valuation of function at a local ring for elliptic curve

script/research/ec/valuate.sage

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+# $ sage -sh
+# $ pip install tabulate
+from tabulate import tabulate
+# P = (2, 4)
+# ord_P(y - 2x) = 2
+# from Washington example 11.4 page 345
+
+K.<x, y> = Integers(11)[]
+Px, Py = K(2), K(4)
+
+assert (3*Px^2 + 4) / (2*Py) == 2
+
+basis = [(x - Px), (y - Py), 1]
+# Return components for basis
+def decomp(f, basis):
+    comps = []
+    r = f
+    for b in basis:
+        a, r = r.quo_rem(b)
+        comps.append(a)
+    assert r == 0
+    return comps
+
+def comp(comps, basis):
+    return sum(a*b for a, b in zip(comps, basis))
+
+f = y - 2*x
+assert comp(decomp(f, basis), basis) == f
+
+# P = (a, b)
+# y² = x³ + Ax + B
+# (y - b)(y + b) = (x - a)³ + C(3,2)a(x - a)² + (3a² + A)(x - a)
+#
+# sage: ((x - a)^3 + binomial(3,2)*a*(x - a)^2 + (3*a^2 + A)*(x - a)).expand()
+# -a^3 + x^3 - A*a + A*x
+# But since (a, b) ∈ E(K) => b² = a³ + Aa + B
+#                         => B = b² - (a³ + Aa)
+#
+# So at every step we replace the component for (y - Py)
+# with the reduction to the component for (x - Px)
+
+A = 4
+B = 0
+
+E = y^2 - x^3 - A*x - B
+
+# f / g
+# Technically we don't need g but we keep track of it anyway
+def apply_reduction(comp_f, comp_g, basis):
+    #a1 = comp_f[1]
+    #comp_f[1] = 0
+
+    b0, b1, _ = basis
+    # so we can replace (y - Py) with this
+    sub_poly_f = b0^2 + binomial(3,2)*Px*b0^1 + (3*Px^2 + A)
+    sub_poly_g = (y + Py)
+    assert E == b1*sub_poly_g - b0*sub_poly_f
+    # b1 == b0 * f / g
+    # so we can replace c b1 with (cf/g) b0
+
+    comp_f[0] = comp_f[0]*sub_poly_g + comp_g[2]*sub_poly_f
+    comp_g[2] *= sub_poly_g
+
+k = 1
+
+table = []
+table.append(("", "f", "g", "k"))
+
+def log(step_name, comp_f, comp_g, k):
+    table.append((step_name, str(comp_f), str(comp_g), k))
+
+comp_f = decomp(f, basis)
+comp_g = [0, 0, 1]
+log("start", comp_f, comp_g, k)
+
+# Reduce
+apply_reduction(comp_f, comp_g, basis)
+log("reduce", comp_f, comp_g, k)
+
+f = comp_f[0]
+# Decompose
+comp_f = decomp(f, basis)
+comp_g = [0, 0, 1]
+log("decomp", comp_f, comp_g, k)
+
+assert comp(comp_f, basis) == (x - 2)^2 - 5*(x - 2) - 2*(y - 4)
+assert comp_f[2] == 0
+k += 1
+
+# Reduce
+apply_reduction(comp_f, comp_g, basis)
+log("reduce", comp_f, comp_g, k)
+
+f = comp_f[0]
+# Decompose
+comp_f = decomp(f, basis)
+comp_g = [0, 0, 1]
+log("decomp", comp_f, comp_g, k)
+
+# Program terminates because remainder is nonzero
+assert comp_f[2] != 0
+
+print(f"basis = {basis}")
+print(tabulate(table))
+print(f"k = {k}")
+