Uniform local models of quasi-unipotent $p$-adic differential equations

Source abstract

We construct uniform analytic models of quasi-unipotent differential modules over Robba rings. For modules of rank at most $r$ and highest ramification break at most $b$, the inner radius of the model annulus depends only on $r,b$ and the residue characteristic $p$. At each radius, the model admits a basis, possibly depending on the radius, in which the connection matrix is bounded by a constant depending only on $r,b,p$. Both bounds are independent of the coefficient field, and these analytic models require no coefficient extension. A prescribed absolute Frobenius structure extends to such a bounded model with the same radius and connection bound. After a finite coefficient extension, we also realize quasi-constant modules, which carry the finite local monodromy, as differential direct summands of sums of pushforwards of rank-one modules, with uniformly bounded cover degrees and highest breaks of the inducing modules.