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]

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

${\begin{aligned}\Gamma (n)&={\text{ker}}({\text{SL}}_{2}(\mathbb {Z} )\to {\text{SL}}_{2}(\mathbb {Z} /n))\\&=\left\{M\in {\text{SL}}_{2}(\mathbb {Z} ):M\equiv {\begin{pmatrix}1&0\\0&1\end{pmatrix}}{\text{ (mod n)}}\right\}\end{aligned}}$

hence it gives a point in $\Gamma (n)\backslash {\mathfrak {h}}$ [4] 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[5]

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, 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)
Mumford, Fogarty & Kirwan 1994, Definition 7.1.

References

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.

Level structure (algebraic geometry)

Level structures on elliptic curves

Example: an abelian scheme

See also

Notes

References

Further reading