Concrete and elementary applications of modular forms to elliptic curves

Question

What are some useful facts/algorithms for elliptic curves that can be obtained (proved completely) using the theory of modular forms without heavy machinery?

It's often been asked what elementary applications of modular forms are (partition congruences, counting solutions to quadratic forms...) and while these are all interesting, to me they don't capture the most compelling number theoretic reason we study modular forms, which is to apply them to elliptic curves (and Galois representations, but that's a different question).

Modular forms are essential to the theory of elliptic curves, but the typical applications rely on very deep theorems (Fermat's last theorem, modularity theorem, Birch-and-Swinnerton Dyer Conjecture, Sato-Tate...). Can we obtain some concrete number theoretic information about elliptic curves without relying on these deep theorems?

Here is an example of what I am looking for. The theory of elliptic functions is useful for elliptic curves because, for instance, they describe all isogenies between elliptic curves over subfields of $\mathbb C$, and can also be used to compute the isogenies explicitly, by writing $\wp (\alpha z)$ as a rational function of $\wp (z)$. This gives us very concrete information.

Treat the word "elementary" relatively--probably nothing here is going to be easy. In particular, we can't say anything about all elliptic curves; we can only say things about particular elliptic curves. One first has to find an elliptic curve associated to a modular form before one can get information, and so probably has to invoke Eichler-Shimura in some form. There are, however, examples where one can compute the elliptic curve associated to a modular form relatively easily. For example, in Knapp's Elliptic Curves he computes that the elliptic curve associated with the eigenform $f\in S_2(\Gamma_0(11))$ is $E: y^2+y=x^3-x^2-10x-20$, and there is a map $X_0(11)\to E(\mathbb C)$. (This is easy because $S_2(\Gamma_0(11))$ is 1-dimensional with genus 1.)

However, after we know that that a specific elliptic curve corresponds to a modular form, what can we concretely do with this knowledge? Saying their $L$-functions correspond is nice but to me, not the type of concrete result I'm looking for.

Answer 1

Here are the things that you can do with the modular form $f$ corresponding to an elliptic curve $E$:

(a) Determine the number of points on $E$ mod $p$ by computing $a_p(f)$ (easy for smallish primes via modular symbols computations).

(b) Compute (perhaps with some effort) a modular parameterization of $E$, and then, by evaluating this at Heegner points, find a point of infinite order on a twist $E_D$ of $E$, in the cases when this twist has rank one.

(c) Compute whether or not $L(E_D,1) = 0$ for every twist $E_D$ of $E$, via modular symbols. If you grant BSD, this tells whether or not the twist $E_D$ has infinitely many points.

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I'm not sure what other facts about $E$ you are expecting to get. What is it you would like to know about an elliptic curve in any case? For most people, the rank (and especially whether or not it is positive) is the main thing, and conjecturally this is what you can get from the $L$-function of $E$, which is essentially inaccessible without modular forms, but is highly computable once you know $f$. (And not just for $E$, but for all its twists.)

Maybe the other thing you might like to know is Sha of $E$, but this is not proven to be finite in general. Nevertheless, modular forms can sometimes be used to witness non-trivial elements of Sha. (Read about the theory of ``the visible part of Sha'', by Cremona and Mazur.)

Answer 2

In addition to guy-in-seoul's excellent answer:

Modularity of an elliptic curve $E$ is absolutely essential to even talk about the $L$-function of $E$ outside of the initial region of convergence of the Euler product. Other than for CM curves, the only known way to analytically continue $L$-functions of elliptic curves uses modularity. Same goes for the functional equation.

Also, modularity allows for a very efficient calculation of the $L$-function of $E$ to desired accuracy. This is due to the fact that modular forms have growth conditions at the cusps, which makes them rapidly decreasing functions of $y$ in the upper-half plane. The integral giving the $L$-function of $E$ as a Mellin transform of a weight $2$ modular form is a very rapidly converging integral. This gives a way of approximating the $L$-function of $E$ which is much better than by truncating the Dirichlet series, or the Euler product (which anyways makes sense only in the initial half-plane of convergence).

Modularity also allows us to make exhaustive lists of elliptic curves over $\mathbb Q$. Since, as you point out, $S_2(11)$ is $1$-dimensional, it follows that there exists exactly one elliptic curve over $\mathbb Q$ of conductor $11$, up to isogeny. This fact is not easy to prove without modularity.

Also, all known constructions of the $p$-adic $L$-functions of $E$ rely on its modularity.