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189-596A: Modular forms


Lecture 1 Friday, September 2. The first lecture week was devoted to discussing some basic concepts of algebraic geometry, such as the notion of a representable functor and of a moduli problem. We tried to assume a minimal background in algebraic geometry, hoping instead that seeing these general concepts in the very concrete setting of elliptic curves and their associated moduli problems is a good way to encounter them for the first time.
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.

Lecture 2 Wednesday, September 7. We then set about to remedy this problem by considering instead the problem of classifying pairs $(E,\omega)$ where $\omega$ is a regular differential on $E$ over $S$. We derived a canonical equation for a pair $(E,\omega)$ over the ring $Z = {\mathbb Z}[1/6]$ and used this to show that the functor classifying such pairs is representable by the ring $$\widetilde M = Z[a_4, a_6, \Delta^{-1}],$$ where $\Delta= -16(a_4^3 + 27 a_6^2)$ is the so -called discriminant function. An element of the ring $\widetilde M$ is called a weakly holomorphic modular form over $Z$. (A terminology that will seem a bit strange to the uninitiated but will be explained later.)

Lecture 3 Friday, September 9. We explained how a weakly holomorphic modular form over $Z$ can be recast as a rule which to every framed elliptic curve $(E,\omega)_{/R}$ over a $Z$-algebra $R$ associates an element $f((E,\omega)_{/R})\in R$, satisfying the base change property $$ f( (E,\omega)_R\otimes_{\varphi} S) = \varphi(f((E,\omega)_R)),$$ for all $Z$-algebra homomorphisms $\varphi:R\longrightarrow S$. We then defined the weight of a modular form and showed that $a_4$ and $a_6$ are modular forms of weight $4$ and $6$ respectively, thereby justifying the choices of subscripts used to label these coefficients.
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 $$ f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^k f(\tau), \quad \mbox{ for all } \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\in {\bf SL}_2({\mathbb Z}).$$

Lecture 4 Monday, September 12. This lecture was devoted to a calculation of the fourier expansion of $G_k(\tau)$: $$ G_k(\tau) = 2\zeta(k) + 2 \frac{(2\pi i)^k}{(k-1)!} \sum_{n=1}^\infty \sigma_{k-1}(n) q^n,$$ where $$ \zeta(k) = \sum_{n=1}^\infty \frac{1}{n^k}$$ is the value of the Riemann zeta-function and $$\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$$ is the divisor function. Using the known formulas for the values of $\zeta(k)$, we concluded for instance that \begin{eqnarray*} G_4(\tau) &=& \frac{2\pi^4}{90} E_4(q), \qquad \qquad E_4(q) = 1+ 240 \sum_{n=1}^\infty \sigma_3(n) q^n, \\ G_6(\tau) &=& \frac{2\pi^6}{945} E_6(q), \qquad \qquad E_6(q) = 1- 504 \sum_{n=1}^\infty \sigma_3(n) q^n. \end{eqnarray*} These explicit formulae led to the conclusion that the framed elliptic curve $\left({\mathbb C}^\times/q^{\mathbb Z}, 2 \frac{dt}{t}\right)$ (depending on the complex parameter $q$) is isomorphic to $$ \left({\mathbb C}^\times/q^{\mathbb Z}, 2\frac{dt}{t}\right) = \left(y^2 = x^3 - \frac{1}{3} E_4(q) + \frac{2}{27} E_6(q), 2 \frac{dx}{y} \right).$$ The equation on the right has coefficients in the ring $Z[[q]]$ of power series with coefficients in $Z={\mathbb Z}[1/6]$, and even in the subring $Z[[q]]_{\le 1}$ of power series that converge on the open unit disc. Furthermore, we even showed that the discriminant of the above equation belongs to ${\mathbb Z}((q))^\times$. This shows that the above curve can be viewed as a framed elliptic curve defined over the ring $Z((q))$, and maybe even (with just a little bit more work) over ${\mathbb Z}((q))$. This framed elliptic curve is called the Tate curve and it plays an important role in the study of modular forms. For instance, we were able to give a purely algebraic definition of the fourier expansion of a modular form (over an arbitrary ring $Z$ in which $6$ is invertible) by the rule $$ f(q) := f\left( \left(E_{q}, \omega_{q}\right)_{/Z((q))}\right),$$ where $$ \left(E_{q},\omega_{q}\right)_{/Z((q))} := \left(y^2 = x^3 - \frac{1}{3} E_4(q) + \frac{2}{27} E_6(q), 2 \frac{dx}{y} \right),$$ viewed as an elliptic curve over $Z((q))$.
We were then able to define a modular form of weight k (over an arbitrary ring $Z$ in which $6$ is invertible) as being a weakly holomorphic modular form $f$ over $Z$ for which $f(q)$ belongs to the power series ring $Z[[q]]$. Such a modular form is said to be a cusp form if $f(q)$ belongs to $q Z[[q]]$.

