The different of the function field of an elliptic curve over F(x) #
Let W be an elliptic Weierstrass curve over a field F, with function field F(W), coordinate
functions x and y, and place at infinity O. The function field is a separable quadratic
extension of the rational function field F(x), and this file computes its different exactly:
Diff(F(W) / F(x)) = div (2y + a₁x + a₃) + 4 O,
in every characteristic. At a place P ≠ O the different exponent is therefore the order of
vanishing of 2y + a₁x + a₃, the partial derivative of the Weierstrass equation in y, and at
O it is 4 + ord_O (2y + a₁x + a₃).
The formula is the divisor identity behind the invariant differential. The divisor of the Weil
differential dx is -2 (x)_∞ + Diff(F(W) / F(x)), the pole divisor of x is 2 O, and the
invariant differential dx / (2y + a₁x + a₃) has divisor zero, so (dx) = div (2y + a₁x + a₃).
Away from characteristic two, 2y + a₁x + a₃ has a pole of order three at O and no other pole,
so d_O = 1 and Diff(F(W) / F(x)) is the divisor of zeros of 2y + a₁x + a₃ plus O. Since
-(x, y) = (x, -y - a₁x - a₃), the function 2y + a₁x + a₃ vanishes at the place of every
rational point of order two. Its zeros have degree 3, so when all four points of order dividing
two are rational, as for y² = x³ - x over ℚ, the different is the sum of their four places:
Diff(F(W) / F(x)) = O + T₁ + T₂ + T₃.
These four places are then exactly the places ramified over F(x): each has ramification index
2, while every other place has different exponent 0 and is unramified. The ramification index
comes from Dedekind's different theorem e = d + 1, which applies because each of these places is
rational and 2 is invertible in F.
In every characteristic the different has degree 4, as the Hurwitz genus formula
2 · 1 - 2 = 2 · (0 - 2) + deg Diff requires.
Main results #
All in the namespace WeierstrassCurve.Affine:
poles_genericX: the pole divisor ofxis2 O.weilDifferentialDivisor_weilDifferentialOfSeparating_genericX: the Weil differentialdxhas divisordiv (2y + a₁x + a₃).different_eq_principal_invariantDifferentialDenom_add: the different ofF(W) / F(x)isdiv (2y + a₁x + a₃) + 4 O.differentExponent_eq_ord_invariantDifferentialDenomanddifferentExponent_infinity_eq_four_add_ord: the different exponents, place by place.degree_different: the different has degree4.ord_infinity_invariantDifferentialDenom,poles_invariantDifferentialDenomanddegree_zeros_invariantDifferentialDenom: away from characteristic two,2y + a₁x + a₃has a triple pole atO, no other pole, and zeros of degree3.differentExponent_infinity_of_two_ne_zeroanddifferent_eq_zeros_add_of_two_ne_zero: away from characteristic two,d_O = 1and the different is the divisor of zeros of2y + a₁x + a₃plusO.different_eq_sum_of_card_eq_four: away from characteristic two, if four rational points ofWhave order dividing two, the different is the sum of their places.differentExponent_pointEquivDegreeOnePlace_of_card_eq_four,differentExponent_eq_zero_of_card_eq_four,ramificationIdx_pointEquivDegreeOnePlace_of_card_eq_four,ramificationIdx_eq_one_of_card_eq_fourandone_lt_ramificationIdx_iff_of_card_eq_four: in that case the places of those four points are exactly the ramified places, each with ramification index2and different exponent1.
References #
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., GTM 254, Springer, 2009, Remark 4.3.7 and Proposition 6.1.3.
- J. Silverman, The Arithmetic of Elliptic Curves, Proposition III.1.5.
The pole divisor of x is 2 O: the coordinate x is regular at every affine place and
has a double pole at the place at infinity.
The divisor of dx is div (2y + a₁x + a₃): the Weil differential dx attached to the
separating element x is 2y + a₁x + a₃ times the invariant differential, which has divisor
zero.
The different of the function field of an elliptic curve over F(x): in every
characteristic, Diff(F(W) / F(x)) = div (2y + a₁x + a₃) + 4 O.
The different exponents away from infinity: at a place P ≠ O, the different exponent
of F(W) / F(x) is the order of 2y + a₁x + a₃ at P.
The different exponent at infinity is 4 + ord_O (2y + a₁x + a₃).
The different of F(W) / F(x) has degree 4, in every characteristic: principal
divisors have degree zero and the place at infinity is rational.
Away from characteristic two #
2y + a₁x + a₃ has a triple pole at infinity away from characteristic two: 2y has a
pole of order three there and a₁x + a₃ one of order at most two.
The pole divisor of 2y + a₁x + a₃ is 3 O away from characteristic two.
The place at infinity is tamely ramified away from characteristic two: its different
exponent over F(x) is 1 = e - 1.
The different away from characteristic two: Diff(F(W) / F(x)) is the divisor of zeros
of 2y + a₁x + a₃ plus the place at infinity.
The zeros of 2y + a₁x + a₃ have degree 3 away from characteristic two: by the product
formula they have the degree of its pole divisor 3 O.
The different when the points of order two are rational: away from characteristic two,
if S is a set of four rational points of order dividing two, then Diff(F(W) / F(x)) is the
sum of their places. This applies when W has full rational 2-torsion, as y² = x³ - x
does.
The different exponent at a point of order dividing two: away from characteristic two, if
S is a set of four rational points of order dividing two, the place of each point of S has
different exponent 1 over F(x).
The different exponent away from the points of order dividing two: away from characteristic
two, if S is a set of four rational points of order dividing two, every place other than the
places of the points of S has different exponent 0 over F(x).
The ramification index at a point of order dividing two: away from characteristic two, if
S is a set of four rational points of order dividing two, the place of each point of S has
ramification index 2 over F(x).
The unramified places when the points of order dividing two are rational: away from
characteristic two, if S is a set of four rational points of order dividing two, every place
of F(W) other than the places of the points of S is unramified over F(x).
The ramified places when the points of order dividing two are rational: away from
characteristic two, if S is a set of four rational points of order dividing two, a place of F(W)
is ramified over F(x) exactly when it is the place of a point of S.