In mathematics, the moduli stack of elliptic curves, denoted as or , is an algebraic stack over classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves . In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme to it correspond to elliptic curves over . The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in .

Properties

Smooth Deligne-Mumford stack

The moduli stack of elliptic curves is a smooth separated Deligne–Mumford stack of finite type over , but is not a scheme as elliptic curves have non-trivial automorphisms.

j-invariant

There is a proper morphism of to the affine line, the coarse moduli space of elliptic curves, given by the j-invariant of an elliptic curve.

Construction over the complex numbers

It is a classical observation that every elliptic curve over is classified by its periods. Given a basis for its integral homology and a global holomorphic differential form (which exists since it is smooth and the dimension of the space of such differentials is equal to the genus, 1), the integrals

give the generators for a -lattice of rank 2 inside of [1] pg 158. Conversely, given an integral lattice of rank inside of , there is an embedding of the complex torus into from the Weierstrass P function[1] pg 165. This isomorphic correspondence is given by

and holds up to homothety of the lattice , which is the equivalence relation

It is standard to then write the lattice in the form for , an element of the upper half-plane, since the lattice could be multiplied by , and both generate the same sublattice. Then, the upper half-plane gives a parameter space of all elliptic curves over . There is an additional equivalence of curves given by the action of the

where an elliptic curve defined by the lattice is isomorphic to curves defined by the lattice given by the modular action

Then, the moduli stack of elliptic curves over is given by the stack quotient

Note some authors construct this moduli space by instead using the action of the Modular group . In this case, the points in having only trivial stabilizers are dense.

Fundamental domains of the action of on the upper half-plane are shown here as pairs of ideal triangles of different colors sharing an edge. The "standard" fundamental domain is shown with darker edges. Suitably identifying points on the boundary of this region, we obtain the coarse moduli space of elliptic curves. The stacky points at and are on the boundary of this region.

Stacky/Orbifold points

Generically, the points in are isomorphic to the classifying stack since every elliptic curve corresponds to a double cover of , so the -action on the point corresponds to the involution of these two branches of the covering. There are a few special points[2] pg 10-11 corresponding to elliptic curves with -invariant equal to and where the automorphism groups are of order 4, 6, respectively[3] pg 170. One point in the Fundamental domain with stabilizer of order corresponds to , and the points corresponding to the stabilizer of order correspond to [4]pg 78.

Representing involutions of plane curves

Given a plane curve by its Weierstrass equation

and a solution , generically for j-invariant , there is the -involution sending . In the special case of a curve with complex multiplication

there the -involution sending . The other special case is when , so a curve of the form

there is the -involution sending where is the third root of unity .

Fundamental domain and visualization

There is a subset of the upper-half plane called the Fundamental domain which contains every isomorphism class of elliptic curves. It is the subset

It is useful to consider this space because it helps visualize the stack . From the quotient map

the image of is surjective and its interior is injective[4]pg 78. Also, the points on the boundary can be identified with their mirror image under the involution sending , so can be visualized as the projective curve with a point removed at infinity[5]pg 52.

Line bundles and modular functions

There are line bundles over the moduli stack whose sections correspond to modular functions on the upper-half plane . On there are -actions compatible with the action on given by

The degree action is given by

hence the trivial line bundle with the degree action descends to a unique line bundle denoted . Notice the action on the factor is a representation of on hence such representations can be tensored together, showing . The sections of are then functions sections compatible with the action of , or equivalently, functions such that

This is exactly the condition for a holomorphic function to be modular.

Modular forms

The modular forms are the modular functions which can be extended to the compactification

this is because in order to compactify the stack , a point at infinity must be added, which is done through a gluing process by gluing the -disk (where a modular function has its -expansion)[2]pgs 29-33.

Universal curves

Constructing the universal curves is a two step process: (1) construct a versal curve and then (2) show this behaves well with respect to the -action on . Combining these two actions together yields the quotient stack

Versal curve

Every rank 2 -lattice in induces a canonical -action on . As before, since every lattice is homothetic to a lattice of the form then the action sends a point to

Because the in can vary in this action, there is an induced -action on

giving the quotient space

by projecting onto .

SL2-action on Z2

There is a -action on which is compatible with the action on , meaning given a point and a , the new lattice and an induced action from , which behaves as expected. This action is given by

which is matrix multiplication on the right, so

See also

References

  1. 1 2 Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-09494-6. OCLC 405546184.
  2. 1 2 Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG].
  3. Galbraith, Steven. "Elliptic Curves" (PDF). Mathematics of Public Key Cryptography. Cambridge University Press via The University of Auckland.
  4. 1 2 Serre, Jean-Pierre (1973). A Course in Arithmetic. New York: Springer New York. ISBN 978-1-4684-9884-4. OCLC 853266550.
  5. Henriques, André G. "The Moduli stack of elliptic curves". In Douglas, Christopher L.; Francis, John; Henriques, André G; Hill, Michael A. (eds.). Topological modular forms (PDF). Providence, Rhode Island. ISBN 978-1-4704-1884-7. OCLC 884782304. Archived from the original (PDF) on 9 June 2020 via University of California, Los Angeles.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.