The pullback of a point divisor along a separable isogeny #
Over a separably closed field a separable isogeny φ : W₁ → W₂ is unramified and splits every
place completely, so every place over the place of a point T of W₂ is the place of a point of
W₁, necessarily one over T. The pullback of the point divisor (T) is therefore the fibre
∑_{φ P = T} (P), with every multiplicity 1, and the fibre over T is the translate of the
kernel by any point over T. This is the divisor computation behind the adjointness of the dual
isogeny for the Weil pairing.
Main results #
TauCeti.Isogeny.divisorPullback_ofPoint_eq_sum:φ^* (T) = ∑_{φ P = T} (P).TauCeti.Isogeny.divisorPullback_ofPoint_sub_eq_sum:φ^* ((T) - (O)) = ∑_{φ K = O} ((P₀ + K) - (K))for anyP₀overT.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, II.3 and III.4.10.
The pullback of a point along a separable isogeny is its fibre: over a separably closed
field, φ^* (T) = ∑_{φ P = T} (P). The isogeny is unramified, and every place over the place of
T is the place of a point, since φ splits it completely.
φ^* ((T) - (O)) is the translate of the kernel minus the kernel: for any P₀ over T, it
is ∑_{φ K = O} ((P₀ + K) - (K)), over a separably closed field.