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]

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^[4]

for each geometric point $s:S\to X$ , $\sigma _{i}(s)$ form a basis for the group of points of order n in ${\overline {X}}_{s}$ ,
$m_{n}\circ \sigma _{i}$ is the identity section, where $m_{n}$ is the multiplication by n.

Notes

↑ Mumford, Ch. 7.
↑ Katz–Mazur, Introduction
↑ Deligne, P.; Husemöller, D. (1987). "Survey of Drinfeld's modules" (PDF). Contemp. Math. 67 (1): 25–91.
↑ Mumford, Definition 7.1.

References

V. Drinfeld, "`Elliptic modules"', Math USSR Sbornik, vol. 23(1974), No.4
Katz, Nicholas M; Mazur, Barry (1985). Arithmetic Moduli of Elliptic Curves. Princeton University Press. ISBN 0-691-08352-5.
M. Harris, R. Taylor, "`The geometry and cohomology of some simple Shimura varieties"', Annals of Mathematical Studies 151, PUP 2001
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.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.

Level structure (algebraic geometry)

Example: an abelian scheme

See also

Notes

References

Further reading