In the end, one needed such a theory to obtain any progress at all in their study it now seems that to carry their study further the machinery of algebraic geometry is needed. Hilbert modular forms are in some sense a generalization of modular forms to higher dimensions Hilbert suggested that their study would help develop a theory of multivariate complex functions. Sections of a certain line bundle correspond to Hilbert modular forms. If the extra structure also includes an embedding of the ring of integers of a totally real number field into the endomorphism ring of the abelian variety, one obtains the Hilbert moduli space. If the extra structure is just a principal polarization and some level structure, then the moduli space is the Siegel moduli space over the complex numbers, this appears as a quotient of a space of matrices. Curves on the moduli space correspond to families of abelian varieties. With sufficient extra structure, this is a fine moduli space, so that every point on the moduli space corresponds to a single abelian variety with extra structure. More precisely, the classification of abelian varieties with suitable extra structure is a moduli problem, and there is an associated moduli space. These conditions, abstracted away from their context, form the definition of a sheaf.Ībelian varieties (with some extra data, such as a principal polarization and possibly a level strucutre) can be classified up to isomorphism by a scheme. Of course one can always restrict an analytic function to a smaller open set if all the restrictions of a function are zero, the function must be zero and if we have a collection of holomorphic functions on different open sets that agree on the overlaps, then they can be pieced together to give a holomorphic function on a union of the open sets.
It is not sufficient to describe the global functions, since a compact complex manifold only has constants as its global holomorphic functions one describes the holomorphic functions on any open set U. However, an alternative way of defining an analytic structure on a manifold is to describe all the holomorphic functions on the manifold. So it is extremely rare that an algebraic object has a neighborhood which is isomorphic to affine space. Unfortunately, the only natural definition for “neighborhood” on an algebraic object uses the Zariski topology, in which every open set is dense (in an irreducible object). From a geometric point of view, this amounts to saying that the property holds for a space if and only if it holds for every point in the space.Ĭlassically, one defines an analytic structure on a topological manifold by specifying a family of coordinate charts ( homeomorphisms from a neighborhood to ℂ n) that are suitably compatible. For example, many properties will hold for a ring if and only if they hold for the localization of that ring at every prime ideal. This can be useful, especially when it allows to apply one’s geometric intuition to purely algebraic problems. However, most results in algebraic geometry rely on difficult results in commutative algebra, so attacking a purely algebraic problem with these techniques usually amounts to reducing one algebraic problem to another. Techniques from algebraic geometry offer the potential of recognizing and taking advantage of this essential similarity. For example, the properties of number fields are very closely parallelled by the properties of nonsingular curves, particularly over finite fields. However, the tools that have been developed are so general they can sometimes be used to view a purely algebraic problem in a geometric light. They are generally used to study the algebraic analogs of geometric objects: curves defined over the rational numbers rather than the complex numbers, for example. Algebraic geometry defines the basic objects and constructs tools closely analogous to all these tools.
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