Lecture 5 Wednesday, September 14. In the last lecture, we showed by a fairly straighforward argument that $\Delta(E_q, 2\frac{dt}{t})$ belongs to ${\mathbb Z}((q))^\times$. More precisely, we showed that it belongs to $q {\mathbb Z}[[q]]^\times$. But we did not obtain much information about what the fourier expansion of $\Delta$ looks like. Our motivating question for the next few lectures is to prove the well known product identity $$ \Delta\left({\mathbb C}^\times/q^{\mathbb Z}, 2\frac{dt}{t}\right) = q \prod_{n=1}^\infty (1-q^n)^{24},$$ which is one of the ``crown jewels" in the subject.
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 $$ d\log \sigma(z) =: \frac{\sigma'(z)}{\sigma(z)} = \xi(z).$$

Lecture 6 Friday, September 16. Today we continued our discussion of the Weierstrass $\xi$ and $\sigma$ functions. These functions have the following properties: In large part, the importance of the $\sigma$ function stems from the fact that it can be used to recover a rational function $f$ on an elliptic curve from its divisor $\sum_{i=1}^r n_i (a_i)$, where we assume that $\sum_{i=1}^r n_i a_i = 0$, via the rule $$ f(z) \sim \prod_{i=1}^r \sigma(z-a_i)^{n_i},$$ where again $\sim$ denotes equality up to a non-zero multiplicative constant in ${\mathbb C}^\times$. We used this to show that $\wp(z)-\wp(a)$, whose divisor is $(a)+(-a)-2(0)$, can be written as $$ \wp(z) - \wp(a) \sim \frac{\sigma(z+a) \sigma(z-a)}{\sigma(z)^2 \sigma(a)^2},$$ leading to a formula for the discriminant in terms of the $\sigma$-function: $$ \Delta(\tau) \sim \frac{1}{\sigma(1/2)^2 \sigma(\tau/2)^2 \sigma((1+\tau)/2)^2}.$$

Lecture 7 Monday, September 19. Today we finished the proof of Jacobi's product formula $$ \Delta(q) = q \prod_1^\infty (1-q^n)^{24}.$$ We did this by writing down an expansion of the $\sigma$-function in terms of $q$ and invoking the expression for $\Delta$ in terms of $\sigma$ obtained on Friday. (Our expansion also included certain factors that depended on $\tau = \log(q)/2\pi i$, but which cancelled out when evaluating the ratio describing $\Delta$.) The evaluation of these expansions is by a calculation which is slightly tedious but fairly "mechanical". The details are to be found in Chapter 1 of Silverman's "Advanced topics" book.

