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François Legrand

Publications and source records attributed to François Legrand.

At least 19 recordsLinked to original sources
arXiv2026

GRANDlib: A simulation pipeline for the Giant Radio Array for Neutrino Detection (GRAND)

The operation of upcoming ultra-high-energy cosmic-ray, gamma-ray, and neutrino radio-detection experiments, like the Giant Radio Array for Neutrino Detection (GRAND), poses significant computational challenges involving the production of numerous simulations of particle showers and their detection, and a high data throughput. GRANDlib is an open-source software tool designed to meet these challenges. Its primary goal is to perform end-to-end simulations of the detector operation, from the interaction of ultra-high-energy particles, through -- by interfacing with external air-shower simulations -- the ensuing particle shower development and its radio emission, to its detection by antenna arrays and its processing by data-acquisition systems. Additionally, GRANDlib manages the visualization, storage, and retrieval of experimental and simulated data. We present an overview of GRANDlib to serve as the basis of future GRAND analyses.

astro-ph.IM↗
arXiv2026

Towards the Giant Radio Array for Neutrino Detection (GRAND): the GRANDProto300 and GRAND@Auger prototypes

The Giant Radio Array for Neutrino Detection (GRAND) is a proposed multi-messenger observatory of Ultra-High-Energy (UHE) particles of cosmic origin. Its main goal is to find the long-sought origin of UHE cosmic rays by detecting large numbers of them and the secondary particles created by their interactions like gamma rays and neutrinos. The GRAND Collaboration plans to achieve this using large arrays of radio antennas that look for the radio signals emitted by the air showers initiated by the interactions of the UHE particles in the atmosphere. Since 2023, three small-scale prototype GRAND arrays have been in operation: GRAND@Nançay in France, GRAND@Auger in Argentina, and GRANDProto300 in China. Together, their goal is to validate the detection principle of GRAND under prolonged field conditions, achieving efficient, autonomous radio-detection of air showers. We describe the hardware, software, layout, and operation of the GRAND prototypes. Using their data, we show a first characterization of the local electromagnetic environment of each site and a measurement of the Galactic synchrotron emission. Despite challenges, the successful operation of the prototypes confirms that the GRAND instrumentation is apt to address the goals of the experiment and lays the groundwork for its ensuing stages.

astro-ph.IM↗
arXiv2025

The Giant Radio Array for Neutrino Detection (GRAND) Collaboration -- Contributions to the 39th International Cosmic Ray Conference (ICRC 2025)

The Giant Radio Array for Neutrino Detection (GRAND) is an envisioned observatory of ultra-high-energy particles of cosmic origin, with energies in excess of 100 PeV. GRAND uses large surface arrays of antennas to look for the radio emission from extensive air showers that are triggered by the interaction of ultra-high-energy cosmic rays, gamma rays, and neutrinos in the atmosphere or underground. In particular, for ultra-high-energy neutrinos, the future final phase of GRAND aims to be sensitive enough to detect them in spite of their plausibly tiny flux. Three prototype GRAND radio arrays have been in operation since 2023: GRANDProto300, in China, GRAND@Auger, in Argentina, and GRAND@Nançay, in France. Their goals are to field-test the GRAND detection units, understand the radio background to which they are exposed, and develop tools for diagnostic, data gathering, and data analysis. This list of contributions to the 39th International Cosmic Ray Conference (ICRC 2025) presents an overview of GRAND, in its present and future incarnations, and a first look at data collected by GRANDProto300 and GRAND@Auger, including the first cosmic-ray candidates detected by them.

astro-ph.IM↗
arXiv2024

The Giant Radio Array for Neutrino Detection (GRAND) Collaboration -- Contributions to the 38th International Cosmic Ray Conference (ICRC 2023)

The Giant Radio Array for Neutrino Detection (GRAND) is an envisioned observatory of ultra-high-energy particles of cosmic origin, with energies in excess of 100 PeV. GRAND uses large surface arrays of autonomous radio-detection units to look for the radio emission from extensive air showers that are triggered by the interaction of ultra-high-energy cosmic rays, gamma rays, and neutrinos in the atmosphere or underground. In particular, for ultra-high-energy neutrinos, the future final phase of GRAND aims to be sensitive enough to discover them in spite of their plausibly tiny flux. Presently, three prototype GRAND radio arrays are in operation: GRANDProto300, in China, GRAND@Auger, in Argentina, and GRAND@Nancay, in France. Their goals are to field-test the design of the radio-detection units, understand the radio background to which they are exposed, and develop tools for diagnostic, data gathering, and data analysis. This list of contributions to the 38th International Cosmic Ray Conference (ICRC 2023) presents an overview of GRAND, in its present and future incarnations, and a look at the first data collected by GRANDProto13, the first phase of GRANDProto300.

