It seems when people talk about modular forms they tend to forget that they are very related to families of elliptic curves. Here I want to explain some simple way to understand the connection. We will consider modular forms for the full modular group .
So consider the simplest family of elliptic curves, the Weierstrass family:
Here and are meant to be some formal parameters. This indeed defines and elliptic curve over the ring , where is the base field, which is supposed to be of characteristic , and is the discriminant:
When we write we in fact mean the corresponding projective variety over with equation
Let us denote the affine chart with coordinate functions by and the point at infinity by since it is the zero point for the addition on the curve.
Now we are going to compute some Laurent series expansions at . First we choose local parameter . Indeed, has pole of order and has pole of order at , therefore has simple zero there. To find expansion of $x$ we solve the following equation in Laurent series:
Rewriting it as
we obtain a polynomial equation in which can be solved by Newton’s method starting with . We obtain
Let us compute the expansion of the invariant differential :
We see that it is possible to integrate this series formally and make it the new local parameter:
Then the expansions of $x$ and $y$ with respect to the new local parameter are:
We also consider the formal integral of :
Consider the power series
If we substitute this power series in place of and find
then we can easily verify that
i.e. we have found a solution for , .
This explains that we should in general put
and define in such a way that
In this way we obtain as a polynomial of , , but in fact it is true that this polynomial is the same polynomial that expresses the Eisenstein series of weight in terms of the Eisenstein series of weights and . So for us modular forms will be homogeneous polynomials of and where weight of is and weight of is .
To define the weight more geometrically let us consider the action of the multiplicative group on :
Then a modular form of weight is a function of which transforms like
If we consider not only functions of , but functions of then we obtain Jacobi forms of index .
Derivatives of modular forms
We want to apply this language to understand some natural operations on modular forms. The first operation is the Euler derivative . This simply takes a modular form of weight and sends it to . It is easy to see that this is exactly the action of the Lie algebra of the multiplicative group. Next we want to reconstruct the Serre derivative.
Suppose we have a derivation on . Let us try to lift it to obtain a derivation of the ring of functions on (which is generated by ). We would have , satisfying a relation
But note that we could simply apply to the Laurent series expansions of term by term (denote it by ) and get a solution to the relation above. Therefore the difference must satisfy
But we also have a solution to the equation above! Namely it is the operator which will be denoted simply by . Therefore we must have a Laurent series which satisfies
Using the fact that commute it is easy to obtain
We expect to be regular functions on . Clearly one can assume to contain only even powers of and to contain only odd powers of – this corresponds to being invariant under the involution . We see that the right hand side is a regular function on which contains only odd powers of . Therefore it is a product of and a polynomial in . So we write
Noting that gives
Next observation is that for any polynomial in we can express it as a derivative of an expression of the form
In fact is the formal integral of and is the formal integral of and these forms generate the first cohomology of . So,
It implies that
But we know that
Looking at the power series expansions we conclude that
So it is natural to consider a derivation for which and a derivation for which . In the former case we obtain
It is easy to see that we have got the Euler operator. In the latter case we obtain
Using our convention one can see that this is the Serre derivative :
It is important that we did not only obtain as a certain canonical derivation which lifts to a derivation on , but we also computed which can be interpreted as a formula which gives the Serre derivatives of all the Eisenstein series.
In the end I would like to mention that using this approach and studying the Gauss-Manin connection one can explain some other things which appear in the theory of modular and quasi-modular forms and seem mysterious, like Bol’s identity and Rankin-Cohen brackets.
The main idea is: “the ring of modular forms, or the ring of quasi-modular forms come naturally equipped with an elliptic curve over it.“
Also here is a useful formula for values of modular forms. If is a modular form of weight and a curve has periods , then
On the left we have the values of as a polynomial of and on the right we have its value as a function on the upper half plane. There is a corresponding formula relating values of quasi-modular forms and periods of differentials of second kind.