The invariants of a genus one curve

By Tom A. Fisher. 2006

58 citations recorded by the source corpus.

It was first pointed out by Weil that we can use classical invariant theory to compute the Jacobian of a genus one curve. The invariants required for curves of degree n=2, 3, 4 were already known to the nineteenth century invariant theorists. We have succeeded in extending these methods to curves of degree n=5, where although the invariants are too large to write down as explicit polynomials, we have found a practical algorithm for evaluating them.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/b28e044d58cbaee588382b05c408897ecafaa83b

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In Computational Theory and Mathematics

1. Samir Siksek
2. Manjul Bhargava
3. Nuno Freitas
4. Bao V. Le Hung
5. A. Shankar
6. Michael Stoll
7. John Cremona
8. Tom A. Fisher

In Geometry and Topology

1. Manjul Bhargava
2. Samir Siksek
3. A. Shankar
4. Nuno Freitas
5. Bao V. Le Hung
6. Michael Stoll
7. John Cremona
8. Tom A. Fisher

In Information Systems

1. Manjul Bhargava
2. Samir Siksek
3. A. Shankar
4. Michael Stoll
5. Nuno Freitas
6. Bao V. Le Hung
7. John Cremona
8. Bjorn Poonen

Citations in the corpus

• Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 (2010, background): “We note that these additional possible invariants ( I, J ) ∈ 116 Z × 132 Z that arise for ternary cubic forms also arise in the case of binary quartics, provided one uses “generalized binary quartics”; see [13] or [20] for details on the construction and uses of these generalized quartics.”
• Elliptic curves over real quadratic fields are modular (2013, methodology): “Using the formulae in [32], the Jacobian elliptic curve of Dc is given by Ec ∶ Y 2 =X(3) + 111cX(2) + 3512c(2)x + 34224.”
• The geometric sieve and the density of squarefree values of invariant polynomials (2014, background): “The dimensions of the representations in (i)–(iv) above are given by 5, 10, 20, and 50, respectively, while the degrees of the corresponding discriminant polynomials are given by 6, 12, 24, and 60, respectively (see [23] for explicit constructions of these invariant polynomials).”
• MINIMISATION AND REDUCTION OF 2-, 3- AND 4-COVERINGS OF ELLIPTIC CURVES (2009, methodology): “irs of quadrics. We use quadrics here, in order to obtain more uniform local results at places with residue characteristic 2. Remark 2.5. There is also a definition of genus one model of degree 5, see [Fi4]. The minimisation and reduction of these models (and possible extensions to larger n) will be ”
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (2013, methodology): “If v ∈ VQp is Qp-soluble and has integral invariants, then it is a further result of Fisher [28] that v is GQp-equivalent to an element in VZp .”