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\begin{center}
{\Huge 189-726B: Modular Forms II} \\ \sk
{\Huge Assignment 5} \\ \sk
{\Large Due: Wednesday, February 13 }
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\noindent
1. Fill in the details of the argument explained in class to
show that the $p$-adic $L$-function
$L_p^\pm(f,s)$ can be expressed as a convergent infinite sum:
$$L_p^\pm(f,s) = \sum_{j=0}^\infty a_j \left(\stackrel{s-1}{j}\right),$$
and give an explicit formula for the coefficients $a_j$ that occur in this
expansion, in terms of the moments of $\mu_f^\pm$ that were
defined in class.
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2. Give an explicit formula for the first derivative of $L_p^\pm(f,s)$ at
$s=1$, in terms of these moments.
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3. Suppose that $N$ is a prime and that $p$ is a primitive root mod $N$.
Show that, if $f$ is a weight $2$ newform on $\Gamma_0(N)$, then the
$L_p^\pm(f,s)$ do not vanish identically on the domain
$(\Z/(p-1)\Z)\times\Z_p$.
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4. What if the assumption on $N$ is relaxed? (I think this problem
is challenging. I remember proving some results in this direction,
but I can't remember
how far I got, or what my proof was. So the idea here is really
to see how far you can go!)
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{\em Remark}. Proving that $L_p^\pm(f,s)$ do not vanish identically
on $(\Z/(p-1)\Z)\times \Z_p$ is (I think) a tractable problem.
It is still not known how to show that $L_p(f,s)$ does not vanish identically
on $\{1\}\times \Z_p$. It is widely believed that this is true.
A proof would make an excellent PhD thesis!
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