Drinfel'd-Ihara Relations for the Crystalline Frobenius

Drinfeld modular curves have many points

Ihara's proof that the reduction of the modular curve $X_0(n)$ at a prime $p$ not dividing $n$ has many points over a quadratic extension is adapted to the drinfeld modular curves $X_0(n)$. In order to do so, some properties of drinfeld modular varieties that have no proof in the literature are proven. The text is more or less self-contained.

Frobenius and monodromy operators in rigid analysis, and Drinfel'd's symmetric space

We define Frobenius and monodromy operators on the de Rham cohomology of $K$-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction $Y$, over a complete discrete valuation ring $K$ of mixed characteristic. For this we introduce log rigid cohomology and generalize the so called Hyodo-Kato isomorphism to versions for non-proper $Y$, for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of $Y$. We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel'd's …

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。