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 .

Level structures on elliptic curves

Classically, level structures on elliptic curves $E = \mathbb/\Lambda$ 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 $\mathbb\oplus \mathbb\cdot \tau$ for $\tau \in \mathfrak$ 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 /math>. In fact, given such a lattice is invariant under the $\Gamma(n) \subset \text_2(\mathbb)$ action on $\mathfrak$, where

$\begin \Gamma(n) &= \text(\text_2(\mathbb) \to \text_2(\mathbb/n)) \\ &= \left\ \end$

hence it gives a point in $\Gamma(n)\backslash\mathfrak$ 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

$e_n\left(\frac, \frac\right) = e^$

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

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 $\sigma_1, \dots, \sigma_$ such that
# for each geometric point $s : S \to X$, $\sigma_(s)$ form a basis for the group of points of order ''n'' in $\overline_s$,
# $m_n \circ \sigma_i$ 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

Notes

References

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Further reading

Notes on principal bundles
* J. Lurie
Level Structures on Elliptic Curves.

Algebraic geometry

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