Lecture 8 Wednesday, September 21. In today's lecture we defined Hecke operators on modular forms, via the rule $$ (T_n f) (E,\omega) = \frac{1}{n} f(E',\omega'),$$ where the sum runs over all framed elliptic curves $(E',\omega')$ which are related to $E$ by an isogeny of degree $n$. We showed that the algebra of Hecke operators is generated by the operators $T_p$ for $p$ a prime, and that they are multiplicative on relatively prime integers $(m,n)$, i.e., $T_{nm} = T_n T_m$. It follows from this that the algebra of Hecke operators is a commutative subring of the endomorphism ring of the space of weakly holomorphic modular forms of weight $k$. We then obtained an explicit formula for $T_p$ on both $\tau$ and $q$-expansions: $$ (T_p f)(\tau) = \frac{1}{p} \sum_{j=0}^{p-1} f\left(\frac{\tau+j}{p}\right) + p ^{k-1} f(p\tau),$$ $$ (T_p f)(q) = \sum_{n} a_{np} q^n + p^{k-1} \sum_n a_n q^{np}.$$ It follows from the latter formula that the Hecke operators preserve the finite dimensional subspaces of modular forms and cusp forms. The subring ${\bf T}_k$ of the endomorphism ring of the space $M_k$ of weight $k$ modular forms is thus a commutative subring of this endomorphism algebra. As we shall see in later lectures, the {\em eigenvectors} for this collection of commuting operators have many appealing arithmetic properties.

Lecture 9 Monday, October 3. Today we sketched the proof that the space $S_k({\mathbb C})$ of weight $k$ cusp forms has a basis consisting of simulatenous eigenvectors for the Hecke operators. We showed that the first fourier coefficient (i.e., the coefficient of $q$) of such an eigenvector is always non-vanishing, and defined a normalised eigenform to be an eigenform for which $a_1=1$. We saw that the fourier coefficients of a normalised eigenform satisfy the same multiplicativity properties as the corresponding Hecke operators, namely, $$ a_{nm} = a_n a_m, \qquad \mbox{ for all } m,n \mbox{ with } \gcd(m,n)=1,$$ and, for all primes $p$, $$ a_{p^{r+1}} = a_p a_{p_r} - p^{k-1} a_{p^{r-1}}.$$
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 $Re(s) \gt 1+k/2$.

Lecture 10 Wednesday, October 5. Today was devoted to recalling the main highlights arising from Hecke's theory giving the analytic continuation and functional equation for the $L$-series $L(f,s)$ attached to a cusp form. The key is to remark that $L(f,s)$ is essentially the Mellin transform of $f$, the latter being defined by $$ M(f,s) = \int_{0}^\infty f(it) t^s \frac{dt}{t}.$$ (It is easy to see that this integral converges when $Re(s) \gt 1+k/2$, using the exponential decay of $f(it)$ as $t\rightarrow \infty$ and the bound $|f(it)| \lt\!\!\lt t^{-k/2}$ as $t\rightarrow 0$.) A direct computation revealed that $$ M(f,s) = (2\pi)^{-s} \Gamma(s) L(f,s), \qquad \mbox{ where } \Gamma(s) = \int_0^{\infty} e^{-t} t^s \frac{dt}{t}.$$ The main step in Hecke's study is to exploit the modularity of $f$ (particularly, its invariance property under the Mobius transformation $z\mapsto -1/z$) to obtain the integral representation for $M(f,s)$ which converges absolutely {\em for all} $s\in {\mathbb C}$: $$ M(f,s) = \int_1^\infty f(it) (t^s + (-1)^{k/2} t^{k-s}) \frac{dt}{t}.$$ It follows that $L(f,s)$ extends to an entire function of $s$, having ``trivial" zeroes at the non-positive integers (arising from the poles of the $\Gamma$-function). An extension of the Riemann hypothesis asserts that the other zeroes of $L(f,s)$ (or, equivlaently, of $M(f,s)$) are concentrated on the axis of symmetry for the functional equation $$ M(f,s) = (-1)^{k/2} M(f, k-s).$$
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.

Lecture 11 Wednesday, October 12. In this lecture we moved on to the topic of (weakly holomorphic) modular forms (of some weight $k$) with level $N$ structure, which we defined algebraically, in the same spirit as in our definition of modular forms on ${\bf SL}_2({\mathbb Z})$, as rules which associate to a triple $(E,\xi,\omega)_{/R}$, where $E$ is an elliptic curve over $E$, $\xi$ is a level $N$ structure on $E$, and $\omega$ a regular differential on $E/R$, an element of $R$ (subject to the usual base change axioms and weight $k$ homogeneity property). Here $R$ ranges over all $Z$ algebras, where the base ring $Z$ is chosen (to avoid technical subtleties) to be one in which $6N$ is invertible.

