Level structure (algebraic geometry)

In algebraic geometry, a level structure on a space X is an extra structure attached to X that shrinks or eliminates the automorphism group of X, by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of X. In applications, a level structure is used in the construction of moduli spaces; a moduli space is often constructed as a quotient. The presence of automorphisms poses a difficulty to forming a quotient; thus introducing level structures helps overcome this difficulty. There is no single definition of a level structure; rather, depending on the space X, one introduces the notion of a level structure. The classic one is that on an elliptic curve (see #Example: an abelian scheme). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in . Classically, level structures on elliptic curves are given by a lattice containing the defining lattice of the variety. From the moduli theory of elliptic curves, all such lattices can be described as the lattice for in the upper-half plane. Then, the lattice generated by gives a lattice which contains all -torsion points on the elliptic curve denoted . In fact, given such a lattice is invariant under the action on , wherehence it gives a point in called the moduli space of level N structures of elliptic curves , which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairinggives a point in the -th roots of unity, hence in . Let be an abelian scheme whose geometric fibers have dimension g. Let n be a positive integer that is prime to the residue field of each s in S. For n ≥ 2, a level n-structure is a set of sections such that for each geometric point , form a basis for the group of points of order n in , is the identity section, where is the multiplication by n. See also: modular curve#Examples, moduli stack of elliptic curves.