Complex multiplication of abelian varieties information
In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century. One of the major achievements in algebraic number theory and algebraic geometry of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension d > 1. The problem is at a deeper level of abstraction, because it is much harder to manipulate analytic functions of several complex variables.
The formal definition is that
the tensor product of End(A) with the rational number field Q, should contain a commutative subring of dimension 2d over Q. When d = 1 this can only be a quadratic field, and one recovers the cases where End(A) is an order in an imaginary quadratic field. For d > 1 there are comparable cases for CM-fields, the complex quadratic extensions of totally real fields. There are other cases that reflect that A may not be a simple abelian variety (it might be a cartesian product of elliptic curves, for example). Another name for abelian varieties of CM-type is abelian varieties with sufficiently many complex multiplications.
It is known that if K is the complex numbers, then any such A has a field of definition which is in fact a number field. The possible types of endomorphism ring have been classified, as rings with involution (the Rosati involution), leading to a classification of CM-type abelian varieties. To construct such varieties in the same style as for elliptic curves, starting with a lattice Λ in Cd, one must take into account the Riemann relations of abelian variety theory.
The CM-type is a description of the action of a (maximal) commutative subring L of EndQ(A) on the holomorphic tangent space of A at the identity element. Spectral theory of a simple kind applies, to show that L acts via a basis of eigenvectors; in other words L has an action that is via diagonal matrices on the holomorphic vector fields on A. In the simple case, where L is itself a number field rather than a product of some number of fields, the CM-type is then a list of complex embeddings of L. There are 2d of those, occurring in complex conjugate pairs; the CM-type is a choice of one out of each pair. It is known that all such possible CM-types can be realised.
Basic results of Goro Shimura and Yutaka Taniyama compute the Hasse–Weil L-function of A, in terms of the CM-type and a Hecke L-function with Hecke character, having infinity-type derived from it. These generalise the results of Max Deuring for the elliptic curve case.
and 23 Related for: Complex multiplication of abelian varieties information
all science. There is also the higher-dimensional complexmultiplication theory ofabelianvarieties A having enough endomorphisms in a certain precise...
He was known for developing the theory ofcomplexmultiplicationofabelianvarieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture...
first abelianvarieties to be studied were those defined over the field ofcomplex numbers. Such abelianvarieties turn out to be exactly those complex tori...
This is a timeline of the theory ofabelianvarieties in algebraic geometry, including elliptic curves. c. 1000 Al-Karaji writes on congruent numbers Fermat...
arithmetic ofabelianvarieties is the study of the number theory of an abelianvariety, or a family ofabelianvarieties. It goes back to the studies of Pierre...
(1961), Complex multiplicationofabelianvarieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Tokyo:...
on abelianvarieties require the idea ofabelianvariety up to isogeny for their convenient statement. For example, given an abelian subvariety A1 of A...
Another class is formed by the abelianvarieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem...
the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers...
higher-dimensional abelianvarieties, so the concept of dual becomes more interesting in higher dimensions. Let A be an abelianvariety over a field k. We...
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is...
basic topological information like the number of holes in certain geometric spaces, complex algebraic varieties, can be understood by studying the possible...
characters. This was to become important once the complex multiplicationofabelianvarieties became established. The Hecke characters in question were...
Jacobian varietyof the Fermat curve has been studied in depth. It is isogenous to a product of simple abelianvarieties with complexmultiplication. The...
general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group...
{A}}_{g}} of principally polarized complexabelianvarietiesof dimension g {\displaystyle g} (a principal polarization identifies an abelianvariety with...
1}G_{i}/G_{i+1}} is a Lie ring, with addition given by the group multiplication (which is abelian on each quotient group G i / G i + 1 {\displaystyle G_{i}/G_{i+1}}...
center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the...
multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number exp(φ i) = cos(φ) + i sin(φ) of...
variety is called a Kummer surface. Shimura, Goro (1998), Abelianvarieties with complexmultiplication and modular functions, Princeton Mathematical Series...
group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups...
homomorphism from the group ofcomplex numbers C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and...
Global Information