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 cusp on the modular curve whose complex points are identified with $\Gamma \backslash {\mathcal H}^*$, where ${\mathcal H}^* := {\mathcal H} \cup {\mathbb P}_1({\mathbb Q})$ is the so-called extended Poincaré upper half plane.

Lecture 12 Monday, October 17. Today was devoted to a further discussion of modular forms with level $N$ structure, with a quick description of how many of the basic notions (Hecke operators, ...) extend from the case of ${\bf SL}_2({\mathbb Z})$ to more general level structures, and a statement (without proofs) of the main results of Atkin-Lehner theory. Then the modular curve $X_\Gamma$ associated to a level $N$ structure, with a special emphasis on the cases where $\Gamma= \Gamma_0(N)$, $\Gamma_1(N)$, and $\Gamma(N)$, were introduced. We concluded with a complete proof that the set $\Gamma \backslash {\mathbb P}_1({\mathbb Q})$ of cusps on $X_\Gamma$ are in natural bijection with suitable equivalence classes of $\Gamma$-level structures on the Tate curve $E_q \simeq {\mathbb C}((q)))^\times/q^{\mathbb Z}$ over ${\mathbb C}$.

Lecture 13 Wednesday, October 19. This lecture started with the statement that the dimension of the space $S_2(\Gamma)$ of cusp forms of weight $2$ and $\Gamma$-level structure (over ${\mathbb C}$, say) is equal to the dimension of the space of regular differentials on $X_\Gamma$, hence, is equal to the genus of $X_\Gamma$. We then explained a general strategy for computing the genus of a branched covering of ${\mathbb P}_1$ (or of a more general curve) using the Riemann Hurwitz formula. We explained how all the quantities in the Riemann-Hurwitz formula can be given a purely group-theoretic expression, in the case of the natural projection $X_\Gamma({\mathbb C}) \rightarrow {\bf SL}_2({\mathbb Z})$, leading to a general strategy for computing the genus of $X_\Gamma$, and hence, the dimension of $X_{\Gamma}$. As an example, we computed the genus of $X_0(p)$ when $p$ is a prime congruent to $1$ or $-1$ modulo $12$.

Lecture 14 Monday, October 24. This lecture started with the goal of computing a basis for the space $S_k(\Gamma)$ of modular forms of weight $k$ relative to a congruence subgroup $\Gamma\subset {\bf SL}_2({\mathbb Z})$, it terms of their $q$ expansions. We started by noting that, at least when $\Gamma = \Gamma_0(N)$, the space $S_k(\Gamma)$ has a basis consisting of modular forms whose $q$ expansions belong to ${\mathbb Q}[[q]]$. (The explanation was a bit terse, and led to a request for further clarifications, which were postponed to lecture 15).

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 $$ \varphi: M_2(\Gamma) \rightarrow H^1(\Gamma,\mathbb C).$$

Lecture 15 Wednesday, October 26. This lecture started with an intuitive explanation of why the space $S_k(\Gamma_1(N))$ has a basis of modular forms with rational fourier expansions, based on the representability of the functor classifying triples $(E,\xi,\omega)$ where $\xi$ is a point of order $N$ on the elliptic curve $E$ and $\omega$ is a regular differential on $E$.
We then pursued our discussion of the map $\varphi$ described in the previous lecture, showing that it is injective. We saw that it is not surjective, since the dimensions of the domain and target are $g+s-1$ and $2g+s-1$ respectively, where $g$ is the genus of $X_\Gamma$ and $s$ is the number of cusps on $X_\Gamma$, i.e, the cardinality of $X_\Gamma - Y_\Gamma$.
We explained that $\varphi$ induces a map from $S_2(\Gamma)$ to the parabolic cohomology $H^1_{\rm par}(\Gamma,\mathbb C)$ consisting of homomorphisms that vanish at all the parabolic elements (stabilisers of cusps). These spaces are of dimension $g$ and $2g$ respectively, so $\varphi$ does not induce an isomorphism between these spaces. However we asserted that $\varphi$ does induce an isomorphism $$ \varphi: S_2(\Gamma) \oplus \overline{S_2(\Gamma)} \longrightarrow H^1_{\rm par}(\Gamma,\mathbb C),$$ by invoking (without proof) the main result of Hodge theory.

