mseval

Function: mseval
Section: modular_symbols
C-Name: mseval
Prototype: GGDG
Help: mseval(M,s,{p}): M being a full modular symbol space, as given by
 msinit, s being a modular symbol from M and p being a path between two
 elements in P^1(Q), return s(p).
Doc: Let $\Delta_0:=\text{Div}^0(\P^1 (\Q))$.
 Let $M$ be a full modular symbol space, as given by \kbd{msinit},
 let $s$ be a modular symbol from $M$, i.e. an element
 of $\Hom_G(\Delta_0, V)$, and let $p=[a,b] \in \Delta_0$ be a path between
 two elements in $\P^1(\Q)$, return $s(p)\in V$. The path extremities $a$ and
 $b$ may be given as \typ{INT}, \typ{FRAC} or $\kbd{oo} = (1:0)$; it
 is also possible to describe the path by a $2 \times 2$ integral matrix
 whose columns give the two cusps. The symbol $s$ is either

 \item a \typ{COL} coding a modular symbol in terms of
 the fixed basis of $\Hom_G(\Delta_0,V)$ chosen in $M$; if $M$ was
 initialized with a nonzero \emph{sign} ($+$ or $-$), then either the
 basis for the full symbol space or the $\pm$-part can be used (the dimension
 being used to distinguish the two).

 \item a \typ{MAT} whose columns encode modular symbols as above. This is
 much faster than evaluating individual symbols on the same path $p$
 independently.

 \item a \typ{VEC} $(v_i)$ of elements of $V$, where the $v_i = s(g_i)$ give
 the image of the generators $g_i$ of $\Delta_0$, see \tet{mspathgens}.
 We assume that $s$ is a proper symbol, i.e.~that the $v_i$ satisfy
 the \kbd{mspathgens} relations.

 If $p$ is omitted, convert a single symbol $s$  to the second form: a vector
 of the $s(g_i)$. A \typ{MAT} is converted to a vector of such.

 \bprog
 ? M = msinit(2,8,1); \\ M_8(Gamma_0(2))^+
 ? g = mspathgens(M)[1]
 %2 = [[+oo, 0], [0, 1]]
 ? N = msnew(M)[1]; #N \\ Q-basis of new subspace, dimension 1
 %3 = 1
 ? s = N[,1]         \\ t_COL representation
 %4 = [-3, 6, -8]~
 ? S = mseval(M, s)   \\ t_VEC representation
 %5 = [64*x^6-272*x^4+136*x^2-8, 384*x^5+960*x^4+192*x^3-672*x^2-432*x-72]
 ? mseval(M,s, g[1])
 %6 = 64*x^6 - 272*x^4 + 136*x^2 - 8
 ? mseval(M,S, g[1])
 %7 = 64*x^6 - 272*x^4 + 136*x^2 - 8
 @eprog\noindent Note that the symbol should have values in
 $V = \Q[x,y]_{k-2}$, we return the de-homogenized values corresponding to $y
 = 1$ instead.