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 SL(2, \mathbf Z).

So consider the simplest family of elliptic curves, the Weierstrass family:

y^2=x^3+ax+b.

Here a and b are meant to be some formal parameters. This indeed defines and elliptic curve over the ring k[a,b,\Delta^{-1}], where k is the base field, which is supposed to be of characteristic {0}, and \Delta is the discriminant:

\Delta=-16(4 a^3 + 27 b^2).

When we write y^2=x^3+ax+b we in fact mean the corresponding projective variety E over Spec\; k[a,b] with equation

\bar y^2 \bar z = \bar x^3 + a \bar x^2 \bar z + b \bar z^3.

Let us denote the affine chart with coordinate functions x, y by U = E\setminus \{0\} and the point at infinity by {0} since it is the zero point for the addition on the curve.

Now we are going to compute some Laurent series expansions at {0}. First we choose local parameter t=-x/y. Indeed, x has pole of order 2 and y has pole of order 3 at {0}, therefore t has simple zero there. To find expansion of $x$ we solve the following equation in Laurent series:

\frac{x^2}{t^2} = x^3 + a x + b.

Rewriting it as

(x t^2)^3 - (x t^2)^2 + a t^4 (x t^2) + b t^6=0

we obtain a polynomial equation in x t^2 which can be solved by Newton’s method starting with x t^2 = 1 + O(t). We obtain

x = t^{-2}-a t^2-b t^4-a^2t^6 - 3 a b t^8+O(t^{10}),

y= -t^{-3}+a t + b t^3 + a^2 t^5 + 3 a b t^7+O(t^9).

Let us compute the expansion of the invariant differential \omega = \frac{dx}{2y}:

\frac{dx}{2y} = (1+2a t^4+3b t^6+6 a^2 t^8+20 a b t^{10}+O(t^{12})) dt.

We see that it is possible to integrate this series formally and make it the new local parameter:

z = \int \frac{dx}{2y} = t+ \frac{2a}5 t^5+\frac{3b}7 t^7+\frac{2 a^2}3 t^9+\frac{20 a b}{11} t^{11}+O(t^{13}).

Then the expansions of $x$ and $y$ with respect to the new local parameter are:

x = z^{-2}-\frac{a}5 z^2-\frac{b}7 z^4+\frac{a^2}{75}z^6 + \frac{3 ab}{385}z^8+O(z^{10}),

y = \frac{\partial}{2\partial z}x = -z^{-3}-\frac{a}5 z -\frac{2b}7 z^3 + \frac{a^2}{25} z^5 + \frac{12 a b}{385} z^7+O(z^9).

We also consider the formal integral of -x dz:

v_0:=-\int x dz = z^{-1} +\frac{a}{15}z^3 +\frac{b}{35} z^5 - \frac{a^2}{525}z^7 - \frac{ab}{1155} z^9 + O(z^11).

Consider the power series

\frac{1}{e^{z}-1} + \frac{1}{2}-\frac{z}{12}= z^{-1} + \sum_{k\geq 2}\frac{B_{2k}}{(2k)!}z^{2k-1}.

If we substitute this power series in place of v_0 and find

x = -\frac{\partial v_0}{\partial z} = \frac{1}{(e^z-1)^2} + \frac{1}{e^z-1} + \frac{1}{12},

y = \frac{\partial x}{2 \partial z} = -\frac{1}{(e^z-1)^3} - \frac{3}{2(e^z-1)^2} - \frac{1}{2(e^z-1)},

then we can easily verify that

y^2=x^3 - \frac{x}{48} + \frac{1}{864},

i.e. we have found a solution for a=-\frac{1}{48}, b=\frac{1}{864}.

This explains that we should in general put

a = -\frac{E_4}{48} \qquad b=\frac{E_6}{864}

and define E_{2k} in such a way that

v_0 = z^{-1} + \sum_{k\geq 2}\frac{B_{2k} E_{2k}}{(2k)!}z^{2k-1}.

In this way we obtain E_{2k} as a polynomial of E_4, E_6, but in fact it is true that this polynomial is the same polynomial that expresses the Eisenstein series of weight 2k in terms of the Eisenstein series of weights 4 and 6. So for us modular forms will be homogeneous polynomials of a and b where weight of a is 4 and weight of b is 6.

