On the p-adic local invariant cycle theorem

Abstract

The aim of this paper is to consider the $p$-adic local invariant cycle theorem in the mixed characteristic case. In the first part of the paper, via case-by-case discussion, we construct the $p$-adic specialization map, and then write out the complete conjecture in $p$-adic case. We proved the theorem in good reduction and semistable reduction cases. In the second part of the paper, by using Berthelot, Esnault and R\{u}lling's trace morphisms in [BER], we first prove the case of coherent cohomology, then we extend it to the Witt vector cohomology, and we then get a result on the Frobenius-stable part of the Witt vector cohomology, which corresponds the slope 0 part of the rigid cohomology, we then get the general $p$-adic local invariant cycle theorem. We also give another approach in the $H^0$ and $H^1$ cases in the general case. In the last part of the paper, based on Flach and Morin's work on the weight filtration in the $l$-adic case, we consider the $p$-adic analogous result (which, together with the $l$-adic's result, serves as a part to prove the compatibility of the Weil-etale cohomology with the Tamagawa number conjecture). This is a direct corollary of the local invariant cycle theorem by taking the weight filtration. And we also consider some typical examples that the weight filtration statement could be verified by direct computations.