We will begin by a 30 minute presentation by

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

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

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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.)

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.

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 $t$).