hep-ex↗
arXiv2024

The Giant Radio Array for Neutrino Detection (GRAND) Collaboration -- Contributions to the 10th International Workshop on Acoustic and Radio EeV Neutrino Detection Activities (ARENA 2024)

This is an index of the contributions by the Giant Radio Array for Neutrino Detection (GRAND) Collaboration to the 10th International Workshop on Acoustic and Radio EeV Neutrino Detection Activities (ARENA 2024, University of Chicago, June 11-14, 2024). The contributions include an overview of GRAND in its present and future incarnations, methods of radio-detection that are being developed for them, and ongoing joint work between the GRAND and BEACON experiments.

astro-ph.IM↗
arXiv2023

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan--Bhattarcharjya in the abelian case.

math.CO↗
arXiv2023

Lüroth's and Igusa's theorems over Division Rings

Let $H$ be a division ring of finite dimension over its center, let $H[T]$ be the ring of polynomials in a central variable over $H$, and let $H(T)$ be its quotient skew field. We show that every intermediate division ring between $H$ and $H(T)$ is itself of the form $H(f)$, for some $f$ in the center of $H(T)$. This generalizes the classical Lüroth's theorem. More generally, we extend Igusa's theorem characterizing the transcendence degree 1 subfields of rational function fields, from fields to division rings.

math.NT↗
arXiv2022

Cloaking, Trapping and Superlensing of Lamb Waves with Negative Refraction

We report on experimental and numerical implementations of devices based on the negative refraction of elastic guided waves, the so-called Lamb waves. Consisting in plates of varying thickness, these devices rely on the concept of complementary media, where a particular layout of negative index media can cloak an object with its anti-object or trap waves around a negative corner. The diffraction cancellation operated by negative refraction is investigated by means of laser ultrasound experiments. However, unlike original theoretical predictions, these intriguing wave phenomena remain, nevertheless, limited to the propagating component of the wave-field. To go beyond the diffraction limit, negative refraction is combined with the concept of metalens, a device converting the evanescent components of an object into propagating waves. The transport of an evanescent wave-field is then possible from an object plane to a far-field imaging plane. Twenty years after Pendry's initial proposal, this work thus paves the way towards an elastic superlens.

physics.class-ph↗
arXiv2022

On Cayley graphs over generalized dicyclic groups

Recently, several works by a number of authors have studied integrality, distance integrality, and distance powers of Cayley graphs over some finite groups, such as dicyclic groups and (generalized) dihedral groups. Our aim is to generalize and/or to give analogues of these results for generalized dicyclic groups. For example, we give a necessary and sufficient condition for a Cayley graph over a generalized dicyclic group to be integral (i.e., all eigenvalues of its adjacency matrix are in $\mathbb{Z}$). We also obtain sufficient conditions for the integrality of all distance powers of a Cayley graph over a given generalized dicyclic group. These results extend works on dicyclic groups by Cheng--Feng--Huang and Cheng--Feng--Liu--Lu--Stevanovic, respectively.

math.CO↗
arXiv2022

On finite embedding problems with abelian kernels

Given a Hilbertian field $k$ and a finite set $\mathcal{S}$ of Krull valuations of $k$, we show that every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/k)$ over $k$ with abelian kernel has a solu\-tion ${\rm{Gal}}(F/k) \rightarrow G$ such that every $v \in \mathcal{S}$ is totally split in $F/L$. Two applications are then given. Firstly, we solve a non-constant variant of the Beckmann--Black problem for solvable groups: given a field $k$ and a non-trivial finite solvable group $G$, every Galois field extension $F/k$ of group $G$ is shown to occur as the specialization at some $t_0 \in k$ of some Galois field extension $E/k(T)$ of group $G$ with $E \not \subseteq \overline{k}(T)$. Secondly, we contribute to inverse Galois theory over division rings, by showing that, for every division ring $H$ and every automorphism $σ$ of $H$ of finite order, all finite semiabelian groups occur as Galois groups over the skew field of fractions $H(T, σ)$ of the twisted polynomial ring $H[T, σ]$.

