Using the Riemann-Roch theorem, we explained that the set of isomorphism classes of elliptic curves over a field $k$ in which $6$ is invertible is in natural bijection with the set of pairs $(a_4,a_6)$ modulo the equivalence relation in which $(a_4,a_6)$ is equivalent to $(b_4,b_6)$ if $a_4 = \lambda^4 b_4$ and $a_6 = \lambda^6 b_6$ for some $\lambda\in k^\times$. The equivalence, of course, assigns to the pair $(a_4,a_6)$ the elliptic curve given by the equation $$ y^2 = x^3 + a_4 x + a_6,$$ in "short Weierstrass form". We used this to explain why the functor on $Z$-algebras (where $Z$ is a base ring in which $6$ is invertible) which to $S$ associates the set of $S$-isomorphism classes of elliptic curves over $S$ fails to be a representable functor.

The purely algebraic definition we have given of a (weakly holomorphic) modular form can seem a bit dry and these objects really start coming to life when considered over specific fields like the field ${\mathbb C}$ of complex numbers, the field of $p$-adic numbers, or fields of characteristic $p$. This lecture was mainly devoted to a review of Weierstrass theory which gives a convenient description of the set of framed elliptic curves over ${\mathbb C}$ in terms of lattices in ${\mathbb C}$. This allowed us to assign to a modular form $f$ as defined in the previous lecture a function on the complex upper half plane by the rule $$f(\tau) = f({\mathbb C}/\langle 1,\tau\rangle, dz).$$ Weierstrass theory leads to explicit formulae for $a_4(\tau)$ and $a_6(\tau)$: $$ \left({\mathbb C}/\langle 1,\tau\rangle, dz\right) = \left( y^2 = x^3 - 15 G_4(\tau) x - 35 G_6(\tau), \frac{1}{2} \frac{dx}{y}\right),$$ where $$ G_k(\tau) = \sum_{(m,n)\ne (0,0)} \frac{1}{(m\tau+n)^k}.$$ The holomorphicity of $G_k(\tau)$ allowed us to conclude that any weakly holomorphic modular form is a holomorphic function on the Poincaré upper half-plane, satisfying the transformation rule $$ f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^k f(\tau), \quad \mbox{ for all } \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\in {\bf SL}_2({\mathbb Z}).$$

We were then able to define a

One of the proofs of this result (which is done, for instance, in Serre's "Course in Arithmetic" is to show that the above infinite product is a modular form of weight twelve by relating its logarithmic derivative to the ``Eisenstein series of weight two" $$ E_1(q) = 1 - 24 \sum_{n=1}^\infty \sigma(n)q^n,$$ and studying the transformation properties of the latter. The approach we will follows is somewhat more long-winded but gains in naturality what it loses in efficiency, and is more in the spirit of our treatment of modular forms emphasising moduli of elliptic curves.

Our starting point was the analytic formula $$ \Delta(\tau) \sim \left(\wp(\frac{1}{2})-\wp(\frac{\tau}{2})\right)^2 \left(\wp(\frac{1}{2})-\wp(\frac{1+\tau}{2})\right)^2 \left(\wp(\frac{\tau}{2})- \wp(\frac{1+\tau}{2})\right)^2,$$ where $\sim$ denotes an equality up to a multiplicative scalar in ${\mathbb C}^\times$. In order to expand $\wp(z)$ in terms of $t= e^{2\pi i z}$, we were led to consider the infinite sums \begin{eqnarray*} X_q(t) &=& \sum_{n=-\infty}^\infty \frac{q^nt}{(1-q^nt)^2} + \left\{ \frac{1}{12} - 2\sum_{n=1}^\infty \frac{q^n}{(1-q^n)^2} \right\},\\ Y_q(t) &=& \sum_{n=-\infty}^\infty \frac{q^nt(1+q^nt)}{(1-q^nt)^3}, \end{eqnarray*} where the constant that appears in the expansion of $X_q(t)$ was not assumed to be known a priori). We observed that $X_q(t)$, viewed as a function of $z$ via the change of variables $t= e^{2\pi i z}$, is an elliptic function having (double) poles only at the points of the lattice $\Lambda_\tau$ spanned by $1$ and $\tau$, and has a Laurent series expansion about $z=0$ of the form $$ X_q(t) = \frac{1}{(2\pi i)^2 z^2} + O(z^2).$$ (The vanishing of the constant term in this expansion is the reason for the complicated constant in the definition of $X_q(t)$.) From this we concluded that $$ \wp_\tau(z) = (2\pi i)^2 X_q(t), \qquad \wp'(\tau) = (2\pi i)^3 Y_q(t).$$

