Reproduce Fisher's order-five elliptic-function normalization¶
Continuously checked
This is the exact source run by make test-examples and the ordinary Python test gate.
Domains: commalg · numthy · root
Source: examples/26_elliptic_descent_functions.py
# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Reproduce Fisher's order-five elliptic-function normalization."""
from noether import Q
from noether.commalg import RationalFunctionField, WeilDivisor
from noether.numthy import EllipticCurve, NthPowerClassGroup
def main() -> None:
coefficients = RationalFunctionField(Q, "lambda")
parameter = coefficients.generator()
curve = EllipticCurve(
coefficients,
[1 - parameter, -parameter, -parameter, 0, 0],
)
point = curve.point(0, 0)
point.verify_exact_order(5)
miller = curve.miller_function(point, 5)
divisor = WeilDivisor(
curve.underlying_curve,
[
(curve.origin().underlying_curve_point.as_closed_point(), 5),
(point.underlying_curve_point.as_closed_point(), -5),
],
)
function = divisor.riemann_roch_space().basis[0]
assert (function / miller.function).as_constant() is not None
evaluation_at_two = function.evaluate_at((2 * point).underlying_curve_point)
evaluation_at_four = function.evaluate_at((4 * point).underlying_curve_point)
assert evaluation_at_two.value is not None
assert evaluation_at_four.value is not None
classes = NthPowerClassGroup(coefficients, 5)
normalization = classes.class_of(evaluation_at_two.value) ** -2 * classes.class_of(
evaluation_at_four.value
)
assert normalization.is_identity()
print("order 5 verified; Riemann--Roch and Miller agree; normalization is trivial")
if __name__ == "__main__":
main()