Lambda-operations for hermitian forms over algebras with involution of the first kind

The Grothendieck-Witt ring of a field is known to be a $\lambda$-ring, where the $\lambda$-operations are induced by the exterior powers of bilinear spaces. We give a similar construction on the mixed Grothendieck-Witt ring of a central simple algebra with involution of the first kind over a field. In doing so we also develop a general framework for pre-$\lambda$-ring structures on semi-rings graded over a monoid. Some explicit computations of even $\lambda$-powers are given in terms of restrictions of trace forms, and we explain how determinants of hermitian forms (which for us are maximal $\lambda$-powers) induce a duality similar to the one well-known for bilinear forms.Comment: 36 page

Mixed Witt rings of algebras with involution of the first kind

The Witt ring $W(K)$ of a field is a central object in the algebraic theory of quadratic forms. In general, we can define a Witt ring for commutative rings with involution. But if $(A,\sigma)$ is a central simple algebra with involution of the first over a field $K$, we only have Witt groups $W^\varepsilon(A,\sigma)$, mainly because $A$ is not commutative, so there is no appropriate tensor product of modules. We use the fact that there is a hermitian Morita equivalence between $(A\otimes_K A,\sigma\otimes \sigma)$ and $(K,Id)$ to define a commutative graded ring structure on $\widetilde{W}(A,\sigma) = W(K) \oplus W^+(A,\sigma)\oplus W^-(A,\sigma)$, the main difficulty being the associativity. We study the basic properties of this ring (the mixed Witt ring of $(A,\sigma)$), as well as the mixed Grothendieck-Witt ring $\widetilde{GW}(A,\sigma)$. In particular, $\widetilde{GW}(A,\sigma)$ is a pre-$\lambda$-ring for some exterior power operations defined through hermitian Morita equivalences. The mixed Witt ring has a fundamental filtration, and the associated graded ring has strong links with modulo 2 Galois cohomology. We also classify the prime ideals of $\widetilde{W}(A,\sigma)$ and relate this to the previous studies on signatures for hermitian forms

The XXL survey: XLVI. Forward cosmological analysis of the C1 cluster sample

We present the forward cosmological analysis of an $XMM$ selected sample of galaxy clusters out to a redshift of unity. Following our previous 2018 study based on the dn/dz quantity alone, we perform an upgraded cosmological analysis of the same XXL C1 cluster catalogue (178 objects), with a detailed account of the systematic errors. We follow the ASpiX methodology: the distribution of the observed X-ray properties of the cluster population is analysed in a 3D observable space (count rate, hardness ratio, redshift) and modelled as a function of cosmology. Compared to more traditional methods, ASpiX allows the inclusion of clusters down to a few tens of photons. We obtain an improvement by a factor of 2 compared to the previous analysis by letting the normalisation of the M-T relation and the evolution of the L-T relation free. Adding constraints from the XXL cluster 2-point correlation function and the BAO from various surveys decreases the uncertainties by 23 and 53 % respectively, and 62% when adding both. Switching to the scaling relations from the Subaru analysis, and letting free more parameters, our final constraints are $\sigma_8$ = $0.99^{+0.14}_{-0.23}$, $\Omega_m$ = 0.296 $\pm$ 0.034 ($S_8 = 0.98^{+0.11}_{-0.21}$) for the XXL sample alone. Finally, we combine XXL ASpiX, the XXL cluster 2-point correlation function and the BAO, with 11 free parameters, allowing for the cosmological dependence of the scaling relations in the fit. We find $\sigma_8$ = $0.793^{+0.063}_{-0.12}$, $\Omega_m$ = 0.364 $\pm$ 0.015 ($S_8 = 0.872^{+0.068}_{-0.12}$), but still compatible with Planck CMB at 2.2$\sigma$. The results obtained by the ASpiX method are promising; further improvement is expected from the final XXL cosmological analysis involving a cluster sample twice as large. Such a study paves the way for the analysis of the eROSITA and future Athena surveys.Comment: 20 pages, 10 figures, accepted for publication in A&A, A&A version has the unabridged abstrac

Mixed Witt rings of algebras with involution

International audienceAlthough there is no natural internal product for hermitian forms over an algebra with involution of the first kind, we describe how to multiply two ε-hermitian forms to obtain a quadratic form over the base field. This allows to define a commutative graded ring structure by taking together bilinear forms and ε-hermitian forms, which we call the mixed Witt ring of an algebra with involution. We also describe a less powerful version of this construction for unitary involutions, which still defines a ring, but with a grading over Z instead of the Klein group. We first describe a general framework for defining graded rings out of monoidal functors from monoidal categories with strong symmetry properties to categories of modules. We then give a description of such a strongly symmetric category Br h (K, ι) which encodes the usual hermitian Morita theory of algebras with involutions over a field K. We can therefore apply the general framework to Br h (K, ι) and the Witt group functors to define our mixed Witt rings, and derive their basic properties, including explicit formulas for products of diagonal forms in terms of involution trace forms, explicit computations for the case of quaternion algebras, and reciprocity formulas relative to scalar extensions. We intend to describe in future articles further properties of those rings, such as a λ-ring structure, and relations with the Milnor conjecture and the theory of signatures of hermitian forms

