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Christian Maire

Publications and source records attributed to Christian Maire.

At least 19 recordsLinked to original sources
arXiv2026

Construction Of Non-Odd Galois Representations With Large Image

In this work, we construct Galois representations with large image that are not GL r -odd (or, equivalently, not regular at infinity). More precisely, for every prime p {\v e} 3, every integer r {\v e} 2, and every integer a P rp1 'p'1q r q{2, r '2s satisfying a '' r pmod 2q, we construct a continuous Galois representation $\rho$\,: G Q __ GL r pQ p q whose image is commensurable with GL r pZ p q and satisfies |trp$\rho$pcqq| '' a, where c denotes a complex conjugation. We also obtain analogous results for the orthogonal group O r pQ p q.

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arXiv2026

Indivisibility of ray class groups of real quadratic fields

Let ${\ell}$, p $\ge$ 5 be primes such that p | (${\ell}$ -1). Let $\Delta$ > 0 be the fundamental discriminant of a real quadratic field in which ${\ell}$ splits. We denote by h - ${\ell}$ ($\Delta$) the order of the minus part (for the Galois action) of the ray class group of Q( $\sqrt$ $\Delta$) of modulus ${\ell}$. In this paper, we study the indivisibility of h - ${\ell}$ ($\Delta$) by p, and prove that under the assumption that this set is non-empty. This lower bound is made unconditional if ${\ell}$ = 2p + 1, i.e. if p is a Sophie Germain prime. Our result can be viewed as being in the continuity of the results of Kohnen-Ono, Ono, Byeon, Beckwith etc. regarding the class numbers of quadratic fields, in the sense that we rely on techniques from the theory of half-integral weight modular forms. Significant difficulties however arise in our study, as we have to study Eisenstein congruences for cuspforms of weight 3 2 , and use a generalized Shimura correspondence of Baruch-Mao. Combined with the results of Lecouturier-Wang, our result has implications eg. for the 5-part of BSD for even quadratic twists of X 0 (11).

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arXiv2025

Maximal $2$-extensions of Pythagorean fields and Right Angled Artin Groups

In this paper, we describe minimal presentations of maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type. For this purpose, we introduce a new class of pro-$2$ groups: $Δ$-Right Angled Artin groups. We show that maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type are $Δ$-Right Angled Artin groups. Conversely, let us assume that a maximal pro-$2$ quotient of an absolute Galois group is a $Δ$-Right Angled Artin group. We then show that the underlying field must be Pythagorean, formally real and of finite type. As an application, we provide an example of a pro-$2$ group which is not a maximal pro-$2$ quotient of an absolute Galois group, although it has Koszul cohomology and satisfies both the Kernel Unipotent and the strong Massey Vanishing properties. We combine tools from group theory, filtrations and associated Lie algebras, profinite version of the Kurosh Theorem on subgroups of free products of groups, as well as several new techniques developed in this work.

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arXiv2025

On s-split p-Hilbert class field towers with prescribed Galois groups

In this work, we show that given a finite p-group G, a number field K having a trivial p-class group Cl K , and a finite set of primes S of K, there exists a finite extension F/K such that the S-split p-Hilbert class field tower L S p (F ) of F has G as its Galois group. This extends results by Ozaki and Hajir-Maire-Ramakrishna.

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arXiv2025

Massey products and unipotent extensions with restricted ramification

We fix a prime p and construct new cases of pro-p extensions of number fields with restricted ramification and splitting, whose Galois groups decompose as coproducts of pro-p absolute Galois groups of local fields. As a consequence, these pro-p extensions satisfy the strong Massey vanishing property and thus admit large unipotent quotients.

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arXiv2024

On the strong Massey property for number fields

Let $n\geq 3$. We show that for every number field $K$ with $ζ_p \notin K$, the absolute and tame Galois groups of $K$ satisfy the strong $n$-fold Massey property relative to $p$. Our work is based on an adapted version of the proof of the Theorem of Scholz-Reichardt.

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arXiv2024

Genus theory, governing field, ramification and Frobenius

In this work we develop, through a governing field, genus theory for a number field $\K$ with tame ramification in $T$ and splitting in $S$, where $T$ and $S$ are finite disjoint sets of primes of $\K$. This approach extends that initiated by the second author in the case of the class group. It allows expressing the $S$-$T$ genus number of a cyclic extension $Ł/\K$ of degree $p$ in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in $Ł/\K$, in the Galois group of the underlying governing extension. For quadratic extensions $Ł/\Q$, the matrices in question are constructed from the Legendre symbols between the primes ramified in $Ł/\Q$ and the primes in $S$.

