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 ). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in .
Level structures on elliptic curves.
Classically, level structures on elliptic curves formula_1 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 formula_2 for formula_3 in the upper-half plane. Then, the lattice generated by formula_4 gives a lattice which contains all formula_5-torsion points on the elliptic curve denoted formula_6. In fact, given such a lattice is invariant under the formula_7 action on formula_8, whereformula_9hence it gives a point in formula_10 called the moduli space of level N structures of elliptic curves formula_11, which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairingformula_12gives a point in the formula_5-th roots of unity, hence in formula_14.
Example: an abelian scheme.
Let formula_15 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 formula_16 such that
See also: modular curve#Examples, moduli stack of elliptic curves.