1Function: mspadicseries
2Section: modular_symbols
3C-Name: mspadicseries
4Prototype: GD0,L,
5Help: mspadicseries(mu, {i=0}): given mu from mspadicmoments,
6 returns the attached p-adic series with maximal p-adic precision, depending
7 on the precision of M (i-th Teichmueller component, if present).
8Doc: Let $\Phi$ be the $p$-adic distribution-valued overconvergent symbol
9 attached to a modular symbol $\phi$ for $\Gamma_0(N)$ (eigenvector for
10 $T_N(p)$ for the eigenvalue $a_p$).
11 If $\mu$ is the distribution on $\Z_p^*$ defined by the restriction of
12 $\Phi([\infty]-[0])$ to $\Z_p^*$, let
13 $$\hat{L}_p(\mu,\tau^{i})(x)
14   = \int_{\Z_p^*} \tau^i(t) (1+x)^{\log_p(t)/\log_p(u)}d\mu(t)$$
15 Here, $\tau$ is the Teichm\"uller character and $u$ is a specific
16 multiplicative generator of $1+2p\Z_p$. (Namely $1+p$ if $p>2$ or $5$
17 if $p=2$.) To explain
18 the formula, let $G_\infty := \text{Gal}(\Q(\mu_{p^{\infty}})/ \Q)$,
19 let $\chi:G_\infty\to \Z_p^*$ be the cyclotomic character (isomorphism)
20 and $\gamma$ the element of $G_\infty$ such that $\chi(\gamma)=u$;
21 then
22 $\chi(\gamma)^{\log_p(t)/\log_p(u)}= \langle t \rangle$.
23
24 The $p$-padic precision of individual terms is maximal given the precision of
25 the overconvergent symbol $\mu$.
26 \bprog
27 ? [M,phi] = msfromell(ellinit("17a1"),1);
28 ? Mp = mspadicinit(M, 5,7);
29 ? mu = mspadicmoments(Mp, phi,1); \\ overconvergent symbol
30 ? mspadicseries(mu)
31 %4 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + 4*5^6 + 3*5^7 + O(5^9)) \
32  + (3 + 3*5 + 5^2 + 5^3 + 2*5^4 + 5^6 + O(5^7))*x \
33  + (2 + 3*5 + 5^2 + 4*5^3 + 2*5^4 + O(5^5))*x^2 \
34  + (3 + 4*5 + 4*5^2 + O(5^3))*x^3 \
35  + (3 + O(5))*x^4 + O(x^5)
36 @eprog\noindent
37 An example with nonzero Teichm\"uller:
38 \bprog
39 ? [M,phi] = msfromell(ellinit("11a1"),1);
40 ? Mp = mspadicinit(M, 3,10);
41 ? mu = mspadicmoments(Mp, phi,1);
42 ? mspadicseries(mu, 2)
43 %4 = (2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + 3^7 + 3^10 + 3^11 + O(3^12)) \
44  + (1 + 3 + 2*3^2 + 3^3 + 3^5 + 2*3^6 + 2*3^8 + O(3^9))*x \
45  + (1 + 2*3 + 3^4 + 2*3^5 + O(3^6))*x^2 \
46  + (3 + O(3^2))*x^3 + O(x^4)
47 @eprog\noindent
48 Supersingular example (not checked)
49 \bprog
50 ? E = ellinit("17a1"); ellap(E,3)
51 %1 = 0
52 ? [M,phi] = msfromell(E,1);
53 ? Mp = mspadicinit(M, 3,7);
54 ? mu = mspadicmoments(Mp, phi,1);
55 ? mspadicseries(mu)
56 %5 = [(2*3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \
57  + (2 + 3^3 + O(3^5))*x \
58  + (1 + 2*3 + O(3^2))*x^2 + O(x^3),\
59  (3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \
60  + (1 + 2*3 + 2*3^2 + 3^3 + 2*3^4 + O(3^5))*x \
61  + (3^-2 + 3^-1 + O(3^2))*x^2 + O(3^-2)*x^3 + O(x^4)]
62 @eprog\noindent
63 Example with a twist:
64 \bprog
65 ? E = ellinit("11a1");
66 ? [M,phi] = msfromell(E,1);
67 ? Mp = mspadicinit(M, 3,10);
68 ? mu = mspadicmoments(Mp, phi,5); \\ twist by 5
69 ? L = mspadicseries(mu)
70 %5 = (2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)) \
71  + (2*3^2 + 2*3^6 + 3^7 + 3^8 + O(3^9))*x \
72  + (3^3 + O(3^6))*x^2 + O(3^2)*x^3 + O(x^4)
73 ? mspadicL(mu)
74 %6 = [2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)]~
75 ? ellpadicL(E,3,10,,5)
76 %7 = 2 + 2*3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 2*3^7 + O(3^10)
77 ? mspadicseries(mu,1) \\ must be 0
78 %8 = O(3^12) + O(3^9)*x + O(3^6)*x^2 + O(3^2)*x^3 + O(x^4)
79 @eprog
80