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 $$E = \mathbb{C}/\Lambda$$ 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 $$\mathbb{Z}\oplus \mathbb{Z}\cdot \tau$$ for $$\tau \in \mathfrak{h}$$ in the upper-half plane. Then, the lattice generated by $$1/n, \tau/n$$ gives a lattice which contains all $$n$$-torsion points on the elliptic curve denoted $$E[n]$$. In fact, given such a lattice is invariant under the $$\Gamma(n) \subset \text{SL}_2(\mathbb{Z})$$ action on $$\mathfrak{h}$$, where $$\begin{align} \Gamma(n) &= \text{ker}(\text{SL}_2(\mathbb{Z}) \to \text{SL}_2(\mathbb{Z}/n)) \\ &= \left\{ M \in \text{SL}_2(\mathbb{Z}) : M \equiv \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \text{ (mod n)} \right\} \end{align}$$ hence it gives a point in $$\Gamma(n)\backslash\mathfrak{h}$$ called the moduli space of level N structures of elliptic curves $$Y(n)$$, which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing"$e_n\left(\frac{1}{n}, \frac{\tau}{n}\right) = e^{2\pi i /n}$"gives a point in the $$n$$-th roots of unity, hence in $$\mathbb{Z}/n$$.

Example: an abelian scheme
Let $$X \to S$$ 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 $$\sigma_1, \dots, \sigma_{2g}$$ such that
 * 1) for each geometric point $$s : S \to X$$, $$\sigma_{i}(s)$$ form a basis for the group of points of order n in $$\overline{X}_s$$,
 * 2) $$m_n \circ \sigma_i$$ is the identity section, where $$m_n$$ is the multiplication by n.

See also: modular curve, moduli stack of elliptic curves.