arXiv subjects

Yves André

Publications and source records attributed to Yves André.

At least 19 recordsLinked to original sources

arXiv2026

Non-abelian Rees construction and pure motives

The classical Rees construction (of common use in commutative algebra and Hodge theory) interpolates between filtrations, viewed as ${\mathbb G}_m$-equivariant vector bundles on the affine line, and their associated gradings. Various non-abelian versions have been proposed, where the multiplicative group ${\mathbb G}_m$ is replaced by an arbitrary reductive group. Building on a construction due to P. O'Sullivan, we present a Galois correspondence between quasi-homogeneous spaces and certain monoidal categories, and apply it to monoidal categories of motives with concrete applications to algebraic cycles. In particular, we give a new proof and generalization of the Clozel-Deligne theorem about numerical equivalence on abelian varieties over finite fields.

math.AG↗

arXiv2025

G-functions, motives, and unlikely intersections -- old and new

In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more ``motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the André-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.

math.NT↗

arXiv2025

On relative integral monodromy of abelian logarithms and normal functions

The relative algebraic monodromy of abelian logarithms (defined as the kernel of a map between algebraic monodromy groups attached to an abelian scheme with and without a section) was computed in \cite{A1}: under natural assumptions, this vector group turns out to be maximal. The relative integral monodromy of abelian logarithms is defined similarly as a kernel of integral monodromy groups, without taking Zariski closures. We show that if the integral monodromy of the abelian scheme is a lattice in the algebraic monodromy (which is not always the case), then the relative integral monodromy of the abelian logarithm is also a lattice in the relative algebraic monodromy. The proof uses a Hodge-theoretic interpretation of sections of abelian schemes. We also consider relative integral monodromy groups in the more general context of normal functions.

math.AG↗

arXiv2023

On the observability of Galois representations and the Tate conjecture

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using $p$-adic Hodge theory. We show that an unconditional result lies behind this implication: the {\it observability} of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen's result. We also discuss another aspect of the Tate conjecture related to the transcendence of $p$-adic periods.

math.AG↗

arXiv2022

Result of the MICROSCOPE Weak Equivalence Principle test

The space mission MICROSCOPE dedicated to the test of the Equivalence Principle (EP) operated from April 25, 2016 until the deactivation of the satellite on October 16, 2018. In this analysis we compare the free-fall accelerations ($a_{\rm A}$ and $a_{\rm B}$) of two test masses in terms of the Eötvös parameter $η({\rm{A, B}}) = 2 \frac{a_{\rm A}- a_{\rm B}}{a_{\rm A}+ a_{\rm B}}$. No EP violation has been detected for two test masses, made from platinum and titanium alloys, in a sequence of 19 segments lasting from 13 to 198 hours down to the limit of the statistical error which is smaller than $10^{-14}$ for $ η({\rm{Ti, Pt}})$. Accumulating data from all segments leads to $η({\rm{Ti, Pt}}) =[-1.5\pm{}2.3{\rm (stat)}\pm{}1.5{\rm (syst)}] \times{}10^{-15}$ showing no EP violation at the level of $2.7\times{}10^{-15}$ if we combine stochastic and systematic errors quadratically. This represents an improvement of almost two orders of magnitude with respect to the previous best such test performed by the Eöt-Wash group. The reliability of this limit has been verified by comparing the free falls of two test masses of the same composition (platinum) leading to a null Eötvös parameter with a statistical uncertainty of $1.1\times{}10^{-15}$.

gr-qc↗

arXiv2022

MICROSCOPE mission: final results of the test of the Equivalence Principle

The MICROSCOPE mission was designed to test the Weak Equivalence Principle (WEP), stating the equality between the inertial and the gravitational masses, with a precision of $10^{-15}$ in terms of the Eötvös ratio $η$. Its experimental test consisted of comparing the accelerations undergone by two collocated test masses of different compositions as they orbited the Earth, by measuring the electrostatic forces required to keep them in equilibrium. This was done with ultra-sensitive differential electrostatic accelerometers onboard a drag-free satellite. The mission lasted two and a half years, cumulating five-months-worth of science free-fall data, two thirds with a pair of test masses of different compositions -- Titanium and Platinum alloys -- and the last third with a reference pair of test masses of the same composition -- Platinum. We summarize the data analysis, with an emphasis on the characterization of the systematic uncertainties due to thermal instabilities and on the correction of short-lived events which could mimic a WEP violation signal. We found no violation of the WEP, with the Eötvös parameter of the Titanium and Platinum pair constrained to $η({\rm Ti, Pt})~=~ [-1.5 \pm 2.3~{\rm (stat)} \pm 1.5~{\rm (syst)}]~\times 10^{-15}$ at $1σ$ in statistical errors.

