Talk:Moduli stack of elliptic curves

Todo

• Include GIT construction (find better substitute of https://arxiv.org/pdf/1111.3032.pdf)
• Arithmetic Moduli of Elliptic Curves. (AM-108) is the authoritative reference
• Mumford 129-139 contains construction for all abelian varieties, special this for elliptic curves
• Mumford GIT page 192 gives ${\mathcal {A}}_{1,(4,8)}$ isomorphic to $x^{4}=y^{4}+z^{4}$ minus twelve points (abelian varieties with level (4,8) structure)
• Silverman also discusses weierstrauss equations over Z
• https://www.math.ucla.edu/~mikehill/Research/surv-douglas2-201.pdf page 51 is an excellent reference for visualization
• Weighted projective space (https://arxiv.org/pdf/1604.02441.pdf)
• https://people.math.umass.edu/~tevelev/66-80.pdf and https://people.math.umass.edu/~tevelev/moduli797.pdf are excellent resources and contain more info, especially with the schotky problem

Characteristic 0

• Add compactification + line bundles over compactification + modular forms as modular functions extending to the compactification

Universal elliptic curve

• Page 16, show isomorphism of semidirect product as a matrix group
• Then give the induced action of this new matrix group on C \times h

Construction in general / over Spec(Z)

• Need Artin's criterion article to be updated.
• Starting at page 59 of pdf of Deligne Rappoport gives construction using Artin's criterion - http://smtp.math.uni-bonn.de/ag/alggeom/preprints/Lesschemas.pdf
• Definition of ${\mathcal {M}}$ is given on pdf page 54
• scheme of curves - pdf page 18
• generalized scheme of elliptic curves - pdf page 36 definition 1.12

Additional references

• TOPOLOGICAL MODULAR FORMS - Paul Goerss - https://sites.math.northwestern.edu/~pgoerss/papers/Exp.1005.P.Goerss.pdf — Preceding unsigned comment added by Wundzer (talk • contribs) 18:11, 19 June 2020 (UTC)