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Hi, I was wondering about good techniques that one can use to show that given certain coefficients, they are the Fourier coefficients of a cusp form, assuming we know the desired weight and level. I am aware of Weil's "converse theorem", but am not aware of any examples of it being used to prove something is a modular form. So, even a pointer to an example of that would be useful. I'm curious if there are other more direct methods as well. Thanks!

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5 Answers 5

up vote 15 down vote accepted

Maybe I can add something:

  1. Wiles famous proof of Fermat's Last Theorem (and the later proof of modularity of all elliptic curves over $\mathbf{Q}$ by Breuil, Conrad, Diamond, Taylor) is a sort of ultimate example in which "given certain coefficients [associated to an elliptic curve by point count], they are [proved to be] the Fourier coefficients of a cusp form". This theorem also produces a general strategy for showing that a list of numbers are the coefficients of a cusp form in some cases -- relate those numbers to the point counts of an elliptic curve. Similar remarks apply for abelian varieties these days, due to Ribet's theorem that Serre's conjecture implies modularity of all abelian varieties over $\mathbf{Q}$ of ${\rm GL}_2$-type.

  2. There is something called the Faltings-Serre method, which is an alternative approach to proving modularity statements in particular cases. A google search for "Faltings Serre" will yield many hits. Here's a particular example: http://www.math.wisc.edu/~boston/ell.pdf . There is also a paper by Edray Goins that explains how to use the Faltings-Serre method to get new results in particular cases that go beyond (1), and it has a good list of references: http://homepage.mac.com/ehgoins/papers/serre-faltings.pdf

  3. Another example is http://wstein.org/papers/artin/, which is a paper Kevin Buzzard and I wrote, in which we show that lists of numbers (traces of Frobenius) associated to certain A_5 Galois representations are the coefficients of weight 1 modular forms. One can get the same result by "pure thought" these days as an application of Khare-Wintenberger's proof of Serre's conjecture. But still, there is some value in the computational techniques introduced in that paper. There are similar ideas in Joe Buhler's Ph.D. thesis and also in Springer Lecture Notes in Math Volume 1585.

  4. There is a Theorem of Sturm (see, e.g., Theorem 9.18 of my modular forms book http://wstein.org/books/modform/stein-modform.pdf and the references) that tends to be very relevant to your question in practice. Sturm's result basically tells you that if the first "few" coefficients of two $q$-expansions of modular forms agree modulo a prime, then all terms agree.

This list could go on and on. Also, there are similar ideas and techniques for Hilbert modular forms.

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How sharp is Sturm's result? –  Dror Speiser Feb 8 '11 at 20:12
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@Dror: Sturm's result in general says one has to check about $2d$ coefficients, where $d$ is the dimension. For two arbitrary cusp forms, that's pretty good (massively better than "effective Chebotarev", say), but for eigenforms it is far from optimal. Under some hypothesis, Sturm reduces the number of coefficients to roughly $2d/2^v$, for some $v$ (this $v$ is $\leq$ the number of primes dividing the level). There is a result in the Buzzard-Stein paper I mentioned that improves on Sturm's result in the case of eigenforms with nontrivial character. –  William Stein Feb 9 '11 at 7:56
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It depends on what you mean by "certain coefficients" and how they're given.

This sort of sounds like you have a (finite) handful of coefficients in a $q$-expansion and want to try to "fill them out" to a modular form of prescribed weight and level. If this is the case, it is pretty easy. This is because the space of such forms is finite-dimensional, and a basis of the space is readily computed (using, for example, the SAGE software). This reduces the problem to basic linear algebra.

If, on the other hand, you have an entire $q$-expansion, it's not so clear. You can truncate and employ the above procedure. I'll just mention three things:

1) If your expansion isn't a modular form, then this will eventually (by taking larger chunks) tell you so.

2) If it is a modular form (but you have no a priori knowledge of this), then it just gives you a lot of evidence that your expansion is a modular form, but doesn't prove it. Of course, if you know more about your coefficients you might be able to go back and prove that it is the form evidenced above.

3) If you do happen to know that your expansion is a modular form of given weight and level and just want to know "what it is," then this method will eventually provably determine what it is (in terms of the basis) by taking large enough chunks.

As for the converse theorem... you'd certainly need all the coefficients, but I have a hard time imagining that method to be very practical (but I'm not very analytically minded).

