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189-596A: Modular forms

Suggested exercises

Exercise session 0. Friday, September 23. In this session Leonardo Colo and Isabella Negrini covered some bacground from sections 1.1. and 1.2 of Silverman's book, concerning such topics as the action of ${\bf SL}_2({\mathbb Z})$ on the upper half plane, the fundamental domain, and the Riemann surface structure of the associated quotient.

Exercise session 1. Friday, October 7. After Isabella concluded her presentation from the previous week, we covered the following exercise from Silverman's "Advanced topics" book, in pages 84-89.

David Lilienfeldt: Exercise 1.8. about divisors on modular curves and the use of Riemann-Roch in obtaining the dimension formula for the space of cusp forms of a given weight.

Exercise session 2. Friday, October 14. The following exercise from Silverman's ``Advanced topics" book will be covered:

James Rickards: Exercise 1.5. about quadratic forms and CM points, which will also be a useful prelude to the sequel to this course, on complex multiplication.

Sami Douba Exercise 1.11 fills one of the few gaps in our coverage so far of the theory of elliptic curves and modular forms, the fact that the set of periods of an elliptic curve is really a lattice, i.e., that given $j\in {\mathbb C}$, there is a $\tau$ in the upper half plane satisfying $j(\tau) = j$. This is known as the uniformisation theorem and is of course at the heart of the complex theory of modular forms.

Exercise session 3. Friday, October 21. The following exercise from Silverman's ``Advanced topics" book will be covered:

Ervin Thiagalingam and Billy Lee: Exercise 1.22. describing the Petersson scalar product on cusp forms of weight $k$, leading to the crucial proof that the Hecke operators are self-adjoint relative to this pairing, and therefore that the Hecke operators are simultaneously diagonalisable. I propose that two volunteers divide the task of presenting this exercise, which is in 6 parts.

Exercise session 4. Friday, October 28.

We will begin by a 30 minute presentation by Farzad Aryan, who has very kindly volunteered to present Riemann's proof of the analytic continuation and functional equation for the zeta function using the theta function and the Poisson summation formula, which I mumbled vaguely about in the lecture of October 5.

Antoine Comeau will be running the following exercise session.

This exercise is meant to fill in the details for the analytic contunation and functional equation of $L$-series of modular forms of general $\Gamma_1(N)$-level.

Let $w_N$ be the Fricke involution that was introduced in class, acting on modular forms of weight $k$.

a) Show that, if $f$ belongs to $S_k(\Gamma_0(N),\chi)$, then $w_Nf$ belongs to $S_k(\Gamma_0(N),\bar\chi)$.

b) Conclude that, if $f$ is a newform, then there is a complex scalar $w_f$ of norm one such that $$ w_N f = w_f \cdot \bar f,$$ where $\bar f$ denotes the modular form obtained from $f$ by conjugating its fourier expansion.

c) Use the result of part b) to write down a functional equation relating $L(f,s)$ to $L(\bar f,s)$.

(Salik Bahar) The following exercise is meant to gain familiarity with basic manipulations of modular forms and Hecke operators.

Let ${\bf T}$ be the $\mathbb Q$-algebra of Hecke operators $T_n$ acting on $S_{24}(\mathbf{SL}_2(\mathbb Z))$. Express ${\bf T}$ as a product of specific number fields, and write down a basis of eigenforms for $S_{24}(\mathbf{SL}_2(\mathbb Z))$. (This fun calculation was done by Hecke.)

Exercise session 5. Friday, November 11. (Leonardo Colo).

The following exercise is meant as a review of the techniques we described in class to calculate the genus of a modular curve.

Let $p$ be an odd prime and let $\lambda\in {\mathbb Z}/p{\mathbb Z}$ be a non-square. Let $\Gamma$ be the set of matrices in ${\bf SL}_2({\mathbb Z})$ defined by $$ \Gamma = \Bigg\{ \left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \mbox { with } a\equiv d, \ \ b \equiv \lambda c, \pmod{p}. \Bigg\}$$
a) Show that $\Gamma$ is a subgroup of ${\bf SL}_2({\mathbb Z})$.

b) Compute the genus of the curve $X_\Gamma$ over $\mathbb C$ whose complex points are identified with $\Gamma \backslash {\mathcal H}^*$.

c) Give a an algebraic description of the level $p$ structure that the curve $X_\Gamma$ classifies, and give a description of the set $X_\Gamma(K)$ of $K$-rational points, where $K$ is a field.

Exercise session 6. Friday, November 25.

1. The first exercise is meant to give you some more experience working with modular symbols. Recall that we already showed in class that the space $\mathcal M(\mathbb Q)^{\Gamma_0(11)}$ of $\Gamma_0(11)$-invariant modular symbols, or equivalently, the space of $M$-symbols for $\Gamma_0(11)$, is three dimensional over $\mathbb Q$.

a) Compute the action of the Hecke operator $T_2$ on this three dimensional space, and show that it breaks $\mathcal M(\mathbb Q)^{\Gamma_0(11)}$ into the direct sum of eigenspaces $V_1$ and $V_2$ of dimensions $1$ and $2$ respectively.

b) Let $T_{-1}$ be the involution on the set of functions on $\mathbb P_1(\mathbb Q)^2$ defined by $$(T_{-1}m) \{a,b\} = m\{ -a,-b\}.$$ Show that $T_{-1}$ descends to a well-defined operation on $\mathcal M(\mathbb Q)^{\Gamma_0(11)}$ and that it commutes with the Hecke operators.

c) Show that $T_{-1}$ decomposes $V_2$ into a direct sum of two one-dimensional eigenspaces.

d) If $p \ne 11$ is a prime and $\lambda_p$ denotes the eigenvalue of $T_p$ acting on $V_2$, show that $$ \lambda_p \equiv p+1 \pmod{5}.$$
e) Explain why questions (a)-(d) imply that there is a unique elliptic curve of conductor $11$, up to isogeny, and that any such curve has a subgroup of order $5$ which is defined over $\mathbb Q$. (This question uses ideas that are outside that scope of the course, but which will be familiar to some of you.)

2. The second exercise is meant to improve your familiarity with modular units. Let $N = p_1 \cdots p_t$ be a square-free integer.

a) Describe the cusps on the modular curve $X_0(N)$. How many are there, and what are their widths?

b) For each divisor $d$ of $N$, show that the function $u_d(\tau) := \Delta(d\tau)/\Delta(\tau)$ is $\Gamma_0(N)$-invariant, and is a modular unit. Compute its divisor.

c) Show that the units $u_d(\tau)$ generate a finite index subgroup of the group of all modular units for $\Gamma_0(N)$. (For this, you might find it helpful to proceed by induction on $t$).