Let $\Delta$ denote the discriminant of the cubic curve given by the formula ($*$). (This is a somewhat complicated expression in the $a_i$s which I won't write down here; but let me note that there are standard expressions $c_4$ and $c_6$ which are certain polynomials in the $a_i$s, such that $1728 \Delta = c_4^3 - c_6^2$.) The formula ($*$) then defines an elliptic curve over $S:= $Spec $\mathbb Z[a_1,a_2,a_3,a_4,a_6,\Delta^{-1}].$ Also the invertible sheaf $\omega$ over $S$ attached to this elliptic curve is canonically trivialized, because
it admits the global section $dx/(2y +a_1x + a_3) = dy/(3x^2 + 2 a_2 x + a_4 - a_1y).$
Thus any modular form of weight $n$, when evaluated on ($*$), gives rise to a global section of the $\mathcal O_S$, which is to say an element of $\mathbb Z[a_1,\cdots,a_6,\Delta^{-1}].$
Furthermore, as Deligne shows and as you noted, given any ellptic curve $E$ over any base $S'$, we may cover $S'$ by open sets $U$ such that $E_{| U}$ is the
pull-back of ($*$) via a map $U \to S$. Thus the value of the modular form on $E$ is determined by its value on $(*)$.
In short, any modular form is determined by giving a certain element of
$\mathbb Z[a_1,\ldots,a_6,\Delta^{-1}]$.
Note that not every element of this ring is actually a modular form, because
the maps $U \to S$ discussed above are not unique. Thus modular forms are those
elements of $\mathbb Z[a_1,\ldots,a_6,\Delta^{-1}]$ which are invariant under the automorphisms of this ring which are induced by "change of Weierstrass equation". Deligne discusses this, and concludes that the ring of modular forms is exactly $\mathbb Z[c_4,c_6,\Delta^{\pm 1}]$. (There is also a question of
holomorphicity of the cusps which I am ignoring here; probably Deligne addresses it by allowing certain singular curves as well, and hence working over $\mathbb Z[a_1,\ldots,a_6]$ rather than $\mathbb Z[a_1,\ldots,a_6,\Delta^{-1}].$ This will give the correct answer of $\mathbb Z[c_4,c_6,\Delta]$, i.e. he doesn't allow $\Delta^{-1}$ as a modular forms, since while this is well-defined on true elliptic curves, it is not well-defined on singular cubic curves.)
Note that one can also replace $\mathbb Z$ by another ring $R$, and restrict attention to $R$-schemes, and hence define the ring of modular forms over $R$. E.g. if you take $R = \mathbb F_2$, you get the ring of modular forms mod $2$. You can check, using Deligne's formlas, that $a_1$ is in variant under change of Weierstrass equation in char. $2$, and thus defines a modular form mod $2$, called the Hasse invariant.
Similarly, you can check that $b_2$ is a well-defined modular form mod $3$ (but only mod $3$). This is the mod $3$ Hasse invariant.
These are good examples to think about to practice using Deligne's (actually Tate's) formulas for the change of Weierstrass equation to define modular forms.
I am unsure of the early motivations for studying elliptic curves, so I will leave that discussion for another to answer.
At any rate, integer factorization is one of the most important problems in applied number theory, and elliptic curves facilitate a sub-exponential factorization algorithm, discovered in 1985 by Hendrik Lenstra.
As you probably already know, the points $(x,y)$ that solve the elliptic curve over a given field can be endowed with a group structure. The algorithm takes advantage of this fact and proceeds as follows:
- Choose a number $n \in \mathbb{N}$ to be factored.
- Choose a random elliptic curve $E(\mathbb{Z}_n)$ and a point $P \in E$.
- Choose a smooth number $e \in \mathbb{N}$. $m!$ for a small $m$ is a common choice.
Compute $eP$. As we do this, the way addition has been defined forces us to compute the inverse of an element modulo $n$, which can be done via the Euclidean algorithm. As we proceed with this step, there are three scenarios we can encounter:
All the calculations could be done since the inverse mentioned above was able to be computed with each addition. In this case, go back to the second bullet above and repeat the whole process with a new elliptic curve.
We arrive at $kP = \infty$ for some $k \leq e$. If this happens, go to the second bullet above and repeat.
We arrive at an addition that could not be computed because the inverse of an element $k \in \mathbb{Z}_n$ did not exist. If this happens, $k$ and $n$ are not coprime, which means $k$ is a nontrivial factor of $n$.
Read more about why this works.
Also, if we count cryptography as a subset of (applied) number theory, then one can also use the group provided by an elliptic curve to carry out discrete-log-based asymmetric cryptosystems like Diffie-Hellman or digital signature schemes like ECDSA. The advantage here is that there are no known algorithms for solving the elliptic curve discrete log problem in sub-exponential time, unlike the $\mathbb{Z}_p$ setting.
Best Answer
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.
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.)