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Regarding your point 2, you can run into undecidability problems if you don't restrict how the coefficients are presented to you (but I haven't heard of this being a problem very often in practice). –  S. Carnahan Feb 8 '11 at 5:17
    
@Ramsey: Point 1) seems vague and only practical if there is something that limits the order of the expansion that needs to be considered. Otherwise one eventually will run into computer resource limitations - which are in fact quite severe in SAGE even at modest weights and levels. –  Laie Feb 8 '11 at 17:33
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@Laie: It is vague, and relies on the fact that if a collection of $q$-expansions are linearly independent, then so are their truncations to the first $N$ terms for large $N$. Perhaps the $N$ required can be huge, I'm not sure. I'd be curious (for several reasons related to these answers) to see what the OP actually has. –  Ramsey Feb 8 '11 at 19:28
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I think Ramsey's answer hits the important points, but I wanted to comment on the converse theorem.

The converse theorem takes as an input a Dirichlet series (more generally, an $L$-function) whose analytic continuation (plus quite a bit more) is already known and tells you that it actually came from an automorphic form. Unfortunately, the only methods that I know for proving analytic continuation require knowing that it already comes from an automorphic form. So it certainly sounds circular, and I've only seen it applied to proving global functoriality (for quasi-split classical groups, small symmetric powers, etc). Let me quickly(!) explain the argument:
Basically (eliding the hard details to the point where I may say things that are literally, but hopefully not morally, incorrect) say you start out with an automorphic representation $\pi$ on some classical group $G$, living inside $GL(N)$. The Langlands-Shahidi method proves that the twisted $L$-functions $L(s,\pi\times \tau)$ are nice (entire, functional equation, etc), with $\tau$ an automorphic representation of $GL(m)$ with $m < N$. Local functoriality gives lifts from $\pi_\nu$ on $G_\nu$ to $\Pi_\nu$ on $GL(N)$ such that the local $L$-factors coincide, in that $L_\nu(s,\pi_\nu\times\tau_\nu)=L_\nu(s,\Pi_\nu\times\tau_\nu)$.
Since $L(s,\Pi\times\tau)=\prod_\nu L_\nu(s,\Pi_\nu\times\tau_\nu)=L(s,\pi\times\tau)$, it is nice for all $\tau$, so the converse theorem applies and $\Pi$ is an automorphic representation of $GL(N)$.

Modularity / potential automorphy methods (e.g. Serre's Conjecture, Modularity Theorem) exist, too. In the weight 2 case, given enough information about the coefficients and a sufficient amount of pluck, it is not inconceivable that you could show that the coefficients actually come from an elliptic curve (or several elliptic curves if your modular form is not an eigenform), then use modularity to deduce that they come from a modular form.

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If you assume that the modular form is a newform with prescribed weight and level (EDIT : you also need to know the eigenvalue of the Atkin-Lehner involution), then its Fourier coefficients can sometimes be guessed using the fact that the associated L-series satisfies a functional equation. The point is that using the functional equation, the L-series can be computed by a rapidly convergent series, so it is only necessary to have few Fourier coefficients at hand to compute an approximation of it. This even gives a very pratical and powerful method for finding modular forms, at least when the level is small. You should look at the following example written by Tim Dokchitser :

http://magma.maths.usyd.edu.au/magma/handbook/text/1392#15270

Dokchitser's algorithm for computing L-functions has been implemented in Sage and Magma. Note that the method outlined here is (to my knowledge) purely experimental, in the sense that it doesn't prove that the resulting q-expansion is modular.

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This "approximate functional equation" technique is surprisingly useful in practice. For instance: sometimes you have the L-function of a variety at all primes except, say, 2, because the special fiber at 2 is horrible. But the method described here allows you to guess the Euler factor at 2 with moral certainty if you know (a long list of) the other Euler factors. –  JSE Feb 8 '11 at 14:36
    
@JSE : in the situation you describe, isn't the method even perfectly rigourous (because Fourier coefficients are integers, it suffices to find approximate them within $1/2$). Ah, maybe the problem is that in general the functional equation of the L-function is only conjectural ? –  François Brunault Feb 8 '11 at 15:20
    
Right, that was the case in the applications I seem to remember (though of course so many more Galois representations are provably modular these days...) –  JSE Feb 9 '11 at 5:57
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( tried posting this earlier today, apologies if it appears twice)

For an example of a direct application of Weil's converse theorem, see Shimura's original formulation of the Shimura lift (Annals 1973). He constructs a map from modular forms of half-integral weight to modular forms of integral weight by defining an L-series built out of the Fourier coefficients of the input form, and spends most of the paper proving the analytic properties needed for the converse theorem.

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