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arXiv2024

On tamely ramified infinite Galois extensions

For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $μ_p \not \subset K$. Our guiding aim is to characterize the finitely generated pro-$p$ quotients of~$G^{\rm ta}$. We give a {unified point of view} by introducing the notion of {\it stably inertially generated} pro-$p$ groups~$G$, for which linear groups are archetypes. This key notion {is compatible} with local {\it tame liftings} as used in the Scholz-Reichardt Theorem. We realize every finitely generated pro-$p$ group~$G$ which is stably inertially generated as a quotient of $G^{\rm ta}$. Further examples of groups that we realize as quotients of $G^{\rm ta}$ include congruence subgroups of special linear groups over ${\mathbb Z}_p[[ T_1,\cdots, T_n ]]$. Finally, we give classes of groups which cannot be realized as quotients of $G^{\rm ta}_{\mathbb Q}$.

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arXiv2023

On the existence of Minkowski units

We investigate the Galois structure of algebraic units in cyclic extensions of number fields and thereby obtain strong new results on the existence of independent Minkowski $S$-units.

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arXiv2022

On tame ${\mathbb Z}/p{\mathbb Z}$ extensions with prescribed ramification

The tame Gras-Munnier Theorem gives a criterion for the existence of a ${\mathbb Z}/{\mathbb Z}$-extension of a number field $K$ ramified at exactly a set $S$ of places of $K$ prime to $p$ (allowing real Archimedean places when $p=2$) in terms of the existence of a dependence relation on the Frobenius elements of these places in a certain governing extension. We give a new and simpler proof of this theorem that also relates the set of such extensions of $K$ to the set of these dependence relations. After presenting this proof, we then reprove the key Proposition 3 using the more sophisticated Wiles-Greenberg formula based on global duality.

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arXiv2022

On Ozaki's theorem realizing prescribed $p$-groups as $p$-class tower groups

We give a streamlined and effective proof of Ozaki's theorem that any finite $p$-group $Γ$ is the Galois group of the $p$-Hilbert class field tower of some number field $\rm F$. Our work is inspired by Ozaki's and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field ${\rm k}_0$ with class number prime to $p$. We construct ${\rm F}/{\rm k}_0$ by a sequence of ${\mathbb Z}/p$-extensions ramified only at finite tame primes and also give explicit bounds on $[{\rm F}:{\rm k}_0]$ and the number of ramified primes of ${\rm F}/{\rm k}_0$ in terms of $\# Γ$.

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arXiv2021

On Galois representations with large image

For every prime number $p\geq 3$ and every integer $m\geq 1$, we prove the existence of a continuous Galois representation $ρ: G_\mathbb{Q} \rightarrow Gl_m(\mathbb{Z}_p)$ which has open image and is unramified outside $\{p,\infty\}$ (resp. outside $\{2,p,\infty\}$) when $p\equiv 3$ mod $4$ (resp. $p \equiv 1$ mod $4$).

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arXiv2021

Deficiency of p-Class Tower Groups and Minkowski Units

Let $p$ be a prime. We define the deficiency of a finitely-generated pro-$p$ group $G$ to be $r(G)-d(G)$ where $d(G)$ is the minimal number of generators of $G$ and $r(G)$ is its minimal number of relations. For a number field $K$, let $K_\emptyset$ be the maximal unramified $p$-extension of $K$, with Galois group $G_\emptyset = Gal(K_\emptyset/K)$. In the 1960s, Shafarevich (and independently Koch) showed that the deficiency of $G_\emptyset$ satisfies $$0\leq \mathrm{Def}({\rm G}_\emptyset) \leq dim (O_K^\times/(O_K^{\times })^p),$$ relating the deficiency of $G_\emptyset$ to the $p$-rank of the unit group $O_K^\times$ of the ring of integers $O_K$ of $K$. In this work, we further explore connections between relations of the group $G_\emptyset$ and the units in the tower $K_\emptyset/K$, especially their Galois module structure. In particular, under the assumption that $K$ does not contain a primitive $p$th root of unity, we give an exact formula for $\mathrm{Def}({\rm G}_\emptyset)$ in terms of the number of independent Minkowski units in the tower. The method also allows us to infer more information about the relations of G$_\emptyset$, such as their depth in the Zassenhaus filtration, which in certain circumstances makes it easier to show that G$_\emptyset$ is infinite. We illustrate how the techniques can be used to provide evidence for the expectation that the Shafarevich-Koch upper bound is "almost always" sharp.

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arXiv2020

Codes from unit groups of division algebras over number fields

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. We prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum-rank metric from division algebras of any degree, and we estimate the size of the alphabet in terms of the degree.

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arXiv2019

On the Shafarevich Group of Restricted Ramification Extensions of Number Fields in the Tame Case

Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X} \stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X} \stackrel{\simeq}{\twoheadleftarrow} \CyB_X \hookleftarrow \Sha_X$. In particular, we show that in the tame case $\Sha_Y$ can {\it increase} with increasing $Y$. This is in contrast with the wild case where $\Sha_Y$ is nonincreasing in size with increasing $Y$.

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arXiv2019

A note on $p$-rational fields and the abc-conjecture

In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.

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