Moduli stack of elliptic curves

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

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Smooth Deligne-Mumford stack

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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References

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  1. ^ 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.
  2. ^ a b 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. ^ a b 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.
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