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
editClassically, level structures on elliptic curves 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 for in the upper-half plane. Then, the lattice generated by gives a lattice which contains all -torsion points on the elliptic curve denoted . In fact, given such a lattice is invariant under the action on , where
hence it gives a point in [4] called the moduli space of level N structures of elliptic curves , which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing
gives a point in the -th roots of unity, hence in .
Example: an abelian scheme
editLet 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 such that[5]
- for each geometric point , form a basis for the group of points of order n in ,
- is the identity section, where is the multiplication by n.
See also: modular curve#Examples, moduli stack of elliptic curves.
See also
editNotes
edit- ^ Mumford, Fogarty & Kirwan 1994, Ch. 7.
- ^ Katz & Mazur 1985, Introduction
- ^ Deligne, P.; Husemöller, D. (1987). "Survey of Drinfeld's modules" (PDF). Contemp. Math. 67 (1): 25–91. doi:10.1090/conm/067/902591.
- ^ 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.
{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) - ^ Mumford, Fogarty & Kirwan 1994, Definition 7.1.
References
edit- Drinfeld, V. (1974). "Elliptic modules". Math USSR Sbornik. 23 (4): 561–592. Bibcode:1974SbMat..23..561D. doi:10.1070/sm1974v023n04abeh001731.
- Katz, Nicholas M.; Mazur, Barry (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. Vol. 151. Princeton University Press. ISBN 978-1-4008-3720-5.
- Mumford, David; Fogarty, J.; Kirwan, F. (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Vol. 34 (3rd ed.). Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.
Further reading
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