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\begin{center}
{\Huge 189-726B: Modular Forms II} \\ \sk
{\Huge Assignment 6} \\ \sk
{\Large Due: Wednesday, February 20 }
\end{center}
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\sk
\noindent
1. Let $P=E_2 = 1-24 \sum_{n=1}^\infty \sigma_1(n) q^n$
be the weight two Eisenstein series.
Prove the claim made in class
that $12 \theta P - P^2$ belongs to the space
$M_4$.
\sk\sk\noindent
2. Let $\Delta = (Q^3-R^2)/ 1728$ be Ramanujan's $\Delta$-function.
Compute $\partial_{12}(\Delta)$ and conclude that $12 P $ is the logarithmic
derivative of $\Delta$. Deduce from this the infinite product
formula for $\Delta$.
\sk\sk\noindent
3. Show that
$$ \theta R = -\Delta \pmod{5}, \qquad
\theta Q = 2\Delta \pmod{7},$$
and conclude the well-known congruences for the Ramanujan $\tau$ function:
$$ \tau(n) \equiv n\sigma_5(n) \pmod{5}, \qquad
\tau(n) \equiv n\sigma_3(n) \pmod{7}.$$
\noindent
{\em At this point, let me correct a mistake I made in class: the coefficient
appearing in the formula for the weight $6$ Eisenstein series $R=E_6$
is $504$, not $540$ (I miscopied...)
So in fact, the only primes that
require special treatment are $2$ and $3$, everything works for $p=5$,
as Shahab said.}
\sk\sk\noindent
4. Show that the mod $5$ modular form $\theta(E_{10})$
has filtration (weight) $12$ and relate it to $\Delta$.
How does this compare to what you calculated in $3$?
\sk\sk\noindent
5. Prove the assertion made in class that $\partial B = -QA$,
where $A$ and $B$ are the homogeneous polynomials of degrees
$p-1$ and $p+1$ respectively satisfying $A(Q,R)=E_{p-1}$ and
$B(Q,R) = E_{p+1}$.
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