## Mordell-Weil lattices in characteristic 2: I. Construction and first properties, Internat (1997)

Venue: | Invent. Math |

Citations: | 10 - 2 self |

### BibTeX

@ARTICLE{Elkies97mordell-weillattices,

author = {Noam D. Elkies},

title = {Mordell-Weil lattices in characteristic 2: I. Construction and first properties, Internat},

journal = {Invent. Math},

year = {1997},

volume = {8},

pages = {343--361}

}

### OpenURL

### Abstract

In a famous 1967 paper [ˇST], ˇSafarevič and Tate constructed elliptic curves of arbitrarily

### Citations

410 | Sphere Packings, Lattices and Groups - Conway, Sloane - 1999 |

97 |
Complexe de de Rham-Witt et Cohomologie Cristalline, Annals
- Illusie
- 1979
(Show Context)
Citation Context ...ion of (1) with x, y in k((t −1 )), the completion of K at the place t = ∞. 1 As noted in [5], Milne had to also assume odd characteristic, but this assumption was later eliminated by work of Illusie =-=[7]-=-, so we may use Milne’s results also in our characteristic-2 setting. 3sBy [4, Thm.2], every element of the φ-Selmer group has a unique representative ξ that is an odd polynomial in t (that is, a k-li... |

94 |
On the Mordell-Weil Lattices
- Shioda
- 1990
(Show Context)
Citation Context ...position, too, in several alternative ways. We could also obtain the formula for ˆh, following Néron [Né], as a sum of local contributions, using for instance the explicit formulas in [Si]; or, as in =-=[Sh1]-=-, [Sh2], using intersection theory, which also provides another interpretation of the estimate (18). Yet another route is to note that ˆh(P) is half the canonical height of the map f : C → E0 associat... |

63 |
On the conjectures of Birch and Swinnerton-Dyer and a geometric analog
- Tate
- 1995
(Show Context)
Citation Context ...ated to the order of the curve’s Tate- ˇ Safarevič group by the conjecture of Birch and Swinnerton-Dyer. In our case of a curve over a function field, this conjecture was formulated by Artin and Tate =-=[9]-=- and proved under certain hypotheses by Milne [8]. 1 In [4, Thm.1] we observed that these hypotheses were satisfied by each of our curves E, and computed the resulting relationship between the order o... |

61 | The arithmetic of elliptic curves - Tate - 1974 |

30 | Computing heights on elliptic curves
- Silverman
- 1988
(Show Context)
Citation Context ...n prove this proposition, too, in several alternative ways. We could also obtain the formula for ˆh, following Néron [Né], as a sum of local contributions, using for instance the explicit formulas in =-=[Si]-=-; or, as in [Sh1], [Sh2], using intersection theory, which also provides another interpretation of the estimate (18). Yet another route is to note that ˆh(P) is half the canonical height of the map f ... |

26 |
On a conjecture of Artin and
- Milne
(Show Context)
Citation Context ...the size of the conjecturally finite Tate- ˇSafarevič group X of E(K). We have seen that for our curve E/K, the order of vanishing equals the rank r as predicted. It then follows from Milne’s theorem =-=[Mi]-=- that X is finite and the rest of the conjecture is also true. The usual statement of the formula uses the definition of the canonical height that includes factors of log |k|, but these powers conveni... |

20 | Some cyclo-quaternionic lattices
- Nebe
- 1998
(Show Context)
Citation Context ...arp, and reported on the computation of the kissing number of the lattice, which was at the time the largest kissing number known in R 64 . Both of these records have since been superseded by G. Nebe =-=[6]-=-, and the kissing number was pushed still higher by a nonlattice packing [3]. Thus n = 6 is now the first case in which the construction of [4] yields a previously unknown lattice of record density. I... |

20 |
Quasi-fonctions et hauteurs sur les variétés abéliennes
- Néron
- 1965
(Show Context)
Citation Context ... degree of x is its total number of poles counted with multiplicity. Remarks. We can prove this proposition, too, in several alternative ways. We could also obtain the formula for ˆh, following Néron =-=[Né]-=-, as a sum of local contributions, using for instance the explicit formulas in [Si]; or, as in [Sh1], [Sh2], using intersection theory, which also provides another interpretation of the estimate (18).... |

14 |
der Vlugt, Reed-Muller codes and supersingular curves
- Geer, van
- 1987
(Show Context)
Citation Context ...mutes with all these automorphisms, we get by Schur’s lemma an alternative proof of the fact that Frobenius acts on Jac(C) by some scalar multiplication. Van der Geer and van der Vlugt investigate in =-=[GV]-=- a family of hyperelliptic curves in characteristic 2 which, like C, have genus 2n−1 and automorphisms by this extraspecial 2-group; much of our discussion of C/k will also apply with little change to... |

6 |
J.A.: On the packing densities of superballs and other
- Elkies, Odlyzko, et al.
- 1991
(Show Context)
Citation Context ...in somewhat higher dimensions listed in [CS].) Note, however, that in any dimension r there exist lattices, not explicitly known for large r, with packing density at least 21−r (Minkowski-Hlawka; cf. =-=[EOR]-=-); for n = 10, 11 this amounts to normalized densities δ larger than those of our Mordell-Weil lattices. Once n ≥ 12, the lower bounds (26), (27) are worse than both the MinkowskiHlawka bounds and the... |

