Computing Galois groups of Eisenstein polynomials over p-adic fields

The most efficient algorithms for computing Galois groups of polynomials over global fields are based on Stauduhar’s relative resolvent method. These methods are not directly generalizable to the local field case, since they require a field that contains the global field in which all roots of the polynomial can be approximated. We present splitting field-independent methods for computing the Galois group of an Eisenstein polynomial over a p-adic field. Our approach is to combine information from different disciplines. We primarily, make use of the ramification polygon of the polynomial, which is the Newton polygon of a related polynomial. This allows us to quickly calculate several invariants that serve to reduce the number of possible Galois groups. Algorithms by Greve and Pauli very efficiently return the Galois group of polynomials where the ramification polygon consists of one segment as well as information about the subfields of the stem field. Second, we look at the factorization of linear absolute resolvents to further narrow the pool of possible groups

p-adic equidistribution of CM points

Let $X$ be a modular curve and consider a sequence of Galois orbits of CM points in $X$, whose $p$-conductors tend to infinity. Its equidistribution properties in $X({\bf C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. We study the equidistribution problem in the Berkovich analytification $X_{p}^{\rm an}$ of $X_{{\bf Q}_{p}}$. We partition the set of CM points of sufficiently high conductor in $X_{{\bf Q}_{p}}$ into finitely many explicit \emph{basins} $B_{V}$, indexed by the irreducible components $V$ of the mod-$p$ reduction of the canonical model of $X$. We prove that a sequence $z_{n}$ of local Galois orbits of CM points with $p$-conductor going to infinity has a limit in $X_{p}^{\rm an}$ if and only if it is eventually supported in a single basin $B_{V}$. If so, the limit is the unique point of $X_{p}^{\rm an}$ whose mod-$p$ reduction is the generic point of $V$. The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasicanonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin--Tate space.Comment: Some improvements in the exposition. 23 pages, 1 new figur

Paraboline variation of pp-adic families of (φ,Γ)(\varphi,\Gamma)-modules

We study the $p$-adic variation of triangulations over $p$-adic families of $(\varphi,\Gamma)$-modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. As an application, we study the ramification of weight parameters over natural $p$-adic families.Comment: 46 pages (larger font than before). Final version. Numerous improvements to expositions and corrections of minor errors. Definitions in Section 4 slightly altered following suggestion of referee. Definitions in Sections 5 and 6 explained in more detail for clarity. Appendix adde

Product formula for p-adic epsilon factors

Let X be a smooth proper curve over a finite field of characteristic p. We prove a product formula for p-adic epsilon factors of arithmetic D-modules on X. In particular we deduce the analogous formula for overconvergent F-isocrystals, which was conjectured previously. The p-adic product formula is the equivalent in rigid cohomology of the Deligne-Laumon formula for epsilon factors in l-adic \'etale cohomology (for a prime l different from p). One of the main tools in the proof of this p-adic formula is a theorem of regular stationary phase for arithmetic D-modules that we prove by microlocal techniques.Comment: Revised version: some proofs and constructions detailed, notation improved, index of notation added ; 88 page