gr-qc↗

arXiv2021

On the canonical, fpqc, and finite topologies on affine schemes. The state of the art

This is a systematic study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the canonical topology and the fpqc topology. The history of the problem takes roots in the foundations of Grothendieck topologies, passes through main strides in Commutative Algebra and leads to new Mathematics up to perfectoids and prisms. We first review the canonical topology of affine schemes and show, keeping with Olivier's lost work, that it coincides with the effective descent topology; covering maps are given by universally injective ring maps, which we discuss in detail. We then give a "catalogue raisonné" of examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology). We discuss a "weakly functorial" aspect of this result. "Splinters" are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. We also investigate their mysterious fpqc analogs, and prove that in prime characteristic, they are all regular. This leads us to the problem of descent of regularity by (non-necessarily flat) morphisms $f$ which are coverings for the fpqc topology, which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma.

math.AG↗

arXiv2020

MICROSCOPE Satellite and its Drag-Free and Attitude Control System

This paper focus on the description of the design and performance of the MICROSCOPE satellite and its Drag-Free and Attitude Control System (DFACS). The satellite is derived from CNES' Myriade platform family, albeit with significant upgrades dictated by the unprecedented MICROSCOPE's mission requirements. The 300kg drag-free microsatellite has completed its 2-year flight with higher-than-expected performances. Its passive thermal concept allowed for variations smaller than 1 $μ$K at the measurement frequency $f_{\rm{EP}}$. The propulsion system provided a 6 axis continuous and very low noise thrust from zero to some hundreds of micronewtons. Finally, the performance of its DFACS (aimed at compensating the disturbing forces and torques applied to the satellite) is the finest ever achieved in low Earth orbit, with residual accelerations along the three axes are lower than $10^{-12} {\rm m/s}^2$ at $f_{\rm{EP}}$ over 8 days.

gr-qc↗

arXiv2020

MICROSCOPE instrument description and validation

Dedicated accelerometers have been developed for the MICROSCOPE mission taking into account the specific range of acceleration to be measured on board the satellite. Considering one micro-g and even less as the full range of the instrument, leads to a customized concept and a high performance electronics for the sensing and servo-actuations of the accelerometer test-masses. In addition to a very accurate geometrical sensor core, a high performance electronics architecture provides the measurement of the weak electrostatic forces and torques applied to the test-masses. A set of capacitive sensors delivers the position and the attitude of the test-mass with respect to a very steady gold coated cage made in silica. The voltages applied on the electrodes surrounding each test-mass are finely controlled to generate the adequate electrical field and so the electrostatic pressures on the test-mass. This field maintains the test-mass motionless with respect to the instrument structure. Digital control laws are implemented in order to enable instrument operation flexibility and a weak position sensor noise. These electronics provide both the scientific data for MICROSCOPE's test of General Relativity and the data for the satellite drag-free and attitude control system (DFACS).

physics.ins-det↗

arXiv2019

Space test of the Equivalence Principle: first results of the MICROSCOPE mission

The Weak Equivalence Principle (WEP), stating that two bodies of different compositions and/or mass fall at the same rate in a gravitational field (universality of free fall), is at the very foundation of General Relativity. The MICROSCOPE mission aims to test its validity to a precision of $10^{-15}$, two orders of magnitude better than current on-ground tests, by using two masses of different compositions (titanium and platinum alloys) on a quasi-circular trajectory around the Earth. This is realised by measuring the accelerations inferred from the forces required to maintain the two masses exactly in the same orbit. Any significant difference between the measured accelerations, occurring at a defined frequency, would correspond to the detection of a violation of the WEP, or to the discovery of a tiny new type of force added to gravity. MICROSCOPE's first results show no hint for such a difference, expressed in terms of Eötvös parameter $δ(Ti,Pt)=[-1\pm{}9{\rm (stat)}\pm{}9{\rm (syst)}] \times{}10^{-15}$ (both 1$σ$ uncertainties) for a titanium and platinum pair of materials. This result was obtained on a session with 120 orbital revolutions representing 7\% of the current available data acquired during the whole mission. The quadratic combination of 1$σ$ uncertainties leads to a current limit on $δ$ of about $1.3\times{}10^{-14}$.

gr-qc↗

arXiv2018

A note on 1-motives

We prove that for $1$-motives defined over an algebraically closed subfield of $\C$, viewed as Nori motives, the motivic Galois group is the Mumford-Tate group. In particular, the Hodge realization of the tannakian category of (Nori) motives generated by $1$-motives is fully faithful.

math.AG↗

arXiv2018

The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao)

