Related papers: Norm relations and computational problems in numbe…
We provide a method for counting number fields of fixed Galois group ordered by arbitrary inertial invariants using analytic techniques from the study of multiple Dirichlet series. We prove unconditional results for infinitely many new…
We construct a norm compatible system of Galois cohomology classes in the cyclotomic extension of the field of rationnals giving rise (conjecturally) to the degree four p-adic L-function of the symplectic group GSp(4). These classes are…
We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…
This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline…
If k is a commutative field and G a reductive (connected) algebraic group over k, we give bounds for the orders of the finite subgroups of G(k); these bounds depends on the type of G and on the Galois groups of the cyclotomic extensions of…
In 2018, Legrand and Paran proved a weaker form of the Inverse Galois Problem for all Hilbertian fields and all finite groups: that is, there exist possibly non-Galois extensions over given Hilbertian base field with given finite group as…
We give a presentation of cyclotomic q-Schur algebras by generators and defining relations. As an application, we give an algorithm for computing decomposition numbers of cyclotomic q-Schur algebras.
We provide a uniform estimate for the $L^1$-norm (over any interval of bounded length) of the logarithmic derivatives of global normalizing factors associated to intertwining operators for the following reductive groups over number fields:…
In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a…
Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of…
We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic…
Recent breakthrough methods \cite{gggz,joux,bgjt} on computing discrete logarithms in small characteristic finite fields share an interesting feature in common with the earlier medium prime function field sieve method \cite{jl}. To solve…
We construct an infinite family of real cyclotomic fields with non-trivial class group. This result generalizes the result in [1] in the sense that our family includes theirs.
For a number field $K$, we extend the notion of the ring class field of an order in $K$ [C. Lv and Y. Deng, SciChina. Math., 2015] to that of an arbitrary number ring in $K$. We give both ideal-theoretic and idele-theoretic description of…
These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups,…
We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
We use $\ell$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various…
This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur…
We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$,…