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.}}

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

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}}

level structure (algebraic geometry)

Level structures on elliptic curves

Example: an abelian scheme

See also

Notes

References

Further reading