Function: mspadicL
Section: modular_symbols
C-Name: mspadicL
Prototype: GDGD0,L,
Help: mspadicL(omsd, {s = 0}, {r = 0}): given
 omsd from mspadicmoments (p-adic distributions attached to an
 overconvergent symbol PHI) returns the value on a
 character of Z_p^* represented by s of the derivative of order r of the
 p-adic L-function attached to PHI.
Doc: Let $\Phi$ be the $p$-adic distribution-valued overconvergent symbol
 attached to a modular symbol $\phi$ for $\Gamma_0(N)$ (eigenvector for
 $T_N(p)$ for the eigenvalue $a_p$). Let $L_p(\Phi,s)$ be the $p$-adic $L$
 function defined by
 $$L_p(\Phi,s)= \int_{\Z_p^*} \chi^s(z) d\mu(z)$$
 where $\mu$ is the distribution on $\Z_p^*$ defined by the restriction of
 $\Phi([\infty]-[0])$ to $\Z_p^*$.

 Let \kbd{omsd} be as returned by \tet{mspadicmoments}
 ($p-1$ distributions attached to $\Phi$ by restriction to the discs
 $a + p\Z_p$, $0 < a < p$).
 Let $s=[s_1,s_2]$ with $s_1 \in \Z \subset \Z_p$ and $s_2 \bmod p-1$ or $s_2
 \bmod 2$ for $p=2$, encoding the $p$-adic character
 $\chi^s := \langle \chi \rangle^{s_1} \tau^{s_2}$; here $\chi$
 is the cyclotomic character from $\text{Gal}(\Q_p(\mu_{p^\infty})/\Q_p)$ to
 $\Z_p^*$, and $\tau$ is the Teichm\"uller character; for convenience, the
 character $[s,s]$ can also be represented by the integer $s$.

 This function returns the value of the $r$-th derivative (in direction
 $\langle \chi \rangle$) of $L_p(\Phi,.)$ at $s$

 When $a_p$ is a $p$-adic unit, the $L$ function takes its values in $\Q_p$.
 When $a_p$ is not a unit, the $L$ function takes its values in the
 two-dimensional $\Q_p$-vector space $D_{cris}(M(\phi))$ where $M(\phi)$ is
 the ``motive'' attached to $\phi$, and we return the two $p$-adic components
 with respect to some fixed $\Q_p$-basis.
 \bprog
 ? M=msinit(3,6,1);
 ? xpm=[5, -3, -1]~;
 ? msissymbol(M,xpm)
 %3 = 1
 ? Mp = mspadicinit(M, 5, 4);
 ? oms = mspadicmoments(Mp, xpm, 1);
 ? mspadicL(oms,0) \\ value on the character chi^0
 %6 = 5 + 2*5^2 + 2*5^3 + 2*5^4 + ...
 ? mspadicL(oms,1) \\ value on the character chi (zero for parity reason)
 %7 = [O(5^13)]~
 ? mspadicL(oms,2) \\ value on the character chi^2
 %8 = 3 + 4*5 + 4*5^2 + 3*5^5 + ...
 ? mspadicL(oms,[0,1]) \\ value on the character tau^1
 %9 = [O(5^13)]~
 ? mspadicL(oms,[0,2]) \\ \\ value on the character tau^2
 %10 = 3 + 5 + 2*5^2 + 2*5^3 + ...
 ? mspadicL(oms,0,1) \\ derivative on chi^0
 %11 = 2*5 + 4*5^2 + 3*5^3 + ...
 @eprog
