Function: ellpadiclambdamu
Section: elliptic_curves
C-Name: ellpadiclambdamu
Prototype: GLD1,L,D0,L,
Help: ellpadiclambdamu(E,p,{D=1},{i=0}): returns the Iwasawa invariants for
 the p-adic L-function attached to E, twisted by (D,.) and the i-th power
 of the Teichmuller character.
Doc: Let $p$ be a prime number and let $E/\Q$ be a rational elliptic curve
 with good or bad multiplicative reduction at $p$.
 Return the Iwasawa invariants $\lambda$ and $\mu$ for the $p$-adic $L$
 function $L_{p}(E)$, twisted by $(D/.)$ and the $i$-th power of the
 Teichm\"uller character $\tau$, see \kbd{ellpadicL} for details about
 $L_{p}(E)$.

 Let $\chi$ be the cyclotomic character and choose $\gamma$
 in $\text{Gal}(\Q_{p}(\mu_{p^{\infty}})/\Q_{p})$ such that $\chi(\gamma)=1+2p$.
 Let $\hat{L}^{(i), D} \in \Q_{p}[[X]]\otimes D_{cris}$ such that
 $$ (<\chi>^{s} \tau^{i}) (\hat{L}^{(i), D}(\gamma-1))
   = L_{p}\big(E, <\chi>^{s}\tau^{i} (D/.)\big).$$

 \item When $E$ has good ordinary or bad multiplicative reduction at $p$.
 By Weierstrass's preparation theorem the series $\hat{L}^{(i), D}$ can be
 written $p^{\mu} (X^{\lambda} + p G(X))$ up to a $p$-adic unit, where
 $G(X)\in \Z_{p}[X]$. The function returns $[\lambda,\mu]$.

 \item When $E$ has good supersingular reduction, we define a sequence
 of polynomials $P_{n}$ in $\Q_{p}[X]$ of degree $< p^{n}$ (and bounded
 denominators), such that
 $$\hat{L}^{(i), D} \equiv P_{n} \varphi^{n+1}\omega_{E} -
    \xi_{n} P_{n-1}\varphi^{n+2}\omega_{E} \bmod \big((1+X)^{p^{n}}-1\big)
    \Q_{p}[X]\otimes D_{cris},$$
 where $\xi_{n} = \kbd{polcyclo}(p^{n}, 1+X)$.
 Let $\lambda_{n},\mu_{n}$ be the invariants of $P_{n}$. We find that

 \item $\mu_{n}$ is nonnegative and decreasing for $n$ of given parity hence
 $\mu_{2n}$ tends to a limit $\mu^{+}$ and $\mu_{2n+1}$ tends to a limit
 $\mu^{-}$ (both conjecturally $0$).

 \item there exists integers $\lambda^{+}$, $\lambda^{-}$
 in $\Z$ (denoted with a $\til$ in the reference below) such that
 $$ \lim_{n\to\infty} \lambda_{2n} + 1/(p+1) = \lambda^{+}
    \quad \text{and} \quad
    \lim_{n\to\infty} \lambda_{2n+1} + p/(p+1) = \lambda^{-}.$$
 The function returns $[[\lambda^{+}, \lambda^{-}], [\mu^{+},\mu^{-}]]$.

 \noindent Reference: B. Perrin-Riou, Arithm\'etique des courbes elliptiques
 \`a r\'eduction supersinguli\`ere en $p$, \emph{Experimental Mathematics},
 {\bf 12}, 2003, pp. 155-186.
