Ticket #3883: sage_trac_3883c.patch

a/sage/schemes/elliptic_curves/ell_generic.py

@@ -1829 +1829 @@
-#         self.__divpoly2[m] = f
-#         return f
-
-
+_
-         r"""
-         Returns the numerator of the x-coordinate of the nth multiple of
-         a point, using torsion polynomials (division polynomials).
@@ -1839 +1839 @@
-         The result is adjusted to be correct for both even and odd n.
-
-         WARNING: -- There may of course be cancellation between the
-
-For more information on how to avoid cancellation,
-see Chris Wuthrich's thesis
+_
+E.g. if a point on an elliptic curve with coefficients in
+ZZ reduces to a singular point modulo a prime, then there will be cancellation, otherwise not, see Chris Wuthrich ``p-adic heights in families of elliptic curves''
-
-         SEE ALSO:
-
+_
-
-         AUTHORS:
-            -- David Harvey (2006-09-24)
@@ -1856 +1856 @@
-
-           sage: (35*P)[0]
-           -804287518035141565236193151/1063198259901027900600665796
-
+_
-           -804287518035141565236193151
-
+_
-           1063198259901027900600665796
-
-           sage: (36*P)[0]
-           54202648602164057575419038802/15402543997324146892198790401
-
+_
-           54202648602164057575419038802
-
+_
-           15402543997324146892198790401
-
-         An example where cancellation occurs:
-           sage: E = EllipticCurve("88a1")
-           sage: P = E([2,2])   # fixed choice of generator
-
+_
-           442446784738847563128068650529343492278651453440
-
+_
-           1427247692705959881058285969449495136382746624
-           sage: n/d
-           310
@@ -1902 +1902 @@
-             return x * cache[n]**2 - cache[-1] * cache[n-1] * cache[n+1]
-
-
-
+_
-         r"""
-         Returns the denominator of the x-coordinate of the nth multiple of
-         a point, using torsion polynomials (division polynomials).
@@ -1912 +1912 @@
-         The result is adjusted to be correct for both even and odd n.
-
-         SEE ALSO:
-
+_
-
-         TODO: the numerator and denominator versions share a calculation,
-         namely squaring $\psi_n$. Maybe would be good to offer a combined
-         version to make this more efficient.
-
-         EXAMPLES:
-
+_
-
-         AUTHORS:
-            -- David Harvey (2006-09-24)
@@ -2036 +2036 @@
-        # the x-coordonate does not depend on the sign of m.  The work
-        # here is done by functions defined earlier:
-
-multiple_x_numerator(m.abs(),x) / self.
+_multiple_x_numerator(m.abs(),x) / self._
-
-        if x_only:
-            # Return it if the optional parameter x_only is set.

a/sage/schemes/elliptic_curves/ell_point.py

@@ -592 +592 @@
-            g = E.division_polynomial(m)
-        else:
-            # The poly g here is 0 at x(Q) iff x(m*Q) = x(P).
-multiple_x_numerator(m) - P[0]*E.
+_multiple_x_numerator(m) - P[0]*E._
-
-            # When 2*P=0, then -Q is a solution iff Q is.  For even m,
-            # no 2-torsion point is a solution, so that g is the