Function: mspadicseries
Section: modular_symbols
C-Name: mspadicseries
Prototype: GD0,L,
Help: mspadicseries(mu,{i=0}): given mu from mspadicmoments,
 returns the attached p-adic series with maximal p-adic precision, depending
 on the precision of M (i-th Teichmueller component, if present).
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 $a_{p}$).
 If $\mu$ is the distribution on $\Z_{p}^{*}$ defined by the restriction of
 $\Phi([\infty]-[0])$ to $\Z_{p}^{*}$, let
 $$\hat{L}_{p}(\mu,\tau^{i})(x)
   = \int_{\Z_{p}^{*}} \tau^{i}(t) (1+x)^{\log_{p}(t)/\log_{p}(u)}d\mu(t)$$
 Here, $\tau$ is the Teichm\"uller character and $u$ is a specific
 multiplicative generator of $1+2p\Z_{p}$, namely $1+p$ if $p>2$ or $5$
 if $p=2$. To explain
 the formula, let $G_{\infty} := \text{Gal}(\Q(\mu_{p^{\infty}})/ \Q)$,
 let $\chi:G_{\infty}\to \Z_{p}^{*}$ be the cyclotomic character (isomorphism)
 and $\gamma$ the element of $G_{\infty}$ such that $\chi(\gamma)=u$;
 then
 $\chi(\gamma)^{\log_{p}(t)/\log_{p}(u)}= \langle t \rangle$.

 The $p$-padic precision of individual terms is maximal given the precision of
 the overconvergent symbol $\mu$.
 \bprog
 ? [M,phi] = msfromell(ellinit("17a1"),1);
 ? Mp = mspadicinit(M, 5,7);
 ? mu = mspadicmoments(Mp, phi,1); \\ overconvergent symbol
 ? mspadicseries(mu)
 %4 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + 4*5^6 + 3*5^7 + O(5^9)) \
  + (3 + 3*5 + 5^2 + 5^3 + 2*5^4 + 5^6 + O(5^7))*x \
  + (2 + 3*5 + 5^2 + 4*5^3 + 2*5^4 + O(5^5))*x^2 \
  + (3 + 4*5 + 4*5^2 + O(5^3))*x^3 \
  + (3 + O(5))*x^4 + O(x^5)
 @eprog\noindent
 An example with nonzero Teichm\"uller:
 \bprog
 ? [M,phi] = msfromell(ellinit("11a1"),1);
 ? Mp = mspadicinit(M, 3,10);
 ? mu = mspadicmoments(Mp, phi,1);
 ? mspadicseries(mu, 2)
 %4 = (2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + 3^7 + 3^10 + 3^11 + O(3^12)) \
  + (1 + 3 + 2*3^2 + 3^3 + 3^5 + 2*3^6 + 2*3^8 + O(3^9))*x \
  + (1 + 2*3 + 3^4 + 2*3^5 + O(3^6))*x^2 \
  + (3 + O(3^2))*x^3 + O(x^4)
 @eprog\noindent
 Supersingular example (not checked)
 \bprog
 ? E = ellinit("17a1"); ellap(E,3)
 %1 = 0
 ? [M,phi] = msfromell(E,1);
 ? Mp = mspadicinit(M, 3,7);
 ? mu = mspadicmoments(Mp, phi,1);
 ? mspadicseries(mu)
 %5 = [(2*3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \
  + (2 + 3^3 + O(3^5))*x \
  + (1 + 2*3 + O(3^2))*x^2 + O(x^3),\
  (3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \
  + (1 + 2*3 + 2*3^2 + 3^3 + 2*3^4 + O(3^5))*x \
  + (3^-2 + 3^-1 + O(3^2))*x^2 + O(3^-2)*x^3 + O(x^4)]
 @eprog\noindent
 Example with a twist:
 \bprog
 ? E = ellinit("11a1");
 ? [M,phi] = msfromell(E,1);
 ? Mp = mspadicinit(M, 3,10);
 ? mu = mspadicmoments(Mp, phi,5); \\ twist by 5
 ? L = mspadicseries(mu)
 %5 = (2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)) \
  + (2*3^2 + 2*3^6 + 3^7 + 3^8 + O(3^9))*x \
  + (3^3 + O(3^6))*x^2 + O(3^2)*x^3 + O(x^4)
 ? mspadicL(mu)
 %6 = [2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)]~
 ? ellpadicL(E,3,10,,5)
 %7 = 2 + 2*3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 2*3^7 + O(3^10)
 ? mspadicseries(mu,1) \\ must be 0
 %8 = O(3^12) + O(3^9)*x + O(3^6)*x^2 + O(3^2)*x^3 + O(x^4)
 @eprog
