Function: ellmodulareqn
Section: elliptic_curves
C-Name: ellmodulareqn
Prototype: LDnDn
Help: ellmodulareqn(N,{x},{y}): given a prime N < 500, return a vector [P, t]
 where P(x,y) is a modular equation of level N. This requires the package
 seadata. The equation is either of canonical type (t=0) or of Atkin type (t=1).
Doc: given a prime $N < 500$, return a vector $[P,t]$ where $P(x,y)$
 is a modular equation of level $N$, i.e.~a bivariate polynomial with integer
 coefficients; $t$ indicates the type of this equation: either
 \emph{canonical} ($t = 0$) or \emph{Atkin} ($t = 1$). This function requires
 the \kbd{seadata} package and its only use is to give access to the package
 contents. See \tet{polmodular} for a more general and more flexible function.

 Let $j$ be the $j$-invariant function. The polynomial $P$ satisfies
 the functional equation,
 $$ P(f,j) = P(f \mid W_{N}, j \mid W_{N}) = 0 $$
 for some modular function $f = f_{N}$ (hand-picked for each fixed $N$ to
 minimize its size, see below), where $W_{N}(\tau) = -1 / (N\*\tau)$ is the
 Atkin-Lehner involution. These two equations allow to compute the values of
 the classical modular polynomial $\Phi_{N}$, such that $\Phi_{N}(j(\tau),
 j(N\tau)) = 0$, while being much smaller than the latter. More precisely, we
 have $j(W_{N}(\tau)) = j(N\*\tau)$; the function $f$ is invariant under
 $\Gamma_{0}(N)$ and also satisfies

 \item for Atkin type: $f \mid W_{N} = f$;

 \item for canonical type: let $s = 12/\gcd(12,N-1)$, then
 $f \mid W_{N} = N^{s} / f$. In this case, $f$ has a simple definition:
 $f(\tau) = N^{s} \* \big(\eta(N\*\tau) / \eta(\tau) \big)^{2\*s}$,
 where $\eta$ is Dedekind's eta function.

 The following GP function returns values of the classical modular polynomial
 by eliminating $f_{N}(\tau)$ in the above functional equation,
 for $N\leq 31$ or $N\in\{41,47,59,71\}$.

 \bprog
 classicaleqn(N, X='X, Y='Y)=
 {
   my([P,t] = ellmodulareqn(N), Q, d);
   if (poldegree(P,'y) > 2, error("level unavailable in classicaleqn"));
   if (t == 0, \\ Canonical
     my(s = 12/gcd(12,N-1));
     Q = 'x^(N+1) * substvec(P,['x,'y],[N^s/'x,Y]);
     d = N^(s*(2*N+1)) * (-1)^(N+1);
   , \\ Atkin
     Q = subst(P,'y,Y);
     d = (X-Y)^(N+1));
   polresultant(subst(P,'y,X), Q) / d;
 }
 @eprog
