Related papers: On Classifying Extensions of $p$-adic Fields

Let $p$ be an irregular prime and $K=\Q(\zeta)$ the $p$-cyclotomic field. Let $\sigma$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in…

Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it…

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair…

Let $S$ be a Banach algebra over $\mathbb{Q}_p$ whose residue fields are finite extensions of $\mathbb{Q}_p$. Given an arithmetic family $V$ of Galois representations, i.e., a finite free $S$-module $V$ with a continuous action of the…

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its…

A finite separable extension of a field is called primitive if there are no intermediate extensions. The most interesting primitive extensions of a local field with finite residue field are the wildly ramified ones, and our aim here is to…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

We prove that the theory of a Henselian valued field of characteristic zero, with finite ramification, and whose value group is a $Z$-group, is model-complete in the language of rings if the theory of its residue field is model-complete in…

Let $K/k$ be a pro-$p$-extension over a number field $k$ whose Galois group is finitely generated and $k_0\subseteq k_1\subseteq\cdots\subseteq k_n\subseteq\cdots$ an ascending sequence of intermediate fields of $K/k$ such that $k_n/k$ is…

We study the number of ramified prime numbers in finite Galois extensions of $\mathbb{Q}$ obtained by specializing a finite Galois extension of $\mathbb{Q}(T)$. Our main result is a central limit theorem for this number. We also give some…

For p = 3 and p = 5, we exhibit a finite nonsolvable extension of the rational numbers which is ramified only at p via explicit computations with Hilbert modular forms.

Let $p\neq2$ be a prime. We show a technique based on local class field theory and on the expansions of certain resultants which allows to recover very easily Lbekkouri's characterization of Eisenstein polynomials generating cyclic wild…

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. If $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of…

Let $K$ be a finite extension of the $p$-adic numbers $\mathbb Q_p$ with ring of integers $\mathcal O_K$, $\mathcal X$ a regular scheme, proper, flat, and geometrically irreducible over $\mathcal O_K$ of dimension $d$, and $\mathcal X_K$…

Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a…

Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p.$ Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ dimensional abelian varieties…

Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper…

We use cell decomposition techniques to study additive reducts of p- adic fields. We consider a very general class of fields, including fields with infinite residue fields, which we study using a multi-sorted language. The results are used…

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 $\mathds{k}$ be an algebraically closed field of characteristic $p$. We give the complete classification of pointed Hopf algebras over $\mathds{k}$ of dimension $p^2q$ for a prime number $q$. The result shows that there are finitely…