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 ${\displaystyle E=\mathbb {C} /\Lambda }$ 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 ${\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau }$ for ${\displaystyle \tau \in {\mathfrak {h}}}$ in the upper-half plane. Then, the lattice generated by ${\displaystyle 1/n,\tau /n}$ gives a lattice which contains all ${\displaystyle n}$-torsion points on the elliptic curve denoted ${\displaystyle E[n]}$. In fact, given such a lattice is invariant under the ${\displaystyle \Gamma (n)\subset {\text{SL}}_{2}(\mathbb {Z} )}$ action on ${\displaystyle {\mathfrak {h}}}$, where

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

hence it gives a point in ${\displaystyle \Gamma (n)\backslash {\mathfrak {h}}}$ ^[4] called the moduli space of level N structures of elliptic curves ${\displaystyle Y(n)}$, which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing

${\displaystyle e_{n}\left({\frac {1}{n}},{\frac {\tau }{n}}\right)=e^{2\pi i/n}}$

gives a point in the ${\displaystyle n}$-th roots of unity, hence in ${\displaystyle \mathbb {Z} /n}$.

Example: an abelian scheme

Let ${\displaystyle X\to S}$ 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 ${\displaystyle \sigma _{1},\dots ,\sigma _{2g}}$ such that ^[5]

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

See also: modular curve#Examples, moduli stack of elliptic curves.

See also

• Siegel modular form
• Rigidity (mathematics)
• Local rigidity

Notes

1. Mumford, Fogarty & Kirwan 1994 , Ch. 7.
2. Katz & Mazur 1985 , Introduction
3. Deligne, P.; Husemöller, D. (1987). "Survey of Drinfeld's modules" (PDF). Contemp. Math. 67 (1): 25–91. doi:10.1090/conm/067/902591.
4. 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.
5. 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. 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)]. 34 (3rd ed.). Springer-Verlag. ISBN   978-3-540-56963-3. MR   1304906.

Further reading

• Notes on principal bundles
• J. Lurie, Level Structures on Elliptic Curves.