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[edit]
Smooth Deligne-Mumford stack[edit]
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[edit]
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[edit]
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
Stacky/Orbifold points[edit]
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[edit]
Given a plane curve by its Weierstrass equation
Fundamental domain and visualization[edit]
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
Line bundles and modular functions[edit]
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
Modular forms[edit]
The modular forms are the modular functions which can be extended to the compactification
Universal curves[edit]
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[edit]
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
SL2-action on Z2[edit]
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
See also[edit]
- Fundamental domain
- Homothety
- Level structure (algebraic geometry)
- Moduli of abelian varieties
- Shimura variety
- Modular curve
- Elliptic cohomology
References[edit]
- ^ a b Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-09494-6. OCLC 405546184.
- ^ a b Hain, Richard (2014-03-25). "Lectures on Moduli Spaces of Elliptic Curves". arXiv:0812.1803 [math.AG].
- ^ Galbraith, Steven. "Elliptic Curves" (PDF). Mathematics of Public Key Cryptography. Cambridge University Press – via The University of Auckland.
- ^ a b Serre, Jean-Pierre (1973). A Course in Arithmetic. New York: Springer New York. ISBN 978-1-4684-9884-4. OCLC 853266550.
- ^ 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.
- Hain, Richard (2008), Lectures on Moduli Spaces of Elliptic Curves, arXiv:0812.1803, Bibcode:2008arXiv0812.1803H
- Lurie, Jacob (2009), A survey of elliptic cohomology (PDF)
- Olsson, Martin (2016), Algebraic spaces and stacks, Colloquium Publications, vol. 62, American Mathematical Society, ISBN 978-1470427986
External links[edit]
- moduli+stack+of+elliptic+curves at the nLab
- "The moduli stack of elliptic curves", Stacks project