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.
[1][2] 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 (Drinfeld 1974).
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
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
{\displaystyle {\begin{aligned}\Gamma (n)&={\text{ker}}({\text{SL}}_{2}(\mathbb {Z} )\to {\text{SL}}_{2}(\mathbb {Z} /n))\\&=\left\{M\in {\text{SL}}_{2}(\mathbb {Z} ):M\equiv {\begin{pmatrix}1&0\\0&1\end{pmatrix}}{\text{ (mod n)}}\right\}\end{aligned}}}
[4] 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 pairing
-th roots of unity, hence in
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[5] See also: modular curve#Examples, moduli stack of elliptic curves.
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