We then turned out attention to the definition of other elliptic functions over ${\mathbb C}$, namely the Weierstrass $\xi$ and $\sigma$ functions. The first is a meromorphic function on $\mathbb C$ having only simple poles at $\Lambda_\tau$ with {\em integer residues}, and satisfying the equation $$ d\xi(z) = - \wp(z) dz.$$ The second (which we didn't yet have a chance to introduce) is a holomorphic function on $\mathbb C$ satisying the equation $$ d\log \sigma(z) =: \frac{\sigma'(z)}{\sigma(z)} = \xi(z).$$

- $\xi(z)$ has simple poles at all $z\in \Lambda_\tau$ and no poles anywhere else, while $\sigma(z)$ has simple zeroes at $z\in\Lambda$ and no poles anywhere else.
- $\xi(z)$ and $\sigma(z)$ satisfy the following transformation properties under translation by
$\omega\in \Lambda_\tau$:
$$ \xi(z+\omega) = \xi(z) + \eta(\omega), \qquad \sigma(z+\omega) = \pm e^{\eta(\omega)(z+\omega/2)}\sigma(z),
$$
where $\eta: \Lambda_\tau\rightarrow \mathbb C$ is the
*quasi-period homomorphism*attached to $\Lambda_\tau$ and the $\pm$ depends on whether $\omega$ belongs to $2\Lambda_\tau$ or not.

We then defined the $L$-series of a cusp form $f$ satisfying $f(q)= \sum a_n q^n$ by re-packaging these fourier coefficients into a Dirichlet series: $$ L(f,s) = \sum_{n=1}^\infty a_n n^{-s}.$$ Using the fact that the function $$ \phi_f(\tau) := y^{k/2} |f(\tau)|$$ is ${\mathbf SL}_2({\mathbb Z})$-invariant and bounded on the fundamental region for ${\mathbf SL}_2({\mathbb Z})$ on the upper half plane, we estabslished the existence of constants $C$ and $C'$ for which $$ |f(\tau)| \le C y^{-k/2}, \qquad a_n(f) \le C' n^{k/2}.$$ The second inequality allowed us to conclude that the infinite sum defining $L(f,s)$ converges absolutely (and uniformly on compact subsets) to an analytic function on the right half plane $Re(s) \gt 1+k/2$.

We then showed that the $L$-function of a normalised eigenform of weight $k$ is distinguished by the fact that it admits an Euler product expansion of the form $$ L(f,s) = \prod_p (1-a_p p^{-s} + p^{k-1-2s})^{-1}.$$ We concluded with some motivational remarks concerning the relation between modular forms and compatible systems of $\ell$-adic Galois representations, hinting at themes that will recur later in the course.

We focused primarily on two kinds of level $N$ structure: of type $0$, or of $\Gamma_0(N)$-type, corresponding to a marked cyclic subgroup (scheme) on $E/R$ of order $N$; of type $1$, or of $\Gamma_1(N)$-type, corresponding to a marked point of order $N$ on $E$, or equivalently, to a monomorphism ${\mathbb Z}/N{\mathbb Z}\rightarrow E$ of group schemes over $R$. We explained that such a modular form of level $N$ admits several $q$ expansions in general, corresponding to the different possible level $N$ structures on the Tate curve $E_q$ over $Z((q))$. We described these level $N$ structures and showed that they are all defined over the ring $Z(\zeta_N)((q^{1/N}))$. This allowed us to define a modular form (resp. a cusp form) to be a rule as above, all of whose $q$ expansions belong to $Z(\zeta_N)[[q^{1/N}]]$ (resp. to $q^{1/N} Z(\zeta_N)[[q^{1/N}]]$.)

