Appendix to ``Deligne's integrality theorem in unequal characteristic and rational points over finite fields'' by Hélène Esnault

Deligne's integrality theorem in unequal characteristic and rational points over finite fields; and Appendix (w/Pierre Deligne)

If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its étale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective regular model carries a rational point. As a consequence, if the Chow group of 0-cycles of $V$ over a large algebraically closed field is trivial, then the mod $p$ reduction of a projective regular model carries a rational point.

Compatible systems and ramification

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on $\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

étale 和 crystalline 伴生体，II

Etale and crystalline companions, II

本文基于Deligne的猜想，证明了在有限域上光滑方案上，任何ℓ-adicWeilsheaf都存在crystalline伴生体，通过Langlands对应和Drinfeld构造，转移了晶格到étale系数对象的性质。

Ramification theory and perfectoid spaces

Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that the modulo s^m of the extension O_L/O_K and modulo t^m of O_E/O_F are isomorphic. Let j=<m be a positive rational number. In this paper, we prove that the ramification of L/K is bounded by j if and only if the ramification of E/F is bounded by j. As an application, we prove that the categories of finite separable extensions of K and F whose ramifications ar…

Higher Ramanujan equations I: moduli stacks of abelian varieties and higher Ramanujan vector fields

We describe a higher dimensional generalization of Ramanujan's differential equations satisfied by the Eisenstein series $E_2$, $E_4$, and $E_6$. This will be obtained geometrically as follows. For every integer $g\ge 1$, we construct a moduli stack $\mathcal{B}_g$ over $\mathbf{Z}$ classifying principally polarized abelian varieties of dimension $g$ equipped with a suitable additional structure: a symplectic-Hodge basis of its first algebraic de Rham cohomology. We prove that $\mathcal{B}_g$ is a smooth Deligne-Mumford stack over $\mathbf{Z}$ of relative dimension $2g^2 + g$ and that $\mathca…