Lecture 16 Monday, October 31. The conclusion of last week's lecture was that to compute $S_2(\Gamma)$ it is enough to understand the structure of ${\mathbb T}_2(\Gamma)$ as a module over itself, and that $\mathbb T_2(\Gamma)$ is isomorphic to the ring generated by the Hecke operators acting on $H^1_{\rm par}(\Gamma,\mathbb C)$. An efficient way of getting a computational handle on the latter space is via the notion of modular symbols introduced by Birch and Manin. We defined the spaces $\mathcal F(R)$ of $R$-valued functions on $\mathbb P_1(\mathbb Q)$, and $\mathcal M(R)$ of $R$ valued modular symbols, defined as the group of functions $$ m: \mathbb P_1(\mathbb Q) \times \mathbb P_1(\mathbb Q) \longrightarrow R$$ satisfying $m\{r,s\} + m\{s,t\} = m\{r,t\}$ and $m\{r,s\} = - m\{s,r\}$. We explained the exact sequence $$ 0 \rightarrow R \rightarrow \mathcal F(R)^\Gamma \rightarrow \mathcal M(R)^\Gamma \rightarrow H^1_{\rm par}(\Gamma,R)\rightarrow 0$$ relating the group of $\Gamma$-invariant modular symbols to $H^1_{\rm par}(\Gamma,R)$ and described the action of the Hecke algebra on $\mathcal M(R)$ induced by the various identifications and inclusions. We then described a strategy for describing the space of $\Gamma$-invariant $\mathbb Z$-valued modular symbols -- a finite rank $\mathbb Z$-module -- efficiently, and doing calculations with them. We concluded by working out the example of modular symbols for $\Gamma_0(11)$.

Lecture 17 Friday, November 4. We completed our discussion of the modular symbol algorithm for computing spaces of cusp forms of weight two, with special emphasis on computing $S_2(\Gamma_0(N))$. This led us naturall to our next topic, a discussion of the Manin-Drinfeld theorem which asserts the finiteness of the cuspidal divisor class group on a modular curve $X_\Gamma$ for any congruence group $\Gamma$. We embarked on this topic by briefly discussing/recalling without proof the statement of Abel's theorem giving an isomorphism (known as the Abel-Jacobi map) $$ AJ: Div^0(X)/P(X) \rightarrow \Omega^1(X)^\vee / p(H_1(X,\mathbb Z))$$ between the group of divisors modulo principal divisors on a compact Riemann surface $X$ (or on an algebraic curve over ${\mathbb C}$) and the quotient $\Omega^1(X)^\vee/p(H_1(X,{\mathbb Z}))$, where $\Omega_1(X)$ is the complex vector space of regular differentials on $X$, the superscript $\vee$ denotes the complex linear dual, and $p:H_1(X,\mathbb Z) \rightarrow \Omega^1(X)^\vee$ is the period map described by integration, $$p(\gamma)(\omega) := \int_\gamma \omega.$$ The Abel-Jacobi map is described by a similar analytic formula, $$ AJ(D)(\omega) = \int_{\partial^{-1}(D)} \omega,$$ where $\partial^{-1}(D)$ denotes any smooth one-chain on $X$ having $D$ as its boundary.

Lecture 18 Monday, November 7. Today we continued our discussion of the Manin-Drinfeld theorem, and described Manin's elegant proof based on the action of Hecke operators on the cusps of the modular curve. We observed that the Manin-Drinfeld theorem asserts that the group of so-called modular units---units in the function field of the open modular curve $Y_\Gamma$ --- have essentially maximal rank, equal to the $\#C_\Gamma-1$, where $C_\Gamma\subset X_\Gamma$ is the set of cusps.