math.NT↗
arXiv2021

On a variant of the Beckmann--Black problem

Given a field $k$ and a finite group $G$, the Beckmann--Black problem asks whether every Galois field extension $F/k$ with group $G$ is the specialization at some $t_0 \in k$ of some Galois field extension $E/k(T)$ with group $G$ and $E \cap \overline{k} = k$. We show that the answer is positive for arbitrary $k$ and $G$, if one waives the requirement that $E/k(T)$ is normal. In fact, our result holds if ${\rm{Gal}}(F/k)$ is any given subgroup $H$ of $G$ and, in the special case $H=G$, we provide a similar conclusion even if $F/k$ is not normal. We next derive that, given a division ring $H$ and an automorphism $σ$ of $H$ of finite order, all finite groups occur as automorphism groups over the skew field of fractions $H(T, σ)$ of the twisted polynomial ring $H[T, σ]$.

math.NT↗
arXiv2021

Galois families of modular forms and application to weight one

We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over an algebraic closure of F_p for p in {3,5,7,11}.

math.NT↗
arXiv2021

A note on finite embedding problems with nilpotent kernel

The first aim of this note is to fill a gap in the literature by proving that, given a global field $K$ and a finite set $\mathcal{S}$ of primes of $K$, every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/K)$ over $K$ with nilpotent kernel has a solution ${\rm{Gal}}(F/K) \rightarrow G$ such that all primes in $\mathcal{S}$ are totally split in $F/L$. We then apply this to inverse Galois theory over division rings. Firstly, given a number field $K$ of level at least $4$, we show that every finite solvable group occurs as a Galois group over the division ring $H_K$ of quaternions with coefficients in $K$. Secondly, given a finite split embedding problem with nilpotent kernel over a finite field $K$, we fully describe for which automorphisms $σ$ of $K$ the embedding problem acquires a solution over the skew field of fractions $K(T, σ)$ of the twisted polynomial ring $K[T, σ]$.

math.NT↗
arXiv2021

Problèmes de plongement finis sur les corps non commutatifs

We extend finite embedding problems over fields, a central notion in inverse Galois theory, to the situation of a skew field $H$ of finite dimension over its center $h$. First, we show that solving a finite embedding problem over $H$ is equivalent to finding a solution to some finite embedding problem over $h$ fulfilling a polynomial constraint. Next, we show that every constant finite split embedding problem over the skew field of fractions $H(t)$ with central indeterminate $t$ has a solution, if $h$ is an ample field. This is a non-commutative analogue of a deep result of Pop. More generally, we solve such finite embedding problems over the skew field of fractions $H(t, σ)$ of the twisted polynomial ring $H[t, σ]$, for some automorphisms $σ$ of $H$ of finite order. Our results extend previous works on the inverse Galois problem over skew fields.

math.NT↗
arXiv2021

Rational pullbacks of Galois covers

The finite subgroups of ${\rm PGL}_2(\mathbb{C})$ are shown to be the only finite groups $G$ with this property: for some integer $r_0$ (depending on $G$), all Galois covers $X\rightarrow \mathbb{P}^1_{\mathbb{C}}$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along non-constant rational maps $\mathbb{P}^1_{\mathbb{C}} \rightarrow \mathbb{P}^1_{\mathbb{C}}$. For $G\subset {\rm PGL}_2(\mathbb{C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A consequence of the converse for inverse Galois theory is that, for $G\not \subset {\rm PGL}_2({\mathbb{C}})$, letting the branch point number grow provides truly new Galois realizations $F/{\mathbb{C}}(T)$ of $G$. Another application is that the ``Beckmann--Black'' property that ``any two Galois covers of $\mathbb{P}^1_{\mathbb{C}}$ with the same group $G$ are always pullbacks of another Galois cover of group $G$'' only holds if $G\subset {\rm PGL}_2({\mathbb{C}})$.

math.NT↗
arXiv2021

On parametric and generic polynomials with one parameter

Given fields $k \subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \in k[T][Y]$ of group $G$ which parametrize all Galois extensions of $L$ of group $G$ via specialization of $T$ in $L$, and the latter are those which are $L$-parametric for every field $L \supseteq k$. We show, for example, that being $L$-parametric with $L$ taken to be the single field $\mathbb{C}((V))(U)$ is in fact sufficient for a polynomial $P(T, Y) \in \mathbb{C}[T][Y]$ to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point.

math.NT↗
arXiv2021

Le problème inverse de Galois sur les corps des fractions tordus à indéterminée centrale

In this article, we show that the Inverse Galois Problem over a skew field $H$ of finite dimension over its center $k$ is equivalent to a variant of the Inverse Galois Problem over $k$ involving a polynomial constraint. As an application, we show that if $k$ contains an ample field, then the Inverse Galois Problem has a positive answer over the skew field $H(t)$ of rational fractions with central indeterminate.

math.NT↗