## Two idle questions about modular curves

This post is an math-blogging experiment in writing down small questions that have occurred to me, and which I haven’t thought about seriously — thus it is highly possible they are poorly formed, or that the answers are obvious.

1. Let f be a cuspform on S_2(Gamma_0(N)) such that A_f has dimension greater than 1. Then the map X_0(N) -> A_f factors through X_0(N)/W, where W is some group of Atkin-Lehner involutions which act as +1 on A_f. Do we know an example of such an f where the map X_0(N)/W -> A_f is not a closed embedding? What if dim A_f is greater than 2? (In some sense, a map from a curve to a three-fold should be less likely to intersect itself “by chance” than a map from a curve to a surface.)
2. In the original proof of Fermat’s Last Theorem, Mazur’s theorem on rational isogenies of elliptic curves over Q was used in a critical way; when E is your Frey curve, you prove that E is modular, then derive a contradiction from the fact that E[p] is an _irreducible_ modular mod p Galois representation with very little ramification. Nowadays, can one write down a proof of Fermat that doesn’t pass through Mazur’s theorem?

## 6 thoughts on “Two idle questions about modular curves”

1. Dear Jordan,

Regarding question 1:

The difference of any two cusps is torsion in A_f. So if A_f(Q) happens to be torsion-free,
these will have to vanish — in other words, all cusps will have to map to zero. I don’t think
this will always be explained just by Atkin-Lehner involutions.

Incidentally, the condition wf = f (for some Atkin-Lehner involution w)
does not imply that the map X_0(N) –> A_f factors through X_0(N)/w.

For example, when N = 15, w_15 acts by -1 (the rank is zero, and so the sign in the
functional equation is +1), so one of w_3 or w_5 acts as 1. (In fact,
it is w_3 that acts as +1, I think.) But X_0(15) is an elliptic curve, there is only one f,
and X_0(15) –> A_f is an isomorphism.

The point is that w_3 acting as +1 on f doesn’t imply that it acts trivially on X_0(15),
just that it acts via translation by a 2-torsion point. (In my paper with Frank on elliptic curves
of odd modular degree, we analyze this phenomenon; see section 2. We only treat
the case when A_f is an elliptic curve, but a similar analysis should work for more general

Of course, you didn’t assert that W is the group of *all* Atkin-Lehner involutions
that act by +1, but anyway, I thought it might be worth pointing out that sometimes
it really is smaller.

Regards,

Matt

2. Z says:

Regarding question 2, isn’t it more or less a cleverly disguised way of asking wether level lowering works in the residually reducible case? I thought about it then realized that if you don’t know, the odds that I would were pretty slim. Anyhow, I’d be interested to see an answer to that more general question.

3. Dear Z,

I interpreted the question in the same way as you, namely: to what extent is level-lowering known for residually reducible representations?

The example to keep in mind always is the case of prime level treated by Mazur in the Eisenein Ideal paper. Following his notation, I will write N for the level (but remember, N is now prime). Mazur works with level Gamma-naught N, and computes all maximal ideals in the Hecke algebra T that give residually reducible representations: there is precisely one of residue characteristic p for each prime p dividing the numerator of (N-1)/12. If p is such a prime, the p-torsion in the Jacobian is then finite at N (in fact, it is a direct sum of Z/pZ and mu_p). Thus it satisfies the hypotheses of level-lowering (other than being
residually reducible). However, we can´t remove N from the level, since there are *no* weight two cuspforms of level 1.

For example, when N = 11, p = 5, and writing E = X-naught 11 (an elliptic curve), we find that E is finite at 11, but can´t be level-lowered. The proof
of level-lowering breaks down because when one reduces modulo N (= 11
in this case), part of the E escapes into the component group. (The Tamagawa number is 5.) Mazur shows that this is what happens in general:
for any Eisenstein prime p, “one half´´ of the Eisenstein part of the p-torsion
in the Jacobian escapes into the component group of mod N fibre of the Neron
model. So to level lower a reducible rhobar, one needs (at least) an additional hypothesis that the redisue characteristic p doesn´t divide the order of the component group at the prime you are trying to remove.

I don´t know how to apply this criterion in pratice (e.g. for a Frey curve coming from a counter-example to FLT).

Regards,

Matthew

4. Z says:

Dear Matthew (such a level of formality and politeness is a fresh departure from the standard blog comment),
Thanks a lot! I was unaware of all this and it is very interesting of course. Funnily, I had been thinking about related questions recently. To wit, I was thinking about the analytic variation of Tamagawa numbers in ordinary families and wondering if they could have a mu invariant.
Regards,
Olivier

5. JSE says:

Just to add to what Matt says in comment 3: in the Frey curve case, any odd prime p dividing the discriminant is indeed going to show up with valuation a multiple of p; in other words, there will be some p in the Tamagawa number, which means that the residually reducible case has to (or rather, as far as I know has to) be dealt with some other way, e.g. using the theorem of Mazur to rule out this situation in advance for large enough p.

Also, Matt, your points in comment 1 are well-taken; the question as I phrased it is for one thing really dependent on A_f up to isomorphism, not only up to isogeny. In other words, if P-Q is torsion on J_0(N), it’s not so crazy to think that it might “by chance” project to 0 on some A_f. I wonder what would happen if one asked a more “isogeny-invariant” version — do we know examples where P-Q is infinite order, but projects to torsion on A_f, for reasons other than Atkin-Lehner?

As for X_0(15), you are right of course, and I feel silly because my student Ozman actually recently grappled with this exact point! (She’s thinking about the arithmetic of twists of modular curves through Atkin-Lehners.) Is there in your opinion a “correct’ version of this question? (I haven’t looked at your paper with Frank yet but will do so, and send it over to my student.)

6. Dear Jordan,

You are right of course that (in the Frey curve context) p will divide the Tamagawa number of E. It has too: E really has semistable reduction at q (say) if q divides the discriminant, and so E[p] can´t be level-lowered “within E´´. What I mean is that when we look at the Neron model of E mod q, it has to be torus together with a component group, and the torus has only order p p-torsion, so one-half of E[p] must escape into the component group (and so in particular, the component group has to have room for it to fit!). (I should add that one-half of E[p] can´t just evaporate mod q, because it is by assumption finite at q, and so by the Neron property extends to characteristic q.)

This is why the condition for E[p] to be finite at q is the same as the condition for p to
divide the order of the connected component group at q.

But what really matters in the level-lowering question is whether p divides the Tamagawa number of the Jacobian J_0(N) in which one is working. (N is now some composite level, say square-free for safety, q is a factor of N, E is an ellpitic curve quotient of J_0(N), say also of conductor N. I´m not sure what the relation is
in general between Tamagawa numbers for J_0(N) and for its quotients. I think William Stein and Amod Agashe have thought about these kinds of questions (and probably
many others have too); it´s related to studying BSD from the viewpoint of “visibility
of Sha´´.

One thing to note is that if E[p] is reducible, then it is sensitive to (p-)isogeny, and
so one can´t necessarily presume that E is an optimal quotient.

It looks like it could be tricky to say anything in general.

One should also bear in mind that a reducible rhobar is not so much information, just a pair of characters with an extension class between them. (And if one is willing to extend scalars in the field of coefficients of the modular form f, one can even choose a lattice in rho_f such that associated rhobar_f is split.) So it´s hard to believe that this kind of object can be powerful enough to force the existence of lower-level modular
forms, or to deduce FLT. But still, it´s interesting to think about it!

Best wishes,

Matt