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 . 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. 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. 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 integralsgive the generators for a -lattice of rank 2 inside of 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 pg 165. This isomorphic correspondence is given byand holds up to homothety of the lattice , which is the equivalence relationIt 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 thewhere an elliptic curve defined by the lattice is isomorphic to curves defined by the lattice given by the modular actionThen, the moduli stack of elliptic curves over is given by the stack quotientNote some authors construct this moduli space by instead using the action of the Modular group .

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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. 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.

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