prime of the coefficient field. Note that if you read Matt’s paper, you might be confused why you can’t also use Serre-Tate theory to prove that elliptic curves with \(a_p = 0\) have CM. I think Florian Herzig gives a nice explanation here of what is going on.

This is also related to the question raised in this post. While that conjecture is not unreasonable, it does skirt the border of conjectures which are actually false, for example, the conjecture that any exceptional splitting in a local Galois representation is caused by (more or less) some global splitting. After all, taken to it’s logical conclusion it would imply not only Lehmer’s conjecture but also (combined with Elkies’ theorem) that all elliptic curves are CM. Greenberg’s conjecture excludes the case of weight one forms, since certainly any form with finite image has many primes for which the local Galois representation is split but the global representation is not if the image is of exceptional type. One can still argue, however, that these forms are potentially CM. On the other hand, non-CM Hilbert modular forms of partial weight one, induced to \(\mathbf{Q}\), also admit some exceptional splitting on inertia. (Note that non-CM Hilbert modular forms actually exist, as follows from the computation of Moy and Specter described here). While these induced forms are not of regular weight (the HT weights are, up to twist \([0,2,2,4]\), the splitting of the local Galois representation is not explained by any correspondences over any finite extension.

I guess the summary is that all of this dicussion points to the fact that Lehmer’s conjecture is not true for any good reason beyond random probability grounds and so is kind of rubbish. Actually this reminds me of an experience one occasionally has after giving a seminar in which you feel like you proved a snazzy result and then the questions from the audience are somewhat deflating. Rest assured, this happens to the best of us — I remember watching a talk online where RLT was talking about his (joint) proof of the Sato-Tate conjecture for \(\Delta\), and the only question from the audience was does this have any implications for Lehmer’s conjecture?

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The Hausdorff Trimester has been indefinitely postponed

Posted on March 18, 2020 by Persiflage

I’m not sure if this is 100% official yet, but the Hausdorff Trimester scheduled for this summer has been (unsurprisingly) postponed. This is probably no great surprise to many of you. We have hopes to reschedule it again (perhaps for 2022) if possible. Together with this update, this is the final unfortunate installment of this series of posts.

On the other hand, mathematics departments have already started to hold online seminars. I was chatting with TG and GB last night and one of them pointed me towards the MIT NT seminar which held it’s first online talk yesterday. If you have zoom installed, which you probably do since you are now expected to teach on zoom next quarter, it is really easy to connect. Bjorn announced that the plan was to continue the seminar online next quarter.

If anyone else plans to hold number theory seminars online, please leave a comment and I will add it to the list.

1. MIT NT seminar (online)

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NSF Proposal, Graduate Fellowship Edition

Posted on March 17, 2020 by Persiflage

Note: I feel as a service to the number theory entertainment complex that I should blog more often in these times, even if it means being less coherent than usual. I might even try to get a few guest posts since I won’t be going to any conferences any time soon…

I recently linked to my first NSF proposal here, but just today I stumbled upon my graduate NSF fellowship application from 1998. There is really only one page which involves any proposal (rather than a list of courses I took or references), and I include the mathematical portion here in full (the only changes from the original are one or two spelling errors and changing fake LaTeX (\rho) to real LaTeX (\(\rho\)).

===============================================================

My research interests center mainly around the study of two dimensional Galois representations, the connection of such representations to Modular forms, and application of these connections to the arithmetic of Elliptic curves. Here are several possible questions which are of interest to me.

1. Serre’s conjectures predict that for any odd, absolutely irreducible Galois representation \(\rho\) into \(\mathrm{GL}_2(\mathbf{F}_q)\), there exists a modular form \(f\) which gives rise to \(\rho\), in the sense of Deligne/Shimura/Deligne-Serre. The characteristic zero representation \(\rho_f\), however, need not be defined into \(\mathrm{GL}_2(W(\mathbf{F}_q))\), (\(W(\mathbf{F}_q)\) = Witt–vectors of \(\mathbf{F}_q\)), but perhaps in \(\mathrm{GL}_2(\mathcal{O})\), for some ramified extension \(\mathcal{O}\) of \(W(\mathbf{F}_q)\). Is there any sense in which one can quantify the ramification of \(\mathcal{O}\) in advance? Is there perhaps a clear cohomological obstruction to a modular lift over \(W(\mathbf{F}_q)\)? Can one quantify this in terms of some \(H^2(G_{\mathbf{Q}},*)\), or perhaps in terms of \(R^{\mathrm{univ}}\), where \(R^{\mathrm{univ}}\) is the universal deformation ring of \(\rho\)? Perhaps if this is too difficult, some qualitative result in this direction?

2. Applications of the above ideas to rational cuspidal eigenforms of Level \(1\). Such forms
are only known to exist if \(\mathrm{dim} S_{2k}(1) = 1\). Can one use ideas from representations to show that no other cuspidal eigenform can be defined over \(\mathbf{Q}\)?

================================================================

My first thought is “I guess I haven’t changed that much as a mathematician over 22 years” followed by “not a bad problem but too optimistic.” The funny thing is that I do think of myself as a number theorist with a certain amount of breadth (despite protestations to the opposite from my most dyspeptic collaborator), so I guess I have to claim that I work on a large circle of ideas and sometimes return to very similar points on the circle. There are also echos in the first proposal of future work with Matt where we studied the ramification of \(\mathcal{O}\) for the reducible representation occurring in weight \(2\) and prime level \(N\), as studied by Mazur. The most definitive result in that paper was computing the exact ramification degree when \(p=2\), where in the case that the ring was a DVR one had \(e = 2^{h-1} – 1\), where \(2^h\) was the order of the \(2\)-part of the class group of \(\mathbf{Q}(\sqrt{-N})\). Other progress on this problem more in the spirit of the formulation above has been done by Lundell and Ramakrishna (MR2770582), although I still think there are many open questions around this problem which are of interest.

On the other hand, the second problem is too optimistic. One reason is related to Buzzard’s observation that, in high weight with \(p\) fixed, the representations all seemed to be defined over rings with very little ramification. (He goes on to make a conjecture along these lines for which nobody has made any progress.) So it seems unlikely to rule out forms of large weight with coefficients in \(\mathbf{Q}\) by showing that there are no such forms over \(\mathbf{Q}_p\) because the latter seems to be false in a strong way. The problem of showing there are no more eigenforms over \(\mathbf{Q}\) when the weight is at least 24, which is very close to Maeda’s Conjecture, is something on which virtually no progress has been made since my proposal, so I guess I don’t have to feel bad for not making any progress myself. On the other hand, I don’t actually think it is an impossible problem. Maybe I should work on it!

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The Arbeitsgemeinschaft has been indefinitely postponed

Posted on March 11, 2020 by Persiflage

An update on the last post: As you probably already know by now if you are a participant, the Arbeitsgemeinschaft has been indefinitely postponed. The recommendation to do so was made by the organizers as universities rapidly began to recommend the cancellation of all work related travel.

I hope that as many of you as possible can cancel your travel plans and get fully reimbursed. For those of you who have difficulties with this, especially those to whom I committed travel funding, please stay tuned. The last message I received from our administration on this matter was the following:

I have forwarded your message to our grants experts. The American Physical Society has asked funding agencies to allow reimbursement for travel expenses that are not reimbursable by airlines and are usually not allowed on grants as they cancelled the very large march meeting the day before it was to begin. I hope to hear if their request has been acted on yet.

Posted in Mathematics, Travel | Tagged | 5 Comments