5 |
Elkies, Mordell–Weil lattices in characteristic 2 II: The Leech lattice as a Mordell– Weil lattice
- D
- 1997
(Show Context)
Citation Context ...s associated with proper factors of q + 1 are not as good as those coming from C itself, but the factor f = 13 of 26 + 1 = 65 turns out to yield a lattice homothetic with the Leech lattice of rank 24 =-=[El]-=-. (iv) Note that when n = 1, the “hyperelliptic” curve C becomes the supersingular elliptic curve E0, and (10) gives its 24 automorphisms. A potentially constant elliptic curve Let L be the commutativ... |

2 |
The determinants of certain Mordell-Weil lattices
- Dummigan
- 1995
(Show Context)
Citation Context ...ons of x(at) with a65 = 1 to obtain a solution in k((t−1 )) of (1) with x a square plus a polynomial of degree 19 or 15, contradicting [4, Thm.2]. (Similar arguments arise in Dummigan’s investigation =-=[2]-=- of the Tate-ˇ Safarevič groups of certain constant elliptic curves related to those of [4].) The evenorder coefficients need not constitute a group, but are still constrained by the invariance under ... |

2 | N.J.A.: On Kissing Numbers in Dimensions 32 to 128
- Edel, Rains, et al.
- 1998
(Show Context)
Citation Context ...hich was at the time the largest kissing number known in R 64 . Both of these records have since been superseded by G. Nebe [6], and the kissing number was pushed still higher by a nonlattice packing =-=[3]-=-. Thus n = 6 is now the first case in which the construction of [4] yields a previously unknown lattice of record density. In this paper we analyze this 128-dimensional lattice MW128. We determine its... |

2 |
On a conjecture of Artin and Tate
- Milne
- 1975
(Show Context)
Citation Context ...č group by the conjecture of Birch and Swinnerton-Dyer. In our case of a curve over a function field, this conjecture was formulated by Artin and Tate [9] and proved under certain hypotheses by Milne =-=[8]-=-. 1 In [4, Thm.1] we observed that these hypotheses were satisfied by each of our curves E, and computed the resulting relationship between the order of the Tate- ˇ Safarevič group X(E) and the discri... |

1 |
Empilements de sphères,Séminaire Bourbaki 6/90, exposé 727, Astérisque 189
- Oesterlé
- 1990
(Show Context)
Citation Context ...ver function fields. This paper is the first of a series devoted to the characteristic-2 family, giving lattices whose rank r is a power of 2. (Oesterlé has already reported on this family briefly in =-=[Oe]-=-.) We begin with a quick summary of lattices in Euclidean space and the associated sphere packings. We then introduce a family of hyperelliptic curves in characteristic 2, and use them to construct po... |

1 |
lattices and sphere
- Mordell-Weil
- 1991
(Show Context)
Citation Context ...ion r via the canonical Néron-Tate height. It seems natural to investigate the structure of this lattice, but curiously it was not until about 1990 that such lattices were examined in earnest. Shioda =-=[Sh2]-=- then found, as part of his analysis of the Néron-Severi groups of Fermat surfaces, a family of elliptic curves in characteristic congruent to 5 mod 6 similar to those of [ ˇST], whose associated latt... |

1 |
for determining the type of a singular fiber in an elliptic pencil
- “Algorithm
- 1975
(Show Context)
Citation Context ...llest integer such that 6z >q, obtaining y 2 1 + t−3z y1 = x 3 1 + tq+1−6z + t −6z a6 with discriminant t−12z . That this is a minimal model at infinity may be checked for instance by Tate’s formulas =-=[Ta3]-=-. These formulas also tell us the local reduction type: if n is even then q ≡ 4 mod 6 and thus q + 1 − 6z =−1, whence we find reduction type II, while if n is odd then q ≡ 2 mod 6, so q + 1 − 6z =−3an... |

1 |
Global fields, codes and sphere packings, proceedings of Journées Arithmétiques 1989, Astérisque
- Tsfasman
- 1991
(Show Context)
Citation Context ...r bounds allow; in particular, X is trivial for n ≤ 8. Acknowledgments John Conway and Neil Sloane’s great treatise [CS] introduced me to the sphere-packing problem; Michael Tsfasman’s beautiful talk =-=[Ts]-=- on arithmetic constructions of lattices at the Journées 3The condition on a means that a33 , the norm of a from k to its 32-element subfield, is one of the 15 nonzero elements of that subfield whose ... |

1 | Mordell-Weil lattices in (x2i,x17,x13,x9,x5,x1) characteristic 2, II: The Leech lattice as a Mordell= (1, a, 1, a+1, a 3 +a 2 +a, a 2 +l), Weil lattice", Invent - Elkies - 1997 |

1 | also [Nebe 1998] G. Nebe, "Some cyclo-quaternionic lattices", in vol. 9 of the Sem. Bourbaki reprint by Soc - North-Holland - 1968 |