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.[1][2] 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).[3]
Level structures on elliptic curves
Classically, level structures on elliptic curves [math]\displaystyle{ E = \mathbb{C}/\Lambda }[/math] 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 [math]\displaystyle{ \mathbb{Z}\oplus \mathbb{Z}\cdot \tau }[/math] for [math]\displaystyle{ \tau \in \mathfrak{h} }[/math] in the upper-half plane. Then, the lattice generated by [math]\displaystyle{ 1/n, \tau/n }[/math] gives a lattice which contains all [math]\displaystyle{ n }[/math]-torsion points on the elliptic curve denoted [math]\displaystyle{ E[n] }[/math]. In fact, given such a lattice is invariant under the [math]\displaystyle{ \Gamma(n) \subset \text{SL}_2(\mathbb{Z}) }[/math] action on [math]\displaystyle{ \mathfrak{h} }[/math], where
[math]\displaystyle{ \begin{align} \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{align} }[/math]
hence it gives a point in [math]\displaystyle{ \Gamma(n)\backslash\mathfrak{h} }[/math][4] called the moduli space of level N structures of elliptic curves [math]\displaystyle{ Y(n) }[/math], which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing
[math]\displaystyle{ e_n\left(\frac{1}{n}, \frac{\tau}{n}\right) = e^{2\pi i /n} }[/math]
gives a point in the [math]\displaystyle{ n }[/math]-th roots of unity, hence in [math]\displaystyle{ \mathbb{Z}/n }[/math].
Example: an abelian scheme
Let [math]\displaystyle{ X \to S }[/math] 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 [math]\displaystyle{ \sigma_1, \dots, \sigma_{2g} }[/math] such that[5]
- for each geometric point [math]\displaystyle{ s : S \to X }[/math], [math]\displaystyle{ \sigma_{i}(s) }[/math] form a basis for the group of points of order n in [math]\displaystyle{ \overline{X}_s }[/math],
- [math]\displaystyle{ m_n \circ \sigma_i }[/math] is the identity section, where [math]\displaystyle{ m_n }[/math] is the multiplication by n.
See also: modular curve#Examples, moduli stack of elliptic curves.
See also
Notes
- ↑ Mumford, Fogarty & Kirwan 1994, Ch. 7.
- ↑ Katz & Mazur 1985, Introduction
- ↑ Deligne, P.; Husemöller, D. (1987). "Survey of Drinfeld's modules". Contemp. Math. 67 (1): 25–91. doi:10.1090/conm/067/902591. http://publications.ias.edu/sites/default/files/Number59.pdf.
- ↑ Silverman, Joseph H., 1955- (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. pp. 439–445. ISBN 978-0-387-09494-6. OCLC 405546184.
- ↑ Mumford, Fogarty & Kirwan 1994, Definition 7.1.
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
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