Searcharxiv⌕ Search

arXiv subjects

Henri Johnston

Publications and source records attributed to Henri Johnston.

At least 19 recordsLinked to original sources
arXiv2026

Algorithms for $p$-rationality and for $p$-saturation of units

Let $K$ be a number field. We give new practical algorithms that determine whether $K$ is $p$-rational. For real cyclotomic fields $K=\mathbb{Q}(ζ_{n})^{+}$, our method avoids computing either the class number or the full unit group. We also use the notion of $p$-rationality to explain the practical efficiency of an algorithm for determining whether a subgroup of the unit group $\mathcal{O}_{K}^{\times}$ is $p$-saturated. This in turn yields a new algorithm for the unconditional verification of unit groups of number fields that substantially outperforms existing unconditional algorithms in practice. Finally, by combining our algorithms for determining $p$-rationality with work of Greenberg, we construct certain Galois representations $ρ: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}_{n}(\mathbb{Z}_{p})$ with open image and further prescribed properties.

math.NT↗
arXiv2026

Applications of representation theory and of explicit units to Leopoldt's conjecture

Let $L/K$ be a Galois extension of number fields and let $G=\mathrm{Gal}(L/K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies Leopoldt's conjecture at $p$ for $L$. We also obtain relations between the Leopoldt defects of intermediate extensions of $L/K$. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers $\mathcal{P}$, there exists an infinite family $\mathcal{F}$ of totally real $S_{3}$-extensions of $\mathbb{Q}$ such that Leopoldt's conjecture for $F$ at $p$ holds for every $F \in \mathcal{F}$ and $p \in \mathcal{P}$.

math.NT↗
arXiv2026

Determination of the stably free cancellation property for orders

Let $K$ be a number field, let $A$ be a finite-dimensional semisimple $K$-algebra, and let $Λ$ be an $\mathcal{O}_{K}$-order in $A$. We give practical algorithms that determine whether $Λ$ has stably free cancellation (SFC). As an application, we determine all finite groups $G$ of order at most $383$ such that the integral group ring $\mathbb{Z}[G]$ has SFC.

math.NT↗
arXiv2024

An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications

Let $p$ be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for every admissible one-dimensional $p$-adic Lie extension whose Galois group has an abelian Sylow $p$-subgroup. Crucially, this result does not depend on the vanishing of any $μ$-invariant. As applications, we deduce the Coates-Sinnott conjecture away from its $2$-primary part and new cases of the equivariant Tamagawa number conjecture for Tate motives.

math.NT↗
arXiv2024

On the existence of free sublattices of bounded index and arithmetic applications

Let $\mathcal{O}$ be a Dedekind domain whose field of fractions $K$ is a global field. Let $A$ be a finite-dimensional separable $K$-algebra and let $Λ$ be an $\mathcal{O}$-order in $A$. Let $n$ be a positive integer and suppose that $X$ is a $Λ$-lattice such that $K \otimes_{\mathcal{O}} X$ is free of rank $n$ over $A$. Then $X$ contains a (non-unique) free $Λ$-sublattice of rank $n$. The main result of the present article is to show there exists such a sublattice $Y$ such that the generalised module index $[X : Y]_{\mathcal{O}}$ has explicit upper bounds with respect to division that are independent of $X$ and can be chosen to satisfy certain conditions. We give examples of applications to the approximation of normal integral bases and strong Minkowski units, and to the Galois module structure of rational points over abelian varieties.

math.NT↗
arXiv2024

Arakelov class groups of random number fields

The main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen--Lenstra--Martinet heuristic on ideal class groups. To that end, we show that Chinburg's Omega(3) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups. As a consequence, we construct a new infinite series of counterexamples to the Cohen--Lenstra--Martinet heuristic, which have the novel feature that their Galois groups are non-abelian.

math.NT↗
arXiv2022

Computation of lattice isomorphisms and the integral matrix similarity problem

Let $K$ be a number field, let $A$ be a finite-dimensional $K$-algebra, let $\mathrm{J}(A)$ denote the Jacobson radical of $A$, and let $Λ$ be an $\mathcal{O}_{K}$-order in $A$. Suppose that each simple component of the semisimple $K$-algebra $A/{\mathrm{J}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on $A$, we give an algorithm that given two $Λ$-lattices $X$ and $Y$, determines whether $X$ and $Y$ are isomorphic, and if so, computes an explicit isomorphism $X \rightarrow Y$. This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: given a number field $K$, a positive integer $n$ and two matrices $A,B \in \mathrm{Mat}_{n}(\mathcal{O}_{K})$, determine whether $A$ and $B$ are similar over $\mathcal{O}_{K}$, and if so, return a matrix $C \in \mathrm{GL}_{n}(\mathcal{O}_{K})$ such that $B= CAC^{-1}$. We give explicit examples that show that the implementation of the latter algorithm for $\mathcal{O}_{K}=\mathbb{Z}$ vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

