Abelian Varieties Of CM-type Research Articles

Finiteness theorems for K3 surfaces and abelian varieties of CM type

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford–Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.

Galois representations attached to abelian varieties of CM type

Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we also describe the full Galois group of $K(A_{\text{tors}})/K$. This makes explicit previous results of Serre and Ribet, and strengthens a theorem of Banaszak, Gajda and Kraso\'n. Our bounds are especially sharp in case the CM type of $A$ is nondegenerate.

Average of the first invariant factor of the reductions of abelian varieties of CM type

For a field of definition [Formula: see text] of an abelian variety [Formula: see text] and prime ideal [Formula: see text] of [Formula: see text] which is of a good reduction for [Formula: see text], the structure of [Formula: see text] as abelian group is: [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We are interested in finding an asymptotic formula for the number of prime ideals [Formula: see text] with [Formula: see text], [Formula: see text] has a good reduction at [Formula: see text], [Formula: see text]. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.

Discrete tomography and the Hodge conjecture for certain abelian varieties of CM-type

We study a problem in discrete tomography on Z, and show that there is an intimate connection between the problem and the study of the Hodge cycles on abelian varieties of CM-type. This connection enables us to apply our results in tomography to obtain several infinite families of abelian varieties for which the Hodge conjecture holds.

Hodge structures on abelian varieties of CM-type

Hodge structures on abelian varieties of CM-type

ABELIAN VARIETIES AND THE GENERAL HODGE CONJECTURE

The Grothendieck version of the general Hodge conjecture is proved for certain simple complex abelian varieties of the 1st type with regard to Albert's classification and for certain abelian varieties of CM-type.

On the Mumford-Tate group of an Abelian variety with complex multiplication

Let (K, @) be a primitive CM-type with [K: O] = 2n (for definitions and previous results see Section 1.1). Fix n, and consider the collection s(n) = {Rank(@)}, where Rank(@) counts the number of independent translates of Qi under the Galois action and (K, @) ranges over all primitive types. The smallest element of S(n), denoted by B(n), is referred to as the sharp lower bound for the rank in dimension n. As was brought to the author’s attention by Ribet, bounds on B(n) of the form p + 1, for p a prime dividing n, follow directly from the proof of Ribet’s Nondegeneracy Theorem [21]. This is recorded as Theorem 1.4. Ribet’s method also gives bounds of the form 2q for q a prime with q2 dividing n, as is observed in Theorem 1.12. When combined with the author’s constructions of Abelian varieties in [S, 91, we obtain the precise value of B(n) for many values of n (Corollaries 1.5 to 1.8 and 1.13). We recall that these constructions use the analytic method of Weil and Shimura, together with new results on the reflex field from an investigation suggested to the author by Shim’ura. The interest of the rank comes from the theory of complex multiplication. If A is an Abelian variety of CM-type (K, @), then the Kubota Rank of A is Rank (@), and controls properties of the classfields constructed from A as in Kubota [ 161 and Ribet [20]. This is connected with the fact that the rank of @ is also the dimension of the Mumford-Tate group of A, and with the relation of this group to the I-adic representations of A, as in Serre [24, 251. The main body of the paper contains results that provide information on S(n). The main result, Theorem 2.5, asserts that when n is odd there is a computable subset S,‘,,,(n) of S(n) that accounts for the ranks of many CM-types on most CM-fields. More precisely, let K, be the maximal totally real subfield of K, and Kg be the Galois closure of K,. We consider the per-

Hodge Classes on Certain Types of Abelian Varieties

The Hodge conjecture for A states that such classes are algebraic in the sense that they are Q-linear combinations of cohomology classes of algebraic subvarieties of codimension p on A. This statement is well known for p = 1 (Lefschetz), and it is trivially true for p = 0. Hence, for a given A, it holds for allp if the ring formed by the Hodge classes under cup product is generated by the Hodge classes of codimension 1. We thus have a simple criterion for the Hodge conjecture to be true for A, which we call the (1,1) criterion. Responding to a question posed by Tate ([14, p. 107]), Mumford found that the (1,1) criterion, while true for all A such that dim(A) c 3, is not necessarily satisfied by abelian varieties of dimension 4. Although Mumford's counterexample involved an abelian variety of CM type [51, Weil [15] has stressed that its most important feature is the action of an imaginary quadratic field k on A in such a way that Lie(A) becomes a free k X C-module. (Abelian varieties of precisely this type play an important role in Deligne's proof [2] that Hodge cycles on an arbitrary abelian variety are absolutely Hodge.) There is speculation that the Hodge conjecture is false, at least in general, for such abelian varieties, but no method is available at present for settling this question either way. Indeed, one knows no example of an algebraic variety for which the Hodge conjecture is false, and only scattered examples (Shioda [12]) of