HEADER

Function: _header_modular_symbols
Class: header
Section: modular_symbols
Doc:
 \section{Modular symbols}

 Let $\Delta_0 := \text{Div}^0(\P^1(\Q))$ be the abelian group of divisors of
 degree $0$ on the rational projective line. The standard $\text{GL}(2,\Q)$
 action on $\P^1(\Q)$ via homographies naturally extends to $\Delta_0$. Given

 \item $G$ a finite index subgroup of $\text{SL}(2,\Z)$,

 \item a field $F$ and a finite dimensional representation $V/F$ of
   $\text{GL}(2,\Q)$,

 \noindent we consider the space of \emph{modular symbols} $M :=
 \Hom_G(\Delta_0, V)$. This finite dimensional $F$-vector
 space is a $G$-module, canonically isomorphic to $H^1_c(X(G), V)$,
 and allows to compute modular forms for $G$.

 Currently, we only support the groups $\Gamma_0(N)$ ($N > 0$ an integer)
 and the representations $V_k = \Q[X,Y]_{k-2}$ ($k \geq 2$ an integer) over
 $\Q$. We represent a space of modular symbols by an \var{ms} structure,
 created by the function \tet{msinit}. It encodes basic data attached to the
 space: chosen $\Z[G]$-generators $(g_i)$ for $\Delta_0$ (and relations among
 those) and an $F$-basis of $M$. A modular symbol $s$ is thus given either in
 terms of this fixed basis, or as a collection of values $s(g_i)$
 satisfying certain relations.

 A subspace of $M$ (e.g. the cuspidal or Eisenstein subspaces, the new or
 old modular symbols, etc.) is given by a structure allowing quick projection
 and restriction of linear operators; its first component is a matrix whose
 columns  form  an $F$-basis  of the subspace.