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.{{harvnb|Mumford|Fogarty|Kirwan|1994|loc=Ch. 7.}}{{harvnb|Katz|Mazur|1985|loc=Introduction}}

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 {{harv|Drinfeld|1974}}.{{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|doi=10.1090/conm/067/902591}}

Level structures on elliptic curves

Classically, level structures on elliptic curves E = \mathbb{C}/\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{Z}\oplus \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 \Gamma(n) \subset \text{SL}_2(\mathbb{Z}) action on \mathfrak{h}, where

\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}

hence it gives a point in \Gamma(n)\backslash\mathfrak{h}{{Cite book|last=Silverman, Joseph H., 1955-|title=The arithmetic of elliptic curves|date=2009|publisher=Springer-Verlag|isbn=978-0-387-09494-6|edition=2nd|location=New York|pages=439–445|oclc=405546184}} 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\left(\frac{1}{n}, \frac{\tau}{n}\right) = e^{2\pi i /n}
gives a point in the n-th roots of unity, hence in \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{{harvnb|Mumford|Fogarty|Kirwan|1994|loc=Definition 7.1.}}

  1. 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,
  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

Notes

{{reflist}}

References

  • {{cite journal |first=V. |last=Drinfeld |title=Elliptic modules |journal=Math USSR Sbornik |volume=23 |year=1974 |issue=4 |pages=561–592 |doi=10.1070/sm1974v023n04abeh001731 |bibcode=1974SbMat..23..561D }}
  • {{cite book

| last1 = Katz

| first1 = Nicholas M.

| author-link = Nick Katz

| last2=Mazur |first2=Barry |author-link2=Barry Mazur

| title = Arithmetic Moduli of Elliptic Curves

| publisher = Princeton University Press

| date = 1985

| isbn =0-691-08352-5 }}

  • {{cite book |first1=Michael |last1=Harris |first2=Richard |last2=Taylor |title=The Geometry and Cohomology of Some Simple Shimura Varieties |series=Annals of Mathematics Studies |volume=151 |url=https://books.google.com/books?id=IokNBAAAQBAJ&pg=PP1 |date=2001 |publisher=Princeton University Press |isbn=978-1-4008-3720-5 }}
  • {{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 | 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}}

Further reading

  • [http://www.math.harvard.edu/~lurie/282ynotes/LectureIII-Cohomology.pdf Notes on principal bundles]
  • J. Lurie, [http://www.math.harvard.edu/~lurie/papers/LevelStructures1.pdf Level Structures on Elliptic Curves. ]

Category:Algebraic geometry

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