Moduli stack of elliptic curves

In mathematics, the moduli stack of elliptic curves is an algebraic stack classifying elliptic curves. In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme S to it correspond to elliptic curves over S.

The moduli stack of elliptic curves is denoted by M_ell or by M_1,1, which is a special case of the moduli stack M_g,n of genus g curves with n marked points.

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

There is a proper morphism of M_1,1 to the affine line, given by the j-invariant of an elliptic curve, which makes the affine line into the coarse moduli space of M_1,1.

References

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, 62, American Mathematical Society, ISBN 978-1470427986

External links

moduli+stack+of+elliptic+curves in nLab
"The moduli stack of elliptic curves", Stacks project

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