Related papers: ABC implies a Zsigmondy principle for ramification

Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This…

We conjecture that a $p$-algebra over a complete discrete valued field $K$ contains a totally ramified purely inseparable subfield if and only if it contains a totally ramified cyclic maximal subfield. We prove the conjecture in several…

Let $K$ be a number field. The $K$-arithmetic type of a rational prime $\ell$ is the tuple $A_{K}(\ell)=(f^{K}_{1},...,f^{K}_{g_{\ell}})$ of the residue degrees of $\ell$ in $K$, written in ascending order. A well known result of Perlis…

For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case.…

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the…

Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $\phi(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely…

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes…

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular,…

Let $F$ be a totally real field and $K$ a finite abelian CM extension of $F$. Using class field theory, we show that our previous result giving a strong form of the Brumer-Stark conjecture implies the minus part of the equivariant Tamagawa…

Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K_{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence…

If the $\ell$-adic cohomology of a projective smooth variety, defined over a $\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational…

For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the…

Let X be a smooth projective variety over a field k. For k separably closed, we prove that the subgroup of unramified classes in the Milnor K-group $K^M_i(k(X))$ of the function field of X is contained in the subgroup of n-divisible…

Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C violates the Hasse principle, i.e., has points everywhere…

In this paper, for any nonic number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^9+ax+b \in \mathbb{Z}[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also…

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…

Abhyankar showed that for a finite tame extension $L_1/K$ and a finite extension $L_2/K$ of $\mathfrak{P}$-adic fields, the condition $[\nu L_1 : \nu K]$ divides $[\nu L_2 : \nu K]$ is sufficient to eliminate ramification, that is, $L_1…

Let $p$ be a prime number and $\zeta_p$ be a primitive $p$-th root of unity in $\bm{C}$. Let $k$ be a field and $k(x_0,\ldots,x_{p-1})$ be the rational function field of $p$ variables over $k$. Suppose that $G=\langle\sigma\rangle \simeq…

In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d <= 5. In analogy with the classical Erdos-Kac theorem, we prove for S_d-extensions that the number of such primes is normally…

Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally…