$p$-adic nearby cycles and $q$-Higgs syntomic complexes with coefficients

Organizers

Hansheng Diao , Heng Du , Yueke Hu , Huajie Li , Bin Xu , Yihang Zhu

Speaker

Abhinandan

Time

Monday, June 8, 2026 10:00 AM - 11:00 AM

Venue

Shuangqing-C654

Abstract

Let $\mathfrak{X}$ be a quasi-compact, separated, smooth $p$-adic formal scheme over the ring of integers $\mathcal{O}$ of the completed algebraic closure $C$ of $\mathbb{Q}_p$. In their joint work, Bhatt, Morrow and Scholze defined a syntomic complex for $\mathfrak{X}$ in terms of the $A\Omega$-complex (which computes the $A_{\inf}$-cohomology of $\mathfrak{X}$), and compared it to the complex of $p$-adic nearby cycles. In this talk, we will look at a generalisation of their result to the case of coefficients. More precisely, we will work with a finite locally free $F$-crystal $\mathcal{F}$ over the relative prismatic site of $\mathfrak{X}/A_{\inf}$, and define a syntomic complex with coefficients in $\mathcal{F}$. Then, we will show that the syntomic complex naturally compares to the complex of $p$-adic nearby cycles with coefficients in the associated $\mathbb{Z}_p$-local system, after truncation in appropriate degrees. The relationship between the two sides is established using $q$-Higgs complexes and $A\Omega$-complexes with coefficients in relative Breuil--Kisin--Fargues modules. This talk will be based on a joint work in progress with Takeshi Tsuji.