Lecture 19 Wednesday, November 9. Following on the existence proof of modular units arising from the Manin-Drinfeld theorem, we gave an algebraic description of a natural collection of modular units on the modular curves $X_1(N)$ and $X(N)$, the so-called Siegel units, from the point of view of the moduli of elliptic curves. An indication of how to calculate the $q$-expansions of these Siegel units was given towards the end of the lecture, using the theory of elliptic functions.

NOTE: There will be no lectures on Monday November 14 and Wednesday November 16, because I will be out of town. But Friday November 18 will be a regular lecture rather than a problem session. We resume with the usual schedule the week after.

Lecture 20 Friday, November 18. This lecture was devoted to a discussion of the connection between the modular symbols introduced in previous weeks, and the special values of the $L$-function $L(f,s)$ and of the twisted variants $L(f,\chi,s)$ at $s=1$ (for $f$ a modular form of weight two). The main result that was proved was the Birch-Manin formula, which asserts that, for any primitive Dirichlet character $\chi$ of conductor $m$, $$ L(f,\chi,1) = - g(\chi) \sum_{a=1}^m \bar\chi(a) m_f\left\{\frac{a}{M}, \infty\right\},$$ where $g(\chi) = \sum_{a=1}^m \chi(a) e^{2\pi i a/M}$ is the Gauss sum associated to $\chi$. The formalism of modular symbols allowed us to conclude from this formula that, for any given modular form $f$ of weight two, the vector space over the maximal cyclotomic extension $\mathbb Q^{\rm ab}$ of $\mathbb Q$ generated by all the central critical values $L(f,\chi,1)$ is finite-dimensional, an interesting and non-trivial statement about the algebraicity properties of the Hecke $L$-functions attached to modular forms and their central critical values.

Lecture 21 Monday, November 21. We continued our discussion of the Birch-Manin formula, focussing now on the special case where $f$ is an eigenform of weight two with rational (hence, integral) fourier coefficients. Such a modular form correspondsd to an elliptic curve over $\mathbb Q$ by the Eichler-Shimura construction, which we described (in a somewhat impressionistic way) in the class. We explained that the modular symbol $m_f$ attached to $f$ can be decomposed into two contributions, the so-called plus and minus modular symbols, which can then be rescaled by ostensibly transcendental periods $\Omega_f^\pm$ to yield $\mathbb Z$-valued modular symbols $m_f^\pm$. These $\mathbb Z$-values modular symbols were used to construct Mazur-Tate theta-elements $\theta_M\in \mathbb Z[G_M]$, where $G_M = (\mathbb Z/M\mathbb Z)^\times = Gal(\mathbb Q(\mu_M)/\mathbb Q)$ which package together the special values $L(f,\chi,1)$ as $\chi$ ranges over the Dirichlet characters of conductor dividing $M$. The Mazur-Tate conjecture, a ``finite level" analogue of the $p$-adic Birch and Swinnerton-Dyer conjecture, was formulated, as the assertion that $\theta_M$ belongs to $I^r$ where $I$ is the augmentation ideal of the integral group ring $\mathbb Z[G_M]$ and $r$ is the Mordell-Weil rank of the elliptic curve attached to $f$. This led us naturally to the definition of the $p$-adic $L$-function of Mazur and Swinnerton Dyer, which is obtained roughly speaking by modifying the elements $\theta_{p^n}$ as $n$ varies in a simple way, so that they become compatible under the natural projections $\mathbb Z_p[G_{p^{n+1}}] \rightarrow \mathbb Z_p[G_{p^n}]$. The resulting element in the Iwasawa algebra $\mathbb Z_p[[G_\infty]] := \lim_\leftarrow \mathbb Z_p[G_{p^n}]$ is the $p$-adic $L$-function of Mazur and Swinnerton-Dyer.