Witt and cohomological invariants of Witt classes

International audienceWe describe all Witt invariants and mod 2 cohomological invariants of the functor I^n as combinations of fundamental invariants; this is related to the study of operations on mod 2 Milnor K-theory. We also study behaviour of these invariants with respect to products, restrictions, similitudes and ramification.On décrit tous les invariants de Witt et les invariants cohomologiques modulo 2 du foncteur I^n comme combinaisons d'invariants fondamentaux; ceci est relié à l'études des opérations sur la K-théorie de Milnor modulo 2. On étudie aussi le comportement de ces invariants vis-à-vis des produits, des restrictions, des similitudes et de la ramification

The cosmological analysis of X-ray cluster surveys V. The potential of cluster counts in the 1<z<21<z<2 range

International audienceCosmological studies have now entered Stage IV according to the Dark Energy Task Force prescription, thanks to new missions (Euclid, Rubin Observatory, SRG/eROSITA) that are expected to provide the required ultimate accuracy in the dark energy (DE) equation of state (EoS). However, none of these projects have the power to systematically unveil the galaxy cluster population at $z>1$. There therefore remains the need for an ATHENA-like mission to run independent cosmological investigations and scrutinise the consistency between the results from the $01$. Such samples will allow a detailed modelling of the evolution of cluster physics along with a standalone cosmological analysis. Our results suggest that survey B has the optimal design as it provides greater statistics. Remarkably, high-$z$ clusters, despite representing 15% or less of the full samples, allow a significant reduction of the uncertainty on the cosmological parameters: $\Delta w_a$ is reduced by a factor of 2.3 and $\Delta f_{NL}^{loc}$ by a factor of 3. Inventorying the high-$z$ X-ray cluster population can play a crucial role in ensuring overall cosmological consistency. This will be the major aim of future new-generation ATHENA-like missions

NGS2: A focal plane array upgrade for the GeMS multiple tip-Tilt wavefront sensor

NGS2 is an upgrade for the multi-natural guide star tip-tilt & plate scale wavefront sensor for GeMS (Gemini Multi-Conjugate Adaptive Optics system). It uses a single Nüvü HNü-512 Electron-Multiplied CCD array that spans the entire GeMS wavefront sensor focal plane. Multiple small regions-of-interest are used to enable frame rates up to 800Hz. This set up will improve the optical throughput with respect to the current wavefront sensor, as well as streamline acquisition and allow for distortion compensation

History of tonsillectomy and risk of oropharyngeal cancer

International audienceObjective: To investigate whether palatine tonsillectomy in youth influences the risk of oropharyngeal cancers (OPC) by assessing the association between history of tonsillectomy and risk of tonsillar, base of tongue (BOT) cancer, and other head and neck cancers (HNC). Materials and Methods: RACKAM was a case-case study comparing frequency of tonsillectomy history in individuals diagnosed with HNC from 2013 to 2018 in 15 centers across France. History of tonsillectomy was defined using combined assessment of patients and rsquo; recollections and surgeons and rsquo; visualizations of tonsil area. OPC subsite-specific odds ratios (OR) of tonsillectomy were calculated using multinomial logistic regression with nonoropharyngeal HNC as reference. Results: 1045 patients were included in the study. Frequency of tonsillectomy was 19.5% in patients with tonsillar cancer (N = 85), 49.3% in BOT (N = 76), 33.8% in other oropharyngeal cancers (N = 202) and 38.0% in non-oropharyngeal HNC (N = 682). History of tonsillectomy was inversely associated with tonsillar cancer (adjusted OR 0.4; 95% CI 0.2 and ndash;0.8), and positively associated with BOT cancer (adjusted OR 1.8; 95% CI 1.1 and ndash;3.1), but was not associated with all OPC combined (adjusted OR 1.1; 95% CI 0.8 and ndash;1.4). Sensitivity analyses considering only patients and rsquo; or surgeons and rsquo; assessments of tonsillectomy provided comparable results. Conclusion: We confirm the long-term protective effect of tonsillectomy performed in youth on future risk of tonsillar cancer, and our study is the second to report a concurrent increased risk of BOT cancer. Our data suggest that tonsillectomy in youth shifts the site of the first diagnosed oropharyngeal tumor and has a limited impact on overall risk of OPC