Related papers: Canonical Valuations and the Birational Section Co…

Let $F$ be a finite extension of $Q_p$, $p>2$. We construct admissible unitary completions of certain representations of $GL_2(F)$ on $L$-vector spaces, where $L$ is a finite extension of $F$. When $F=Q_p$ using the results of Berger,…

We verify a special case of a conjecture of G. Carlsson that describes the $\l$-adic $K$-theory of a field $F$ of characteristic prime to $\l$ in terms of the representation theory of the absolute Galois group $G_F$. This conjecture is…

Assuming the finiteness of the Shafarevich-Tate group of elliptic curves over number fields we make several observations on the birational Grotendieck anabelian setion conjecture. We prove that the birational setion conjecture for curves…

Let $\mathcal{P}: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point,…

We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory…

We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus…

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does…

We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory…

Canonical matrices of (a) bilinear and sesquilinear forms, (b) pairs of forms, in which every form is symmetric or skew-symmetric, and (c) pairs of Hermitian forms are given over finite fields of characteristic not 2 and over finite…

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the…

Let $A$ be an affinoid integral domain over a non-Archimedean field $K$, and let $L$ be its field of fractions. We prove that the normalization of $A$ can be reconstructed from $L$ by taking the intersection of all maximal discrete…

We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over Q if it holds over number fields.

Author's generalization of one-dimensional class field theory to theory of abelian totally ramified p-extensions of a complete discrete valuation field with arbitrary non-separably p-closed residue field and its applications are described.

Canonical differential calculus is defined for finitely generated abelian group with an involution existing consistently. Two such canonical calculi are found out. Fermionic representation for canonical calculus is defined based on…

The Elementary Type Conjecture in Galois theory provides a concrete inductive description of the finitely generated maximal pro-$p$ Galois groups $G_F(p)$ of fields $F$ containing a root of unity of order $p$. We describe several variants…

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring $A$ with field of fractions $K$ and for a finite Galois extension $L$ of $K$, the integral closure…

We study the behaviour of forking in valued fields, and we give several sufficient conditions for parameter sets in a Henselian valued field of residue characteristic zero to be an extension base. Notably, we consider arbitrary (potentially…

In this exposition we discuss the theory of algebraic extensions of valued fields. Our approach is mostly through Galois theory. Most of the results are well-known, but some are new. No previous knowledge on the theory of valuations is…

Let F be a Henselian valued field with char(F) = p and D a semi-ramified, "not strongly degenerate" p-algebra. We show that all Galois subfields of D are inertial. Using this as a tool we study generic abelian crossed product p-algebras,…

Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the…