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Optimality and Uniqueness of the Leech Lattice Among Lattices
 arXiv:math.MG/04 03263v1 16
, 2004
"... Abstract. We prove that the Leech lattice is the unique densest lattice in R 24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R 24 can exceed the Leech lattice’s density by a factor of ..."
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Cited by 33 (3 self)
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Abstract. We prove that the Leech lattice is the unique densest lattice in R 24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R 24 can exceed the Leech lattice’s density by a factor of more than 1 + 1.65 · 10 −30, and we give a new proof that E8 is the unique densest lattice in R 8. 1.
Ordinary elliptic curves of high rank over ¯ Fp(x) with constant jinvariant
"... constant jinvariant ..."
Lattices of algebraic cycles on Fermat varieties in positive characteristics
 Proc. London Math. Soc
"... Abstract. Let X be the Fermat hypersurface of dimension 2m and of degree q + 1 defined over an algebraically closed field of characteristic p>0, where q is a power of p, and let NL m (X) be the free abelian group of numerical equivalence classes of linear subspaces of dimension m contained in X. By ..."
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Cited by 3 (1 self)
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Abstract. Let X be the Fermat hypersurface of dimension 2m and of degree q + 1 defined over an algebraically closed field of characteristic p>0, where q is a power of p, and let NL m (X) be the free abelian group of numerical equivalence classes of linear subspaces of dimension m contained in X. By the intersection form, we regard NL m (X) as a lattice. Investigating the configuration of these linear subspaces, we show that the rank of NL m (X) is equal to the 2m th Betti number of X, that the intersection form multiplied by (−1) m is positive definite on the primitive part of NL m (X), and that the discriminant of NL m (X) is a power of p. Let L m (X) be the primitive part of NL m (X) equipped with the intersection form multiplied by (−1) m. In the case p = q = 2, the lattice L m (X) is described in terms of certain codes associated with the unitary geometry over F2. The lattice L 2 (X) is isomorphic to the laminated lattice of rank 22. This explains Conway’s identification ·222 ∼ = PSU(6, 2) geometrically. The lattice L 3 (X) is of discriminant 2 16 · 3, minimal norm 8, and kissingnumber 109421928. 1.
Points of low height on elliptic curves and surfaces
, 2006
"... I: Elliptic surfaces over P 1 with small d ..."
RIGID LATTICES ARE MORDELLWEIL
, 2004
"... Abstract. We say a lattice Λ is rigid if it its isometry group acts irreducibly on its ambient Euclidean space. We say Λ is MordellWeil if there exists an abelian variety A over a number field K such that A(K)/A(K)tor, regarded as a lattice by means of its height pairing, contains at least one copy ..."
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Abstract. We say a lattice Λ is rigid if it its isometry group acts irreducibly on its ambient Euclidean space. We say Λ is MordellWeil if there exists an abelian variety A over a number field K such that A(K)/A(K)tor, regarded as a lattice by means of its height pairing, contains at least one copy of Λ. We prove that every rigid lattice is MordellWeil. In particular, we show that the Leech lattice can be realized inside the MordellWeil group of an elliptic curve over a number field. 1.
The identification of three moduli spaces
, 1999
"... Abstract. It is one of the wonderful “coincidences ” of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G yield two Gcovers of the moduli space ( P1) of 6 conf ..."
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Abstract. It is one of the wonderful “coincidences ” of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G yield two Gcovers of the moduli space ( P1) of 6 configurations of six points on the projective line modulo PGL2, via the 3 and 2torsion of the Jacobians of the double and triple cyclic covers of P 1 branched at those six points. Remarkably these two covers are isomorphic. This was proved over C by transcendental methods in [HW]. We give an algebraic proof valid over any field not of characteristic 2 or 3 that contains the cube roots of unity. We then explore the connection between this Gcover S of ( P1) and the elliptic surface 6 y 2 = x 3 + sextic(t), whose MordellWeil lattice is E8 with automorphisms by a central extension of G. 0. Introduction. The moduli spaces of the title all cover the moduli space ( P1 6) of unordered sextuples of distinct points on P1 modulo the action of PGL2, or equivalently of sextic
unknown title
, 2008
"... Ordinary elliptic curves of high rank over Fp(x) with constant jinvariant ..."
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Ordinary elliptic curves of high rank over Fp(x) with constant jinvariant
RESEARCH STATEMENT
"... A seminal paper in Arithmetic Geometry is L.J. Mordell’s [Mor22]“On the rational solutions of the indeterminate equations of the third and fourth degrees”. In this paper we can find two statements that have greatly influenced not only number theory, but mathematics in general. First he proves that t ..."
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A seminal paper in Arithmetic Geometry is L.J. Mordell’s [Mor22]“On the rational solutions of the indeterminate equations of the third and fourth degrees”. In this paper we can find two statements that have greatly influenced not only number theory, but mathematics in general. First he proves that the group of rational points