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