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Level structure (algebraic geometry)

释义

Example: an abelian scheme
See also
Notes
References
Further reading

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 an abelian scheme). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in {{harv|Drinfeld|1974}}.^[3]

Example: an abelian scheme

Let 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^[4]

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.

Notes

1. ^{{harvnb|Mumford|loc=Ch. 7.}}
2. ^{{harvnb|Katz–Mazur|loc=Introduction}}
3. ^{{cite journal|url=http://publications.ias.edu/sites/default/files/Number59.pdf|title=Survey of Drinfeld's modules|last1=Deligne|first1=P.|last2=Husemöller|first2=D.|year=1987|journal=Contemp. Math|volume=67|issue=1|pages=25–91}}
4. ^{{harvnb|Mumford|loc=Definition 7.1.}}

References

V. Drinfeld, "`Elliptic modules"', Math USSR Sbornik, vol. 23(1974), No.4
{{cite book

| last = Katz
| first = Nicholas M
| authorlink = Nick Katz
|author2=Mazur, Barry |authorlink2=Barry Mazur
| title = Arithmetic Moduli of Elliptic Curves
| publisher = Princeton University Press
| date = 1985
| location =
| pages =
| url =
| doi =
| id =
| 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
{{Cite book| last1=Mumford | first1=David | author1-link=David Mumford | last2=Fogarty | first2=J. | last3=Kirwan | first3=F. | author3-link=Frances Kirwan | title=Geometric invariant theory | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)] | isbn=978-3-540-56963-3 |mr=1304906 | year=1994 | volume=34}}

Example: an abelian scheme

See also

Notes

References

Further reading