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 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 for in the upper-half plane. Then, the lattice generated by gives a lattice which contains all -torsion points on the elliptic curve denoted . In fact, given such a lattice is invariant under the action on , where

hence it gives a point in ^{[4]} called the moduli space of level N structures of elliptic curves , which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing

gives a point in the -th roots of unity, hence in .

## 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

^{[5]}

- 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.

## See also

## 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.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.**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.