The ramification of p-adic representations coming from geometry

Abstract: The purpose of this talk is to explain a geometric analogue of Sen's classical theorem, which describes the close relationship between $p$-adic Lie filtrations and ramification filtrations for $p$-adic fields. Let $X$ be a smooth variety over a perfect field $k$ with characteristic $p>0$, let $D\subset X$ be a reduced divisor with smooth normal crossings, and let $U=X\backslash D$. Consider a continuous representation $\rho:\pi_1(U) \to GL(\Z_p)$, which gives rise to an $p$-adic Lie tower of \'etale covers $U_n \to U$. We may associate to each cover a Swan divisor $sw(U_n/U)$, supported on $D$, using the ramification filtration of Abbes-Saito. In general, the growth of these divisors can be arbitrarily wild. Instead, we restrict ourselves to representations that are ordinary geometric (e.g. $\rho$ arises as the $p$-adic Tate module of a family of ordinary Abelian varieties). Our main result states that for $\rho$ ordinary geometric, there exists integers $c_1>c_0>0$ such that $c_1p^{2n} D > sw(U_n/U) > c_0 p^{2n}D$. This says that even though $\rho$ has infinite monodromy, the Swan conductors $sw(U_n/U)$ grow as `slowly as possible'.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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