On Fermat varieties

Shioda, Tetsuji; Katsura, Takeshi

doi:10.2748/tmj/1178229881

Cited by 140 publications

(92 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In fact, general results are available in the literature on the zeta function of Fermat hypersurfaces over finite fields (starting with [W]; see also [SK,Section 3]), although they do not seem to have been spelled out for cubics. Let us first define…”

Section: 3mentioning

confidence: 99%

“…In other words, the eigenvalues of the Frobenius morphism of F p acting on the first cohomology group are ω andω. They are therefore the Jacobi sums denoted by j(1, 2) and j(2, 1) in [SK,(3.1)], and also the generators of the prime ideals p andp in Z[ζ] (ζ = exp(2iπ/3)) such that (p) = pp.…”

Section: 3mentioning

confidence: 99%

“…The ideal (j(α)) in Z[ζ] is invariant under permutations of the α i and its decomposition is computed by Stickelberger (see [SK,(3.10)]):…”

Section: 3mentioning

confidence: 99%

See 2 more Smart Citations

Lines on Cubic Hypersurfaces Over Finite Fields

Debarre¹,

Laface²,

Roulleau³

2017

Geometry Over Nonclosed Fields

View full text Add to dashboard Cite

Abstract. We show that smooth cubic hypersurfaces of dimension n defined over a finite field F q contain a line defined over F q in each of the following cases:• n = 3 and q ≥ 11;• n = 4, and q = 2 or q ≥ 5;• n ≥ 5. For a smooth cubic threefold X, the variety of lines contained in X is a smooth projective surface F (X) for which the Tate conjecture holds, and we obtain information about the Picard number of F (X) and the 5-dimensional principally polarized Albanese variety A(F (X)).

show abstract

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

See 1 more Smart Citation

Lines on Cubic Hypersurfaces Over Finite Fields

Debarre¹,

Laface²,

Roulleau³

2017

Geometry Over Nonclosed Fields

View full text Add to dashboard Cite

show abstract

“…It is shown below that it can be constructed directly in terms of E 4 via the twist map [19]. Alternatively, one can use the Shioda-Katsura decomposition [30,31] to reconstruct the cohomology of S 4 from the cohomology of the quartic plane curve…”

Section: Elliptic Curves From K3 Surfacesmentioning

confidence: 99%

The Langlands program and string modular K3 surfaces

Schimmrigk

2007

Nuclear Physics B

View full text Add to dashboard Cite

ABSTRACT:A number theoretic approach to string compactification is developed for CalabiYau hypersurfaces in arbitrary dimensions. The motivic strategy involved is illustrated by showing that the Hecke eigenforms derived from Galois group orbits of the holomorphic two-form of a particular type of K3 surfaces can be expressed in terms of modular forms constructed from the worldsheet theory. The process of deriving string physics from spacetime geometry can be reversed, allowing the construction of K3 surface geometry from the string characters of the partition function. A general argument for K3 modularity follows from mirror symmetry, in combination with the proof of the Shimura-Taniyama conjecture.

show abstract

“…Let N X be the lattice of numerical equivalence classes of algebraic cycles of codimension 2 on X ⊗ F 2 . The rank of N X is equal to the 4th Betti number b 4 (X) = 23 of X; that is, X is supersingular (see Shioda and Katsura [10], Tate [11]). Corollary 1.3.…”

Section: Introductionmentioning

confidence: 99%

On Edge’s correspondence associated with $$\cdot 222$$ · 222

Shimada

2017

European Journal of Mathematics

View full text Add to dashboard Cite

Abstract. We describe explicitly the correspondence of Edge between the set of planes contained in the Fermat cubic 4-fold in characteristic 2, and the set of lattice points T of the Leech lattice Λ 24 such that OABT is a regular tetrahedron, where O is the origin of Λ 24 , and A and B are fixed points of Λ 24 such that OAB is a regular triangle of edge length 2. Using this description, we present Conway's isomorphism from PSU(6, 4) to ·222 in terms of matrices.

show abstract

On Fermat varieties

Cited by 140 publications

References 9 publications

Lines on Cubic Hypersurfaces Over Finite Fields

Lines on Cubic Hypersurfaces Over Finite Fields

The Langlands program and string modular K3 surfaces

On Edge’s correspondence associated with $$\cdot 222$$ · 222

Contact Info

Product

Resources

About