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We introduce the \emph{Frobenius slope envelope}~$\Env^i(X)$ of a smooth proper $\mathcal{O}_K$–scheme~$X$ in mixed characteristic, defined as the lower convex hull of the crystalline Newton polygon of \[ D_i := H^i_{\mathrm{cris}}(X_k/W(k))\otimes_{W(k)}K_0 \] and the Hodge–Tate polygon of the $p$-adic Galois representation \[ V^i := H^i_{\et}(X_K,\Q_p). \] Under good reduction $(N=0)$, we prove the unconditional dominance \[ \Brk(V^i)\;\preceq\;\Env^i(X), \] hence $\Swan_i(X/K)=0$ in our Artin–Swan normalization. In the semistable setting, we obtain an explicit Swan bound \[ \Swan_i(X/K)\;\le\;\sum_{\lambda} m_\lambda\bigl(C_i\,\lambda+\nu_i\bigr), \] where $\lambda$ runs over Frobenius slopes of~$D_i$ with multiplicities~$m_\lambda$, $\nu_i$ is the nilpotency index of~$N$ on the $(\varphi,N,\Fil)$–module attached to~$V^i$, and $C_i>0$ depends only on~$i$ (identified via Serre’s upper/lower numbering conversion and Deligne’s monodromy-filtration bounds in our normalization). We establish functoriality and a Künneth-type \emph{Minkowski additivity} \[ \Env^{i+j}(X\times Y)\;=\;\Env^{i}(X)\;\boxplus\;\Env^{j}(Y), \] and prove openness of the bounded-envelope locus in families. Conditionally on a canonical break-control from $(\varphi,N,\Fil)$–data, we give an equality criterion \[ \Brk(V^i)=\Env^i(X) \] (split slope filtration compatible with Hodge filtration and minimal monodromy~$\nu_i=1$). Worked cases (ordinary vs.\ supersingular) and base-change behavior (Herbrand reindexing) illustrate sharpness. Arithmetic applications include conductor control in modular/Shimura families and consequences for local factors of~$L$-functions.
Kundnani R. T., Kant Sh., Alam K. 2025. Frobenius Slope Envelopes and Ramification Bounds in Mixed Characteristic. PREPRINTS.RU. https://doi.org/10.24108/preprints-3113860
