Level structure (algebraic geometry)

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 ${\displaystyle E=\mathbb{C}/\mathrm{\Lambda }}$ 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 ${\displaystyle \mathbb{Z}\oplus \mathbb{Z}\cdot \tau }$ for ${\displaystyle \tau \in \mathfrak{h}}$ in the upper-half plane. Then, the lattice generated by ${\displaystyle 1/n,\tau /n}$ gives a lattice which contains all ${\displaystyle n}$-torsion points on the elliptic curve denoted ${\displaystyle E[n]}$. In fact, given such a lattice is invariant under the ${\displaystyle \mathrm{\Gamma }(n)\subset {\text{SL}}_2(\mathbb{Z})}$ action on ${\displaystyle \mathfrak{h}}$, where

${\displaystyle \begin{array}{cc}\mathrm{\Gamma }(n)& =\text{ker}({\text{SL}}_2(\mathbb{Z})\to {\text{SL}}_2(\mathbb{Z}/n))\\ & =\left\{M\in {\text{SL}}_2(\mathbb{Z}):M\equiv \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\text{ (mod n)}\right\}\end{array}}$

hence it gives a point in ${\displaystyle \mathrm{\Gamma }(n)\mathrm{\setminus }\mathfrak{h}}$ called the moduli space of level N structures of elliptic curves ${\displaystyle Y(n)}$, which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing

${\displaystyle e_n\left(\frac{1}{n},\frac{\tau }{n}\right)=e^{2\pi i/n}}$

gives a point in the ${\displaystyle n}$-th roots of unity, hence in ${\displaystyle \mathbb{Z}/n}$.