To define the weight more geometrically let us consider the action of the multiplicative group on E:

(a,b,x,y)\longrightarrow (\lambda^4 a, \lambda^6 b, \lambda^2 x, \lambda^3 y), \qquad (\lambda\in k^\times).

Then a modular form f of weight k is a function of a, b which transforms like

f \longrightarrow \lambda^k f.

If we consider not only functions of a, b, but functions of a, b, x, y then we obtain Jacobi forms of index {0}.

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 \delta_e. This simply takes a modular form f of weight k and sends it to kf. 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 \partial on k[a,b]. Let us try to lift it to obtain a derivation of the ring of functions on U (which is generated by a,b,x,y). We would have \partial^* x, \partial^* y satisfying a relation

2 y \partial^* y = (3 x^2 + a) \partial^* x + x \partial a + \partial b.

But note that we could simply apply \partial to the Laurent series expansions of x,y term by term (denote it by \partial x, \partial y) and get a solution to the relation above. Therefore the difference must satisfy

2y (\partial y-\partial^* y) = (3 x^2 + a) (\partial x - \partial^* x).

But we also have a solution to the equation above! Namely it is the operator \frac{d}{dz} which will be denoted simply by '. Therefore we must have a Laurent series \alpha which satisfies

\partial y = \partial^* y + \alpha y',\qquad \partial x = \partial^* x+ \alpha x'.

Using the fact that ', \partial commute it is easy to obtain

\alpha' x'=2\partial^* y - (\partial^* x)'.

We expect \partial^* x, \partial^* y to be regular functions on U. Clearly one can assume \partial^* x to contain only even powers of z and \partial^* y to contain only odd powers of z – this corresponds to \partial^* being invariant under the involution (x,y)\longrightarrow (x,-y). We see that the right hand side is a regular function on U which contains only odd powers of z. Therefore it is a product of y and a polynomial in x, a, b. So we write

\alpha' x' = 2 y P(x).

Noting that x' = 2y gives

\alpha' = P(x).

Next observation is that for any polynomial in x we can express it as a derivative of an expression of the form

y Q(x) + A z + B v_0 \qquad (Q\in k[a,b][x], \; A\in k[a,b],\; B\in k[a,b].)

In fact z is the formal integral of \omega=dz and v_0 is the formal integral of -\eta = -\frac{x dx}{2y} = -x dz and these forms generate the first cohomology of E. So,

\alpha = y Q(x) + A z + B v_0.

It implies that

\partial x = R(x) + (A z + B v_0) x' \qquad (R\in k[a,b][x]).

But we know that

\partial x = -\frac{\partial a}{5} z^2 - \frac{\partial b}{7} z^4 + O(z^6).

Looking at the power series expansions we conclude that

\partial x = A( z x'+ 2x) + B(v_0 x' + 2 x^2 + \frac {4a}3).

So it is natural to consider a derivation for which (A,B)=(1,0) and a derivation for which (A,B) = (0,1). In the former case we obtain

\alpha = z,\; \partial a = 4 a, \; \partial b = 6b,\; \partial^*x = 2x,\; \partial^* y=3y.

It is easy to see that we have got the Euler operator. In the latter case we obtain

\alpha = v_0,\; \partial a = 6 b,\; \partial b = - \frac{4 a^2}{3},\; \partial^* x = 2 x^2 + \frac{4a}3,\; \partial^* y = 3xy.

Using our convention a = -\frac{E_4}{48} \; b=\frac{E_6}{864} one can see that this is the Serre derivative \delta_s:

\delta_s E_4 = -\frac{E_6}{3},\qquad \delta_s E_6 = -\frac{E_4^2}{2}.

It is important that we did not only obtain \delta_s as a certain canonical derivation which lifts to a derivation on k[U], but we also computed \delta_s x 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 f is a modular form of weight k and a curve y^2 = x^3 + a_0 x + b_0 has periods \omega_1, \omega_2, then

f(a_0,b_0) = f(\frac{\omega_1}{\omega_2}) \left(\frac{\omega_2}{2\pi i}\right)^{-k}.

On the left we have the values of f as a polynomial of a, b 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.

 

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