#### Vol. 2, No. 2, 2020

 Recent Issues Volume 4, Issue 4 Volume 4, Issue 3 Volume 4, Issue 2 Volume 4, Issue 1 Volume 3, Issue 4 Volume 3, Issue 3 Volume 3, Issue 2 Volume 3, Issue 1 Volume 2, Issue 4 Volume 2, Issue 3 Volume 2, Issue 2 Volume 2, Issue 1 Volume 1, Issue 4 Volume 1, Issue 3 Volume 1, Issue 2 Volume 1, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN (electronic): 2576-7666 ISSN (print): 2576-7658 Author Index To Appear Other MSP Journals
On $p$-adic vanishing cycles of log smooth families

### Shuji Saito and Kanetomo Sato

Vol. 2 (2020), No. 2, 309–335
##### Abstract

In this paper, we will prove that the sheaf of $p$-adic vanishing cycles on a regular log smooth family is generated by Milnor symbols, assuming that the base dvr contains a primitive $p$-th root of unity. Our result generalizes the surjectivity results of Bloch and Kato (Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152) and Hyodo (Invent. Math. 91:3 (1988), 543–557) to a regular log smooth case.

##### Keywords
$p$-adic vanishing cycles, Milnor $K$-groups, log smooth families
##### Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 11G45, 14F30