Lecture 22 Wednesday, November 23. This lecture gave more details on the Mazur-Swinnerton Dyer $p$-adic $L$-function and its construction as a $p$-adic Mellin transform (sometimes called the Mazur-Mellin transform) of the measure on $G_\infty$ associated to the element in the Iwasawa algebra constructed in the previous class. For many of us, a lot of this material was a review of what was covered in the seminar on Iwasawa theory, and the goal was to explain how the ideas used to construct $p$-adic analogues of Dirichlet $L$-functions (the Kubota-Leopolodt $p$-adic $L$-functions) also extend to the setting of Hecke $L$-functions attached to normalised newforms.

Lecture 23 Monday, November 28. Today we embarked on a new topic: the theory of modular forms mod $p$. We first introduced the algebra of modular forms over the field $\mathbb F_p$ with $p$ elements, which is isomorphic, when $p\ge 5$, to the graded ring $\mathcal M(\mathbb F_p) := \mathbb F_p[E_4,E_6]$, where $E_4$ and $E_6$ are the normalised Eisenstein series given by the formulae $$ E_4(q) = 1 + 240 \sum_{n=1}^\infty \sigma_3(n) q^n, \qquad E_6(q) = 1 -504 \sum_{n=1}^\infty \sigma_5(n) q^n. $$ We considered the $q$-expansion morphism $$ q-exp: \mathcal M(\mathbb F_p) \rightarrow \mathbb F_p[[q]],$$ which we observed to be injective on each of the graded pieces of $\mathcal M(\mathbb F_p)$. The most interesting feature of the theory is that this $q$-expansion map fails to be injective on $\mathcal M(\mathbb F_p)$. The image of the $q$ expansion map is called the algebra of modular forms mod $p$, and denoted $\bar{\mathcal M}$. Our goal was to determine the structure of $\bar{\mathcal M}$, which is a non-trivial quotient of $\mathcal M(\mathbb F_p)$. The main theorem in the theory of $\bar {\mathcal M}$ is that the kernel $I$ of the $q$-expansion map on $\mathcal M(\mathbb F_p)$ is equal to the ideal generated by $A(E_4,E_6)-1$, where $A(X,Y)$ is a homogenous polynomial of degree $p-1$ (in which $X$ and $Y$ are assigned the degrees $4$ and $6$, as usual) satisfying $A(E_4,E_6) = E_{p-1}$. The polynomial $A\in \mathbb F_p[X,Y]$ is called the Hasse invariant in characteristic $p$. By arguments of pure commutative algebra we showed that

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.

Lecture 24 Wednesday, November 30. Today we gave a modular description of the Hasse invariant as a modular form over $\mathbb F_p$, defining it as the rule which to a pair $(E,\omega)$ defined over a field $k$ (or even a ring) of characteristic $p$, associates the quantity $A(E,\omega)$ defined by the rule $$ (\varphi_p^\vee)^*(\omega) = A(E,\omega) \omega^{(p)},$$ where $\varphi_p: E\rightarrow E^{(p)}$ is the Frobenius morphism, an inseperable isogeny of degree $p$, and $\varphi_p^\vee: E^{(p)} \rightarrow E$ is its dual isogeny. We showed that $A =E_{p-1}$ as a modular form over $\mathbb F_p$ by observing that $A$ is also of weight $p-1$, and that it has the same $q$-expansion as $E_{p-1}$, namely, $A(q) = 1$ (a direct calculation on the Tate curve over $\mathbb F_p((q))$).

It follows from this more conceptual description of $A$ that the zeroes of $A(E,\omega)$ correspond precisely to the supersingular elliptic curves in characteristic $p$, i.e., those for which $\varphi_p^\vee$ (as well as the multiplication-by-$p$ map) are purely inseperable.

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 $p-1$.

Lecture 25 Monday, December 5.
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 $\overline{\mathbb F_p}$.

Lecture 26 Wednesday, December 7.
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 integrating modular forms of weight two, using this as a motivating question for the introduction of weak harmonic Maass forms and mock modular forms of weight zero. This was meant as a preparation for the Montreal-Toronto conference that was starting the next day.