John Jones - MAT 598 Homework

1. Determine the number of points above $i$. (Hint, the answer to this and the next part may have cases depending on the congruence class of $p$ modulo something.)
2. Determine the number of points above $\rho$.
3. For $3\le p \le 19$, determine the genus of $X_0(p)$.

If you want to use gp, you can use the file modforms.gp. Just read it in, and you can use the following functions. Note, if you reset the series precision with default(seriesprecision, 500), then functions which generate $E_4$, $E_6$, and $\widetilde{\Delta}$ will do so in the new precsion.

You would have to produce relevant bases yourself, but the functions e4(), e6(), and delt() should give you the raw materials you need. For example, for cusp forms in weight 24, you could use basis=[delt()*delt(), delt()*e4()^3] and for all forms in weight 24, you could use basis=[delt()*delt(), delt()*e4()^3, e4()^6].

The basic functions are domat(n,k,basis) where you give the values of $n$ and $k$ for $T_n(k)$, and then you supply the basis analogous to the ones above, and similarly dopoly(n,k,basis) to get the characteristic polynomial of the Hecke operator.

1. For weight 40 cusp forms (so $S_{40}$), compute matricies for $T_{40}(j)$, where $1\le j \le 10$, and $j=15$. Verify that $T_k(r)T_k(s)=T_k(rs)$ for \((r,s)=(2,3), (2,5), (3,5)\).
2. Verify that $T_{40}(2)$ and $T_{40}(3)$ define the same field.
3. Verify that $T_{40}(2)$ has all real eigenvalues.
4. Verify the three recursions for $T_{40}(p^n)$ involving the matricies already computed.
5. Extend your basis to switch to $M_{40}$, and compute $T_k(2)$, and factor its characteristic polynomial. Verify that you get the same cubic factor as before and a linear factor. Verify that the root of the linear part is what we predicted for eigenvalues for Eisenstein series.

1. Prove that if $N>0$, the reduction map $SL_2(\Z)\to SL_2(\Z/N\Z)$ is surjective.
2. Prove that in this map, $\Gamma_0(N)$ surjects onto the subgroup of $SL_2(\Z/N\Z)$ of matricies of the form $\begin{pmatrix} a & b \\ 0 & d \end{pmatrix}$.

1. Pick a basis for $S_{24}$ where the $q$-expansions of basis elements are rational. Give the first few terms of these $q$-expansions (and explain where you got them).
2. Compute the matrix for $T_{24}(2)$ on this basis.
3. Find the characteristic polynomial, $f(x)$, for this matrix.
4. Determine the number of real roots of $f(x)$, and whether or not the polynomial is irreducible over $\Q$.

You are encouraged to use a computer to help with this computation, preferably gp, but it is your choice. Parts of your solution will probably be printouts from the computation session.

Here are some suggestions on carrying out the computation. The theorem which allowed us to compute the dimension of $S_{24}$ over $\C$ did so by producing a basis. It turns out that you can scale this basis to get forms with integral $q$-expansions. (See below for more on basis picking.)

You can compute the first few coefficients of $T_{24}(2) f$ by the explicit formula for the action of Hecke operators in terms of $q$-expansions. If working by hand, you should think about how many coefficients you really need. If you are writing a program, note that the formula given for $n>1$ works for all $n\ge 0$.

You will need to take these $q$-expansions and express them as linear combinations of your basis elements. Your basis is probably already in a good form for doing this, but you can make it trivial by modifying the basis $\{f_1, f_2\}$ so that $c_i(f_j) = \delta_{i,j}$ for $1\leq i,j\leq 2$.

Some useful gp functions: eta(q) to help get the expansion for $\widetilde{\Delta}$, sum(...) to do summations, polcoeff(f1, 5, q) to get the coefficient of $q^5$ in the expansion f1, gcd(a,b), divisors(n), charpoly for a characteristic polynomial, polsturm to find the number of roots for a polynomial. Also, gp will do matrix computations if needed. Finally, there are lots of commands for determining if a quadratic polynomial is irreducible (polisirreducible, factor, or compute its poldisc and see if it is a square).

1. Suppose $\sum_{j=1}^n f_j =0$ where each $f_j$ is a modular form of weight $j$. Prove that $f_j=0$ for all $j$. (Hint: use $\tau\mapsto -1/\tau$).
2. Prove $G_4(\tau)$ and $G_6(\tau)$ are algebraically independent. (Hint: divide away common factors of $G_4$ and then evaluate at $\rho$.)

1. Show directly that $G_6(i)=0$
2. Similarly, show $G_4(\rho)=0$.
3. Prove $j(i)=1728$.
4. Prove $j(\rho)=0$.

1. Prove that $\mathcal{R}$ is a subring of $\C$ which contains $\Z$.
2. Prove that if $\alpha\in\mathcal{R}$, then $\alpha$ satisfies a monic quadratic equation over $\Z$ (deduce $\alpha\cdot 1=a + b\tau$ and $\alpha\cdot \tau = c + d\tau$ and eliminate $\tau$).
3. Prove that if $\alpha\in\mathcal{R}-\Z$, then $\Q(\alpha)=\Q(\tau)$.
4. Prove that if $\mathcal{R}\neq \Z$, then $\Q(\tau)$ is a quadratic imaginary field, and $\mathcal{R}$ is a subring of finite index in the ring of integers of $\Q(\tau)$.