Related papers: On Abhyankar's lemma about ramification indices

We study the following question: given a global field $F$ and finite group $G$, what is the minimal $r$ such that there exists a finite extension $K/F$ with $\mathrm{Aut}(K/F)\cong G$ that is ramified over exactly $r$ places of $F$? We…

Consider a primitive polynomial $f$ in two variables, thought of as a map from the affine plane to the affine line. We study the minimimal compactification of $f$; from our result one deduces in particular that if one of the fibers of $f$…

Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of…

We prove that number fields with arbitrary degree but weak ramification satisfy the Leopoldt conjecture on the l-adic rank of the group of units

Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in…

It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is…

Classically the ramification filtration of the Galois group of a complete discrete valuation field is defined in the case where the residue field is perfect. In this paper, we define without any assumption on the residue field, two…

We show that compatible systems of $\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite…

In this paper, at first, we show that for a ramified regular local ring $S$, which is an Eisenstein extension of an unramified regular local ring $R$, when an ideal $I$ of $S$ is extended from an ideal $J$ of $R$, the punctured spectrum of…

In this paper we consider birational properties of ramification in excellent local rings. We give an example showing that local monomialization (and weak local monomialization) can fail for extensions of algebraic local rings in algebraic…

The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven…

We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given…

We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points…

We count abelian number fields ordered by arbitrary height function whose generator of tame inertia is restricted to lie in a given subset of the Galois group, and find an explicit formula for the leading constant. We interpret our results…

For any prime power $q$, a polynomial $f(X)\in\F_q[X]$ is ``exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one…

We introduce the ramified partition algebra, which is a physically motivated and natural generalization of the partition algebra. We investigate its representation theory and demonstrate quasi--heredity under certain conditions. Under these…

Let $G$ be a finite nilpotent group and $K$ a number field with torsion relatively prime to the order of $G$. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of…

This is the second of two papers on the uniform asymptotics for real double Hurwitz numbers with triple ramification. Using the modified tropical correspondence theorem established in the first paper of this series, we introduce a…

In this paper, we study the ramification of extensions of a function field generated by division points of rank 2 Drinfeld modules. Also conductors of certain rank 2 Drinfeld modules are defined as analogues of those for elliptic curves. A…

Double ramification loci parametrise marked curves where a weighted sum of the markings is linearly trivial; higher-rank loci are obtained by imposing several such conditions simultaneously. We obtain closed formulae for the orbifold Euler…