math.NT↗
arXiv2020

Computing isomorphisms between lattices

Let K be a number field, let A be a finite dimensional semisimple K-algebra and let Lambda be an O_K-order in A. It was shown in previous work that, under certain hypotheses on A, there exists an algorithm that for a given (left) Lambda-lattice X either computes a free basis of X over Lambda or shows that X is not free over Lambda. In the present article, we generalise this by showing that, under weaker hypotheses on A, there exists an algorithm that for two given Lambda-lattices X and Y either computes an isomorphism X -> Y or determines that X and Y are not isomorphic. The algorithm is implemented in Magma for A=Q[G], Lambda=Z[G] and Lambda-lattices X and Y contained in Q[G], where G is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of Q with Galois group isomorphic to Q_8 x C_2, the direct product of the quaternion group of order 8 and the cyclic group of order 2.

math.NT↗
arXiv2019

On the p-adic Stark conjecture at s=1 and applications

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s=1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s=1 for E/F implies Stark's conjecture at s=1 for E/F. This leads to a `prime-by-prime' descent theorem for the `equivariant Tamagawa number conjecture' (ETNC) for Tate motives at s=1. As an application of these results, we provide strong new evidence for special cases of the ETNC for Tate motives and the closely related `leading term conjectures' at s=0 and s=1.

math.NT↗
arXiv2018

On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture

We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer-Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant. In combination with the authors' previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.

math.NT↗
arXiv2016

Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian p-adic Lie extensions. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant.

math.NT↗
arXiv2015

On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a wide class of interesting extensions, including cases in which the full ETNC in not known. Similarly, we construct annihilators of higher dimensional algebraic K-groups of the ring of integers in L.

math.NT↗
arXiv2014

Explicit integral Galois module structure of weakly ramified extensions of local fields

Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say that L/K is weakly ramified when G_2 is trivial. Let O_L be the valuation ring of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring O_K[G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every free generator of P_L over O_K[G] is also a free generator of O_L over its associated order in the group algebra K[G]. Along the way, we prove a `splitting lemma' for local fields, which may be of independent interest.

math.NT↗
arXiv2013

Noncommutative Fitting invariants and improved annihilation results

To each finitely presented module M over a commutative ring R one can associate an R-ideal Fit_R(M) which is called the (zeroth) Fitting ideal of M over R and which is always contained in the R-annihilator of M. In an earlier article, the second author generalised this notion by replacing R with a (not necessarily commutative) o-order Lambda in a finite dimensional separable algebra, where o is an integrally closed complete commutative noetherian local domain. To obtain annihilators, one has to multiply the Fitting invariant of a (left) Lambda-module M by a certain ideal H(Lambda) of the centre of Lambda. In contrast to the commutative case, this ideal can be properly contained in the centre of Lambda. In the present article, we determine explicit lower bounds for H(Lambda) in many cases. Furthermore, we define a class of `nice' orders Lambda over which Fitting invariants have several useful properties such as good behaviour with respect to direct sums of modules, computability in a certain sense, and H(Lambda) being the best possible.

math.RA↗
arXiv2010

Computing generators of free modules over orders in group algebras II

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. In a previous article, we gave a necessary and sufficient condition for X to be free of given rank d over A. In the case that (i) the Wedderburn decomposition of E[G] is explicitly computable and (ii) each component is in fact a matrix ring over a field, this led to an algorithm that either gives elements that either gives an A-basis for X or determines that no such basis exists. In the present article, we generalise the algorithm by weakening condition (ii) considerably.

math.NT↗
arXiv2010

A non-abelian Stickelberger theorem

Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z_(p)[G] that annihilates the p-part of the class group of L.

math.NT↗
arXiv2010

On the restricted Hilbert-Speiser and Leopoldt properties

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if, for every tame G-Galois extension L/K, the ring of integers O_L is free as an O_K[G]-module. If O_L is free over the associated order A_{L/K} for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-Speiser of type G. We show that the converse does not hold in general, but that a modified version does hold for many number fields K (in particular, for K/Q Galois) when G=C_p has prime order. We give examples with G=C_p to show that even the modified converse is false in general, and that the modified converse can hold when the original does not.

math.NT↗