Covariants and Contravariants for Genus One Models

The functions in this section implement the invariant theory developed in [Fisher, 2006].

Hessian(model): ModelG1 -> ModelG1

We write \(X_n\) for the (affine) space of genus one models of degree \(n\). The module of covariants \(X_n \to X_n\) 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.

CoveringCovariants(model): ModelG1 -> [ RngMPolElt ]

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).

Contravariants(model): ModelG1 -> ModelG1, ModelG1

We write \(X_n\) for the (affine) space of genus one models of degree \(n\), and \(X_n^*\) for its dual. The module of contravariants \(X_n \to X_n^*\) 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.

HesseCovariants(model, r): ModelG1, RngIntElt -> ModelG1, ModelG1

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 \(r \equiv 1 \pmod{n}\) then the covariants evaluated are the identity map and the Hessian. If \(r \equiv -1 \pmod{n}\) then the covariants evaluated are the contravariants. If \(n=5\) there are two further possibilities. We identify \(X_5= \wedge^2 V \otimes W\) where \(V\) and \(W\) are 5-dimensional vector spaces. Then the covariants evaluated for \(r \equiv 2,3 \pmod{5}\) take values in \(\wedge^2 W \otimes V^*\) and \(\wedge^2 W^* \otimes V\) respectively.

HessePolynomials({n, r, }{invariants : parameters}): RngIntElt, RngIntElt, [RngElt] -> RngElt, RngElt, RngElt

Variables: [ RngMPolElt ]                    Default: 

The Hesse polynomials \(D(x,y), c_4(x,y), c_6(x,y)\). 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 \(r \equiv 1 \pmod{n}\).