Hodge Cycles Research Articles

On the standard conjecture for complex Abelian schemes over smooth projective curves

We reduce the Hodge conjecture for Abelian varieties to the question of the existence of an algebraic isomorphism for all and all principally polarized complex Abelian schemes of relative dimension over smooth projective curves. If the canonically defined Hodge cycles are algebraic for all integers , then the Grothendieck standard conjecture on the algebraicity of the operators and holds for . We prove for an Abelian scheme under the assumption that for some geometric fibre of non-exceptional dimension.

A Note on Algebraic Γ-Monomials and Double Coverings

We discuss algebraic Γ-monomials of Deligne. Deligne used the theory of Hodge Cycles to show that algebraic Γ-monomials generate Kummer extensions of certain cyclotomic fields. Das, using a double complex of Anderson and Deligne's results, showed that certain powers of algebraic Γ-monomials and certain square roots of sine monomials generate abelian extensions of Q. Das also gave one example of a nonabelian double covering of a cyclotomic field generated by the square root of a sine monomial. In this note, we will produce infinitely many examples of non-abelian double coverings of cyclotomic fields of Das type. The construction of the examples depends in an interesting way on a lemma of Gauss figuring in an elementary proof of quadratic reciprocity.

The Hodge conjecture for general Prym varieties

The space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group.

Algebraic gamma monomials and double coverings of cyclotomic fields

We investigate the properties of algebraic gamma monomials—that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex S K {\mathbb {SK}} , to compute H ∗ ( ± , U ) H^*(\pm , {\mathbb {U}}) , where U {\mathbb {U}} is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension K / F K/F , we define a double covering of K / F K/F to be an extension K ~ / K \tilde {K}/K of degree ≤ 2 \leq 2 , such that K ~ / F {\tilde {K}}/F is Galois. We demonstrate that each class a ∈ H 2 ( ± , U ) {\mathbf {a}}\in H^2(\pm , {\mathbb {U}}) gives rise to a double covering of Q ( ζ ∞ ) / Q {\mathbb {Q}}(\zeta _ \infty )/{\mathbb {Q}} , by Q ( ζ ∞ , sin ⁡ a ) / Q ( ζ ∞ ) {\mathbb {Q}}(\zeta _ \infty ,\sqrt {\sin {\mathbf {a}}})/{\mathbb {Q}}(\zeta _ \infty ) . When a {\mathbf {a}} lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if a {\mathbf {a}} represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of Q ( ζ ∞ , sin ⁡ a ) / Q {\mathbb {Q}}(\zeta _ \infty ,\sqrt {\sin {\mathbf {a}}})/{\mathbb {Q}} is abelian and hence sin ⁡ a ∈ Q ( ζ ∞ ) \sqrt {\sin {\mathbf {a}}} \in {\mathbb {Q}}(\zeta _ \infty ) . The sin ⁡ a \sqrt {\sin {\mathbf {a}}} may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

ABELIAN VARIETIES OF TYPE III AND THE HODGE CONJECTURE

We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.

Density of members with extra Hodge cycles in a family of Hodge structures

Density of members with extra Hodge cycles in a family of Hodge structures

Hodge cycles on the jacobian variety of the Catalan curve

The jacobian variety of the Catalan curve yq = xp - 1is shown to be nondegenerate. As an application, a 0-1-matrix whose determinantcomputes the relative class number of the pq-th cyclotomic field is constructed.

Hodge cycles on Kuga fiber varieties

AbstractWe determine the dimension of the space of Hodge cycles for the generic fibers of the Kuga fiber varieties associated to certain quaternion algebras.

A remark on Hodge cycles on moduli spaces of rank 2 bundles over curves.

Article A remark on Hodge cycles on moduli spaces of rank 2 bundles over curves. was published on January 1, 1996 in the journal Journal für die reine und angewandte Mathematik (volume 1996, issue 470).

Invariants and Hodge cycles, III

This chapter focuses on invariants and Hodge cycles. It discusses that the problem of determining the space of Hodge cycles is reducible to the problem of determining the space of group invariants. However, unlike the ordinary invariants-theory, this invariants-theory is a bit complicated, because it relates also to a combinatrix of the Galois-group. The chapter describes the character-algebra of SL (2, C ). It presents that when a representation P of SL (2, C ) is a linear combination of 1, X 2 , X 4 ,….with non-negative coefficients, P is said to be even . It presents the character-algebra of a product of SL 2 ( C ). Chemistry and abelian scheme is described in the chapter.

Invariants and Hodge cycles, IV

Invariants and Hodge cycles, IV

Fields of definition for some Hodge cycles

Fields of definition for some Hodge cycles

Branching Rules and Hodge Cycles on Certain Abelian Varieties

Branching Rules and Hodge Cycles on Certain Abelian Varieties

Tate's Conjecture for K3 Surfaces of Finite Height

This paper extends the proof [16] of the Tate conjecture for ordinary K3 surfaces over a finite field to the more general case of all K3's of finite height. As in [16], our method is to find a of the K3 to characteristic zero with sufficiently many Hodge cycles. In the ordinary case, the so-called canonical lifting of Deligne and Illusie [7] did the job, and a study of the Galois action on p-adic etale cohomology revealed the Hodge cycles. Here we use more general quasi-canonical liftings, and the action of the crystalline Weil group on

On some arithmetic problems related to the Hodge cycles on the fermat varieties

On some arithmetic problems related to the Hodge cycles on the fermat varieties

On some arithmetic problems related to the hodge cycles on the fermat varieties

On some arithmetic problems related to the hodge cycles on the fermat varieties

Periods of linear algebraic cycles

In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the homology class of such algebraic cycles. This is known as the variational Hodge conjecture.