## The Jacobian and Formal Group of a Curve of Genus 2 over an Arbitrary Ground (1990)

Venue: | Math. Proc. Cambridge Philos. Soc. 107 |

Citations: | 18 - 3 self |

### BibTeX

@INPROCEEDINGS{Flynn90thejacobian,

author = {E. V. Flynn},

title = {The Jacobian and Formal Group of a Curve of Genus 2 over an Arbitrary Ground},

booktitle = {Math. Proc. Cambridge Philos. Soc. 107},

year = {1990},

pages = {107}

}

### Years of Citing Articles

### OpenURL

### Abstract

The ability to perform practical computations on particular cases has greatly influenced the theory of elliptic curves. First, it has allowed a rich sub-branch of the Mathematics of Computation to develop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fields, and — more recently — the application of

### Citations

818 |
The arithmetic of elliptic curves
- Silverman
- 1986
(Show Context)
Citation Context ... for a given P, P ′ ∈ N with the same s1 and s2, the power series σ3, . . . , σ15 uniquely determine s3, . . . , s15, and so P = P ′. We finally note, as an analogue of an elliptic curves result (see =-=[12]-=-, p.111), that the local power series σi are homogeneous with respect to wt1 and wt2, since each of the equations used in the recursive substitutions are homogeneous. §3. The Formal Group Associated w... |

333 |
lectures on theta
- Mumford, “Tata
- 1984
(Show Context)
Citation Context ...n F (X) is a quintic; we do not impose this restriction, and the theory presented will apply to the general case. This renders unavailable much of the analytic theory of the Jacobian, such as that in =-=[10]-=-. In particular, we may not assume the existence of a Weierstrass point defined over K. As a group, the Jacobian of a curve of genus 2 has long been well understood in terms of divisor classes modulo ... |

231 |
Fundamentals of Diophantine geometry
- Lang
- 1983
(Show Context)
Citation Context ...d similarly for n-weightless). Furthermore, (φ1 ≈ φ2 and ψ1 ≈ ψ2) ⇒ φ1ψ1 ≈ φ2ψ2 and (φ1ψ ≈ φ2ψ and ψ weightless) ⇒ φ1 ≈ φ2. We shall also make use of the following generalisation of Gauss’ Lemma (see =-=[5]-=-, p.55). Lemma 3.4. Let φ = ψθ, where φ, ψ and θ are power series in several variables defined over K. If θ and φ are defined over R, and θ is weightless (so that ψ = φ/θ is d-weightless), then ψ is d... |

109 |
Algebras and Lie Groups
- Lie
- 1991
(Show Context)
Citation Context ...ups. 8( ) F1 Lemma 3.1. There is a formal group law F = defined over K such that u = F(s, t) F2 in some neighbourhood NF of O. For the proof of Lemma 3.1 it is necessary to sift through Chapter 4 of =-=[11]-=- (or some equivalent), which shows that every analytic group is locally just a formal group over K. The result we need is on p. 4.22 of [11], that any analytic group chunk contains an open subgroup wh... |

83 |
On the equations defining abelian varieties
- Mumford
- 1966
(Show Context)
Citation Context ...dependent, and to check non-singularity at the origin. We note that Theorem 1.2. is a special case of a Theorem of Mumford, that Jacobian varieties may be defined by the intersection of quadrics (see =-=[9]-=-). However, we shall try, at this stage, to give the reader a brief idea of the techniques used to derive such identities. We first introduce two weights on X, Y, and fi with respect to which C is hom... |

32 |
Introduction to Algebraic and Abelian Functions
- Lang
- 1982
(Show Context)
Citation Context ... − ∞ (where ∞ is some fixed Weierstrass point over K), then Θ + + Θ − is equivalent over K to 2Θ. Now, Θ is ample, and the Riemann-Roch Theorem gives that ℓ(nΘ) = n 2 (n ≥ 1). A theorem of Lefschetz (=-=[6]-=-, p.105) implies that a basis for L(3Θ) gives a projective embedding of the Jacobian into P8 . David Grant has independently developed this point of view in [4], which assumes a Weierstrass point over... |

19 |
Abelian varieties over p-adic ground fields
- Mattuck
- 1955
(Show Context)
Citation Context ...u = s2 t2 ( u1 u2 ) are local parameters for a, b and c, respectively. We first note that, if we view J(C) as an abelian variety over K, the field of fractions of R, then it is an analytic group (see =-=[7]-=-). We therefore have the following from the general theory of analytic groups. 8( ) F1 Lemma 3.1. There is a formal group law F = defined over K such that u = F(s, t) F2 in some neighbourhood NF of O... |

14 |
The Mordell-Weil group and curves of genus 2. Arithmetic and geometry. Papers dedicated to I.R
- Cassels
- 1983
(Show Context)
Citation Context ...not assume the existence of a Weierstrass point defined over K. As a group, the Jacobian of a curve of genus 2 has long been well understood in terms of divisor classes modulo linear equivalence (see =-=[1]-=-). The canonical equivalence class of divisors of the form (x, y) + (x, −y) , denoted by O, gives the group identity. Any other element of the Jacobian is represented by an unordered pair of points {(... |

14 |
Speeding up the Pollard and elliptic curve methods of factorization
- Montgomery
- 1987
(Show Context)
Citation Context ...lop, devoted to elliptic curves: the search for curves of large rank, large torsion over number fields, and — more recently — the application of elliptic curves to the factorisation of large integers =-=[8]-=-. Second, computation with special curves has motivated, and formed a testing ground for, many of the deep conjectures of the general theory. For abelian varieties of higher dimension, there is a exte... |

11 |
Formal Groups in Genus 2
- Grant
- 1990
(Show Context)
Citation Context ... = n 2 (n ≥ 1). A theorem of Lefschetz ([6], p.105) implies that a basis for L(3Θ) gives a projective embedding of the Jacobian into P8 . David Grant has independently developed this point of view in =-=[4]-=-, which assumes a Weierstrass point over K and uses the quintic form for F (X). In our case, we wish only to use maps defined over K, and so no analogue of L(3Θ) is available. We do, however, have an ... |

3 |
Arithmetic of curves of genus 2. Number Theory and Applications
- Cassels
- 1989
(Show Context)
Citation Context ...at infinity (that is to say, of degree at most 2 in each of X1, X2) , a pole of any order at O, and are regular elsewhere. Such functions form a 16-dimensional vector space. A basis has been given in =-=[2]-=-; we find it convenient to adopt the following slightly different basis and notation (where (x1, y1), (x2, y2) ∈ C). Definition 1.1. Let the map J : D(C) → P15 take D = {(x1, y1), (x2, y2)} ∈ D(C) to ... |

1 | Curves of Genus 2
- Flynn
- 1989
(Show Context)
Citation Context ...d therefore (without appeal to Lemma 3.1) divide the localisations of ν1(a, b) and ν2(a, b). Progress towards an explicit expression of this type for the global group law is described in Chapter 3 of =-=[3]-=-. An alternative proof of Theorem 3.5 along these lines, as well as being more direct, would be likely to provide a more efficient method of computing terms of the formal group. 13Appendix A. A Set o... |