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 (Drinfeld 1974).
Level structures on elliptic curves Classically, level structures on elliptic curves E = C / Λ {\displaystyle 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 Z ⊕ Z ⋅ τ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau } for τ ∈ h {\displaystyle \tau \in {\mathfrak {h}}} in the upper-half plane. Then, the lattice generated by 1 / n , τ / n {\displaystyle 1/n,\tau /n} gives a lattice which contains all n {\displaystyle n} -torsion points on the elliptic curve denoted E [ n ] {\displaystyle E[n]} . In fact, given such a lattice is invariant under the Γ ( n ) ⊂ SL 2 ( Z ) {\displaystyle \Gamma (n)\subset {\text{SL}}_{2}(\mathbb {Z} )} action on h {\displaystyle {\mathfrak {h}}} , where Γ ( n ) = ker ( SL 2 ( Z ) → SL 2 ( Z / n ) ) = { M ∈ SL 2 ( Z ) : M ≡ ( 1 0 0 1 ) (mod n) } {\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}}} hence it gives a point in Γ ( n ) ∖ h {\displaystyle \Gamma (n)\backslash {\mathfrak {h}}} called the moduli space of level N structures of elliptic curves Y ( n ) {\displaystyle Y(n)} , which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing e n ( 1 n , τ n ) = e 2 π i / n {\displaystyle e_{n}\left({\frac {1}{n}},{\frac {\tau }{n}}\right)=e^{2\pi i/n}} gives a point in the n {\displaystyle n} -th roots of unity, hence in Z / n {\displaystyle \mathbb {Z} /n} .
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