The lecture ended with the assertion that set of all $q$-expansions for modular forms with $\Gamma$-level structure, where $\Gamma= \Gamma_0(N)$ or $\Gamma_1(N)$, taken modulo the automorphisms of $Z(\zeta_N)((q^{1/N}))/Z(\zeta_N)((q))$, is in natural bijection with the quotient $ \Gamma\backslash {\mathbb P}_1({\mathbb Q})$. Such an equivalence class of called a

The first observation that allows the efficient calculation of $S_k(\Gamma)$ is the existence of a perfect Hecke-equivariant pairing $${\mathbb T}_k(\Gamma) \times S_k(\Gamma) \rightarrow \mathbb C$$ sending $(T,f)$ to the fourier coefficient $a_1(Tf)$, where ${\mathbb T}_k(\Gamma)$ is the $\mathbb C$-subalgebra of End$(S_k(\Gamma))$ generated by all the Hecke operators. The resulting isomorphism from the dual of ${\mathbb T}_k$ to $S_k(\Gamma)$ sends the element $\varphi$ to the modular form $f$ whose $q$ expansion is $\sum_{n=1}^\infty \varphi(T_n) q^n$. (Again, assuming here that $\Gamma= \Gamma_0(N)$ or $\Gamma_1(N)$ for simplicity.)

The second tool used to gain a handle on the structure of ${\mathbb T}_k(\Gamma)$ without computing $S_k(\Gamma)$ a priori, which we described when $k=2$, is to exploit the analytically defined integration pairing $$ M_2(\Gamma) \times \Gamma_{ab} \longrightarrow \mathbb C,$$ where $\Gamma_{ab}:= \Gamma/[\Gamma,\Gamma]$ is the maximal abelian quotient of $\Gamma$. This pairing is defined by $$ \langle f, \gamma \rangle := \int_{\tau_0}^{\gamma \tau_0} 2\pi i f(\tau) d\tau.$$ This expression does not depend on the path from $\tau_0\in {\mathcal H}$ to its translate under $\Gamma$, nor does it depend on the base point $\tau_0$. We ended by showing that the above pairing induces a natural $\mathbb C$-linear homomorphism $$ \varphi: M_2(\Gamma) \rightarrow H^1(\Gamma,\mathbb C).$$

We then pursued our discussion of the map $\varphi$ described in the previous lecture, showing that it is

We explained that $\varphi$ induces a map from $S_2(\Gamma)$ to the

a) it is enough to show that $A(X,Y) -1$ is irreducible to show that it generates $I$ (by an argument invoking the fact that all nested chains of prime ideals in $\mathbb F_p[X,Y]$, a ring of Krull dimension two, have length $\le 2$).

b) It is then enough to show that the homogenous polynomial $A(X,Y)$ has no multiple factors. (By a geometric argument in which Bezout's theorem was the main ingredient.) The argument actually showed that $A(X,Y)-1$ is absolutely irreducible, i.e., stays irreducible in $\bar{\mathbb F}_p[X,Y]$ when $A(X,Y)$ has no multiple factors.

We are now left with the problem of showing that the Hasse invariant $A(X,Y)$ has no mutiple factors, a task which will rely on an elegant modular description of the Hasse invariant.

It follows from this more conceptual description of $A$ that the zeroes of $A(E,\omega)$ correspond precisely to the

From this, we concluded that the zeroes of $A$ are defined over $\mathbb F_q$ with $q=p^2$, and correspond to the elliptic curves $E: y^2 = f(x)$ for which the coefficient of $x^{q-1}$ in $f(x)^{(q-1)/2}$ is $0$ (modulo $p$, of course). Denoting this coefficient as $A_q(E,dx/y)$, we saw that $A_q$ is a modular form of weight $q-1 = (p-1)(p+1)$ which has the same zeroes as the modular form $A$ of weight $p-1$.

