Level structure (algebraic geometry)

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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

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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

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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]

  1. for each geometric point  ,   form a basis for the group of points of order n in  ,
  2.   is the identity section, where   is the multiplication by n.

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

See also

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Notes

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  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.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  5. ^ Mumford, Fogarty & Kirwan 1994, Definition 7.1.

References

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Further reading

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