Jacob Lurie, Elliptic Cohomology I: Spectral Abelian Varieties, (pdf)
It is described as the first in a series of papers whose ultimate aim is to carry out the details of the program outlined in A Survey of Elliptic Cohomology.
This first paper provides a discussion of abelian varieties in the setting of spectral algebraic geometry. It sets out to answer the question:
What does it mean to give an elliptic curve (or, more generally, an abelian variety) over a cohomology theory $A$? To what extent do such objects behave like their counterparts in classical algebraic geometry?
It is followed up by the further parts
Jacob Lurie, Elliptic Cohomology II: Orientations, 2018. 288pp (pdf)
Jacob Lurie, Elliptic Cohomology III: Tempered Cohomology, 2019. 286pp (pdf)
Jacob Lurie, Elliptic Cohomology IV: Equivariant elliptic cohomology, to appear.
Contents
1 Abelian Varieties in Spectral Algebraic Geometry
1.1 Varieties over $E_{\infty}$-Rings
1.2 Abelian Group Objects of $\infty$-Categories
1.3 Commutative Monoid Objects of $\infty$-Categories
1.4 Abelian Varieties
1.5 Strict Abelian Varieties
2 Moduli of Elliptic Curves
2.1 Comparing Abelian Varieties with Strict Abelian Varieties
2.2 Deformation Theory of Abelian Varieties
2.3 Deformation Theory of Strict Abelian Varieties
2.4 The Moduli Stack of Elliptic Curves
3 Cartier Duality
3.1 Coalgebra Objects of $\infty$-Categories
3.2 Duality between Algebra and Coalgebra Objects
3.3 Bialgebra Objects of $\infty$-Categories
3.4 The Spectrum of a Bialgebra
3.5 The Affine Line
3.6 Smash Products of $E_{\infty}$-Spaces
3.7 The Cartier Dual of a Functor
3.8 Duality for Bialgebras
3.9 Duality for Hopf Algebras
4 Biextensions and the Fourier-Mukai Transform
4.1 Line Bundles and Invertible Sheaves
4.2 Biextensions of Abelian Varieties
4.3 Digression: Tannaka Duality
4.4 Biextensions: Tannakian Perspective
4.5 Categorical Digression
4.6 The Convolution Product
4.7 The Fourier-Mukai Transform
5 Duality Theory for Abelian Varieties
5.1 Perfect Biextensions
5.2 Dualizing Sheaves
5.3 Multiplicative Line Bundles
5.4 The Functor $\mathcal{P}ic^0_X$
5.5 Representability of the Functor $\mathcal{P}ic^m_X$
5.6 Existence of the Dual Abelian Variety
6 $p$-Divisible Groups
6.1 Finite Flat Group Schemes
6.2 Epimorphisms and Monomorphisms
6.3 Cartier Duality for Finite Flat Group Schemes
6.4 $p$-Torsion Objects of $\infty$-Categories
6.5 $p$-Divisible Groups
6.6 Cartier Duality for $p$-Divisible Groups
6.7 The $p$-Divisible Group of a Strict Abelian Variety
6.8 Comparison of Duality Theories
7 The Serre-Tate Theorem
7.1 Deformation Theory of the Functor $R \mapsto BT_h(R)$
7.2 The Case of a Trivial Square-Zero Extension
7.3 Proof of the Serre-Tate Theorem
7.4 Application: Lifting Abelian Varieties from Classical to Spectral Algebraic Geometry