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The functions in this section implement the invariant theory developed
in [Fis].
We write Xn for the (affine) space of genus one models
of degree n. The module of covariants Xn to Xn is a free
module of rank 2 over the ring of invariants. The generators
are the identity map and a second covariant which we term the
Hessian. (In the cases where n=2 or 3 this is the determinant of a matrix
of second partial derivatives.) This function evaluates the Hessian
of the given genus one model.
The covariants that define the covering map from the given
genus one model to its Jacobian
(this is the same as the defining equations of the nCovering).
We write Xn for the (affine) space of genus one models
of degree n, and Xn * for its dual. The module of contravariants
Xn to Xn * is a free module of rank 2 over the ring of invariants.
This function evaluates the generators P and Q at the given genus one model.
Evaluates a pair of covariants, which depend on an integer r,
at the genus one model of degree prime to r.
The pencil spanned by these genus one models is a family of
genus one curves invariant under the same representation of
the Heisenberg group. (In other words, the universal family
above a twist of X(n).)
If r ≡ 1 mod (n) then the covariants evaluated are the identity
map and the Hessian. If r ≡ - 1 mod (n) then the
covariants evaluated are the contravariants. If n=5 there
are two further possibilities. We identify X5= ^2 V tensor W
where V and W are 5-dimensional vector spaces. Then the
covariants evaluated for r ≡ 2, 3 mod (5) take values
in ^2 W tensor V * and ^2 W * tensor V
respectively.
Variables: [ RngMPolElt ] Default:
The Hesse polynomials D(x, y), c4(x, y), c6(x, y).
These polynomials give the invariants for the pencil of genus
one models computed by HesseCovariants.
The RubinSilverbergPolynomials are closely related to
these formulae in the case r ≡ 1 mod (n).
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