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 (Drinfeld 1974).

Level structures on elliptic curves

Classically, level structures on elliptic curves

$E=\mathbb{ℂ}/\mathrm{\Lambda }$

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

$\mathbb{\mathbb{Z}}\oplus \mathbb{\mathbb{Z}}\cdot \tau$

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

$\mathrm{\Gamma }(n)\subset {\text{SL}}_2(\mathbb{\mathbb{Z}})$

action on

$\mathfrak{𝔥}$

, where

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

hence it gives a point in

$\mathrm{\Gamma }(n)\mathrm{\setminus }\mathfrak{𝔥}$

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

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

gives a point in the

$n$

-th roots of unity, hence in

$\mathbb{\mathbb{Z}}/n$

.

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

• Drinfeld, V. (1974). "Elliptic modules". Math USSR Sbornik 23 (4): 561–592. doi:10.1070/sm1974v023n04abeh001731. Bibcode: 1974SbMat..23..561D. 
• Katz, Nicholas M. (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN 0-691-08352-5. 
• Harris, Michael; Taylor, Richard (2001). The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies. 151. Princeton University Press. ISBN 978-1-4008-3720-5. https://books.google.com/books?id=IokNBAAAQBAJ&pg=PP1. 
• Mumford, David; Fogarty, J.; Kirwan, F. (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. 34 (3rd ed.). Springer-Verlag. ISBN 978-3-540-56963-3. 

Further reading

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