Today we studied the Hasse invariant more deeply, and showed that it has the same zeroes as the weight $p-1$ modular form $A_p$ defined by the relation $$ A_p\left(y^2=f(x), dx/y\right) = \mbox{Coefficient of $x^{p-1}$ in } f(x)^{\frac{p-1}{2}}.$$ We showed, by a direct computation, that the value of $A_p$ on the Legendre family $y^2 = x (x-1) (x-\lambda)$ is equal to $$ A_p(\lambda) = \pm \sum_{j=0}^m {m\choose j}^2 \lambda^j, \qquad m:= (p-1)/2$$ by a direct calculation. We then showed that $A_p(\lambda)$ satisfies the Picard-Fuchs differential equation $$ \lambda(\lambda-1) A_p''(\lambda) + (2\lambda-1) A_p'(\lambda) + \frac{1}{4} A_p(\lambda) = 0.$$ Presumably, this can be checked by a direct calculation, boiling down to elementary manipulations of binomial coefficients mod $p$. I did not attempt the exercise, but rather presented a proof which (I hope) makes up in memorableness what it lacks in conciseness.

We started by showing that the Picard-Fuchs equation is the very same differential equation satisfied by the class of the regular differential $\omega$ and its derivatives $\omega'$ and $\omega''$ relative to the parameter $\lambda$, in the de Rham cohomology of $E_\lambda$ (over $\mathbb C$, say). This followed from a direct calculation of $\omega$, $\omega'$ and $\omega''$, combined with a calculation of $d g$ where $g = x^{1/2} (x-1)^{1/2} (x-\lambda)^{-3/2}$. From this it follows that the Picard-Fuchs equation is the differential equation satisfied by the periods of $\omega$ against a class in the integral homology of $E_\lambda$, viewed as a function of $\lambda$.

The reason why the differential equation satisfied by the periods of $E_\lambda$ is precisely the same equation that is satisfied by its Hasse invariant is quite beautiful, and arises from the circumstance that $$ y^p \omega = y^{p-1} dx = (a_0(\lambda) + a_1(\lambda) x + \cdots + a_{p-1}(\lambda) x^{p-1} + \cdots ) dx,$$ where $a_{p-1} = A_p$ by definition. The fact that $\partial/\partial_\lambda y^p = 0$ in characteristic $p$, and that the coefficient of $x^{p-1} dx $ in any exact differential is equal to $0$, thus implies that the coefficient $a_{p-1}(\lambda)$ satisfies the same Picard-Fuchs equation as $\omega$.

The fact that $A_p(\lambda)$ has no multiple roots now follows easily from the fact that $A_p$ satisfies a linear ODE of order $2$, since differentiating the equation successively relative to $\lambda$ gives by induction, for all $n\ge 2$, a non-trivial linear relation between the derivative $A_p^{(n)}(\lambda_0)$ and all the lower order derivatives, provided $\lambda_0(\lambda_0-1)\ne 0$. It was then checked by an easy direct calculation that $A_p(0)\ne 0$ and $A_p(1) \ne 0$, concluding our proof of the fact that $A_p(\lambda)$ has no multiple roots.

In conclusion, the roots of $A_p(\lambda)$ are all distinct, hence there are $m= (p-1)/2$ distinct supersingular $\lambda$-invariants over $\overline{\mathbb F_p}$.

In this concluding lecture, we discussed the derivatives of modular forms, and the various ways in which the fact that $f'(z)$ fails to be modular of weight $k+2$ when $f$ is of weight $k$ can be remedied by modifying $f'(z)$ by a simple factor. The first approach involves the function $y$, which is non-holomorphic but has a simple transformation property under fractional linear transformations, and leads to the Shimura Maass derivative operator on so-called ``nearly holomorphic modular forms". The second, which involves the weight two Eisenstein series $E_2$, which is holomorphic but not quite modular of weight two, leads to the derivative operators of Ramanujan. Serre observed that, because $E_2$ is a genuine modular form mod $p$, this leads to the conclusion that the derivative operator $ 2\pi i d/dz$, which is given on $q$ expansions as $q d/dq$, does preserve the space of modular forms mod $p$, and even of $p$-adic modular forms. The lecture concluded with a brief discussion of the question of