Narrow Results

In this work, we present the integral trace form ${\mathrm{\text{Tr}}}_{𝕄/\mathbb{Q}}(x^2)$ of a cyclic extension $𝕄/\mathbb{Q}$ with degree $pq$, where $𝕄=𝕂𝕃$, $p$ and $q$ are distinct odd primes, the conductor of $𝕄$ is a square free integer, and $x$ belongs to the ring of algebraic integers ${{𝒪}}_{𝕄}$ of $𝕄$. The integral trace form of $𝕄/\mathbb{Q}$ allows one to calculate the packing radius of lattices constructed via the canonical (or twisted) homomorphism of submodules of ${{𝒪}}_{𝕄}$.

We explore whether a root lattice may be similar to the lattice $\mathcal{𝒪}$ of integers of a number field $K$ endowed with the inner product $(x,y):={\mathrm{\text{Trace}}}_{K/\mathbb{Q}}(x\cdot 𝜃(y))$, where $𝜃$ is an involution of $K$. We classify all pairs $K$, $𝜃$ such that $\mathcal{𝒪}$ is similar to either an even root lattice or the root lattice ${\mathbb{Z}}^{[K:\mathbb{Q}]}$. We also classify all pairs $K$, $𝜃$ such that $\mathcal{𝒪}$ is a root lattice. In addition to this, we show that $\mathcal{𝒪}$ is never similar to a positive-definite even unimodular lattice of rank $\le 48$, in particular, $\mathcal{𝒪}$ is not similar to the Leech lattice. In Appendix B, we give a general cyclicity criterion for the primary components of the discriminant group of $\mathcal{𝒪}$.

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p^A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL₂(5)⁵.10, and field discriminant 5⁶⁹. A closely related polynomial has degree 120, Galois group of the form SL₂(5)⁵.20, and field discriminant 5³¹¹. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5^-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5⁴.

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer–Siegel theorem both for number fields and function fields.

We tabulate polynomials in ℚ[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and ∞. We explain how these polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification.

We prove the Hasse principle and weak approximation for varieties defined by the smooth complete intersection of three quadratics in at least 19 variables, over arbitrary number fields.

Consider a non-split one-dimensional torus defined over a number field $K$. For a finitely generated group $G$ of rational points and for a prime number $\ell$, we investigate for how many primes $𝔭$ of $K$ the size of the reduction of $G$ modulo $𝔭$ is coprime to $\ell$. We provide closed formulas for the corresponding Dirichlet density in terms of finitely many computable parameters. To achieve this, we determine in general which torsion fields and Kummer extensions contain the splitting field.

We give a list of ${\mathrm{\text{PGL}}}_2({𝔽}_{\ell })$ number fields for $\ell \ge 11$ which have rational companion forms. Our list has 53 fields and seems likely to be complete. Some of the fields on our list are very lightly ramified for their Galois group.

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if $G$ is a finitely generated and torsion-free multiplicative subgroup of a number field $K$ having rank $r$, then the ratio between $n^r$ and the Kummer degree $[K({\zeta }_n,\sqrt[n]{G}):K({\zeta }_n)]$ is bounded independently of $n$. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

Let $K$ be a number field, and let ${\alpha }_1,\dots ,{\alpha }_r$ be elements of $K^{\times }$ which generate a subgroup of $K^{\times }$ of rank $r$. Consider the cyclotomic-Kummer extensions of $K$ given by $K({\zeta }_n,\sqrt[n_1]{{\alpha }_1},\dots ,\sqrt[n_r]{{\alpha }_r})$, where $n_i$ divides $n$ for all $i$. There is an integer $x$ such that these extensions have maximal degree over $K({\zeta }_g,\sqrt[g_1]{{\alpha }_1},\dots ,\sqrt[g_r]{{\alpha }_r})$, where $g=gcd(n,x)$ and $g_i=gcd(n_i,x)$. We prove that the constant $x$ is computable. This result reduces to finitely many cases the computation of the degrees of the extensions $K({\zeta }_n,\sqrt[n_1]{{\alpha }_1},\dots ,\sqrt[n_r]{{\alpha }_r})$ over $K$.