A Fourier expansion principle for coherent cohomology of powers of Tate curves

Source abstract

In the case of the totally degenerate split degenerations given by fibered powers of Tate curves over $\mathbf{Z}[[q^{1/w}]]$, this article proves a Fourier expansion principle that applies to higher coherent (equivariant) cohomology classes of rigidified line bundles on the $\mathbf{Z}((q^{1/w}))$-fiber. We do not require any ampleness hypotheses. This generalizes a previous Fourier expansion principle due to Faltings--Chai that applies only to sections of ample line bundles; that version was used as a key step in their proof of their classification of polarized degenerations and thus in the construction of toroidal compactifications of moduli spaces of polarized abelian varieties. This article can be considered as a theory of Fourier expansions for algebraic theta cocycles associated to possibly indefinite quadratic forms, and has application to arithmetic: in joint work with Venkatesh, we have used it to produce an algebraic theory of Kudla--Millson indefinite modular theta functions.