Given a point $ξ$ on a complex abelian variety $A$, its abelian logarithm can be expressed as a linear combination of the periods of $A$ with real coefficients, the Betti coordinates of $ξ$. When $(A, ξ)$ varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often $ξ$ takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when $ξ$ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira-Spencer map associated to $(A, ξ)$ (assuming $A$ without fixed part, and $\mathbb{Z} ξ$ Zariski-dense in $A$), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension $\leq 3$, and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if $A\to S$ is a principally polarized abelian scheme of relative dimension $g$ which has no non-trivial endomorphism (on any finite covering), and if the image of $S$ in the moduli space $\mathcal{A}_g$ has dimension at least $g$, then the Betti map of any non-torsion section $ξ$ is generically a submersion, so that $ξ^{-1}A_{tors}$ is dense in $S(\mathbb{C})$.

math.AG↗

arXiv2017

The MICROSCOPE mission: first results of a space test of the Equivalence Principle

According to the Weak Equivalence Principle, all bodies should fall at the same rate in a gravitational field. The MICROSCOPE satellite, launched in April 2016, aims to test its validity at the $10^{-15}$ precision level, by measuring the force required to maintain two test masses (of titanium and platinum alloys) exactly in the same orbit. A non-vanishing result would correspond to a violation of the Equivalence Principle, or to the discovery of a new long-range force. Analysis of the first data gives $δ\rm{(Ti,Pt)}= [-1 \pm 9 (\mathrm{stat}) \pm 9 (\mathrm{syst})] \times 10^{-15}$ (1$σ$ statistical uncertainty) for the titanium-platinum Eötvös parameter characterizing the relative difference in their free-fall accelerations.

astro-ph.IM↗

arXiv2016

On the Kodaira-Spencer map of abelian schemes

Let $A$ be an abelian scheme over a smooth affine complex variety $S$, $\varOmega_A$ the $\sO_S$-module of $1$-forms of the first kind on $A$, $\sD_S\varOmega_A$ the $\sD_S$-module spanned by $\varOmega_A$ in the first algebraic De Rham cohomology module, and $θ_\partial: \varOmega_A \to \sD_S\varOmega_A/\varOmega_A$ the Kodaira-Spencer map attached to a tangent vector field $\partial$ on $S$. We compare the rank of $\sD_S\varOmega_A/\varOmega_A$ to the maximal rank of $θ_\partial$ when $\partial$ varies: we show that both ranks do not change when one passes to the "modular case", \ie when one replaces $S$ by the smallest weakly special subvariety of $\sA_g$ containing the image of $S$ (assuming, as one may up to isogeny, that $A/S$ is principally polarized), we then analyse the "modular case" and deduce, for instance, that {\it for any abelian pencil of relative dimension $g$ with Zariski-dense monodromy in $Sp_{2g}$}, {\it the derivative with respect to a parameter of a non zero abelian integral of the first kind is never of the first kind}.

math.AG↗

arXiv2016

Groupes de Galois motivques et périodes

In the mid sixties, A. Grothendieck envisioned a vast generalization of Galois theory to systems of polynomials in several variables, motivic Galois theory, and introduced tannakian categories on this occasion. In characteristic zero, various unconditional approaches were later proposed. The most precise one, due to J. Ayoub, relies on Voevodsky theory of mixed motives and on a new tannakian theory. It sheds new light on periods of algebraic varieties, and shows in particular that polynomial relations between periods of a pencil of algebraic varieties always arise from Stokes formula.

math.AG↗

arXiv2012

Uniform sheaves and differential equations

Real blow-ups and more refined "zooms" play a key role in the analysis of singularities of complex-analytic differential modules. They do not change the underlying topology, but the uniform structure. This suggests to revisit the cohomology theory of differential modules with help of a suitable new notion of uniform sheaves based on the uniformity rather than the topology. We also investigate the $p$-adic situation (in particular, an analog of real blow-ups) from this uniform viewpoint.

math.AG↗

arXiv2012

Idées galoisiennes

Classical applications of Galois theory concern algebraic numbers and algebraic functions. Still, the night before his duel, Galois wrote that his last mathematical thoughts had been directed toward applying his "theory of ambiguity to transcendental functions and transcendental quantities". We outline some more or less recent ideas and results in this direction.

math.HO↗

arXiv2010

On nef and semistable hermitian lattices, and their behaviour under tensor product

We study the behaviour of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces. One approach to show that semistable vector bundles in characteristic zero are preserved by tensor product is based on the notion of nef vector bundles. We revisit this approach and show how far it can be transferred to hermitian lattices. J.-B. Bost conjectured that semistable hermitian lattices are preserved by tensor product. Using properties of nef hermitian lattices, we establish an inequality which improves on earlier results in that direction. On the other hand, we show that, in contrast to nef vector bundles, nef hermitian lattices are not preserved by tensor product. We axiomatize our method in the general context of monoidal categories, and give an elementary proof of the fact that semistable multifiltered vector spaces (which play a role in diophantine approximation) are preserved by tensor product.

math.NT↗