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 #Example: an abelian scheme). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in Template:Harv.

Level structures on elliptic curves

Classically, level structures on elliptic curves $E=ℂ/\mathrm{\Lambda }$ 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 $\mathbb{Z}\oplus \mathbb{Z}\cdot \tau$ for $\tau \in 𝔥$ 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 $\mathrm{\Gamma }(n)\subset {\text{SL}}_2(\mathbb{Z})$ action on $𝔥$, where

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

hence it gives a point in $\mathrm{\Gamma }(n)\mathrm{\setminus }𝔥$ 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

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

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 ${\bar{X}}_s$,
2. $m_n\circ {\sigma }_i$ is the identity section, where $m_n$ is the multiplication by n.

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

See also

• Siegel modular form
• Rigidity (mathematics)
• Local rigidity

References

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Further reading

• J. Lurie, Notes on principal bundles
• J. Lurie, Level Structures on Elliptic Curves.