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| André motive | |
|---|---|
| Name | André motive |
| Field | Algebraic geometry |
| Introduced | 1990s |
| Founder | Yves André |
André motive
The André motive is a conjectural and formal object introduced to refine the theory of motives and to provide an unconditional tannakian framework linking Hodge theory, étale cohomology, and algebraic cycles; it sits alongside concepts such as Grothendieck motive, Tannakian category, Hodge conjecture, and Tate conjecture. Conceived by Yves André in response to limitations in the constructions of Jean-Pierre Serre, Alexander Grothendieck, and Pierre Deligne, the André motive systematizes realizations related to periods, Galois representations, l-adic cohomology, and mixed Hodge structures.
André motives originate in work by Yves André building on the framework of Grothendieck's motives and the tannakian approach of Saavedra Rivano and Pierre Deligne, with antecedents in conjectures of Alexander Grothendieck and developments by Jean-Pierre Serre and John Tate. André formulated a category of motives—often called the category of André motives—designed to bypass dependence on unresolved conjectures such as the Hodge conjecture and the Standard conjectures on algebraic cycles by using the theory of motivated cycles inspired by ideas from Mumford–Tate group theory, Lefschetz theorem on hyperplane sections, and the study of algebraic cycles on abelian varieties.
The formal construction of André motives employs the language of Tannakian categorys, where objects are smooth projective varieties over a base field and morphisms are given by motivated correspondences defined via algebraic cycles and operations such as Künneth decomposition, Lefschetz operator, and Hodge projector. Realization functors send an André motive to concrete realizations: the Betti cohomology realization links to Hodge structure, the étale cohomology realization links to Galois representations of the absolute Galois group, and the de Rham cohomology realization links to Gauss–Manin connections; these functors respect compatibilities highlighted in the work of Pierre Deligne and Grothendieck. The tannakian group associated to the category of André motives generalizes the Mumford–Tate group and provides a motivic Galois group whose representations recover the various cohomological realizations, paralleling constructions by Ulf Persson and Christophe Soulé.
André motives share properties with Grothendieck motives while remaining unconditional: they form a semisimple tannakian category under standard hypotheses related to Weil cohomology theorys, admit tensor products and duals as in the formalism of Tannaka duality, and their motivic Galois groups control period relations akin to those studied by Kontsevich and Maxim Kontsevich in the theory of periods. Fundamental examples include motives attached to abelian varietys, K3 surfaces, and Shimura variety cohomology; in these cases André motives often coincide with expected Grothendieck motives as studied by Mikhail S. Rapoport, Gerd Faltings, and Richard Taylor. For elliptic curves with complex multiplication, the André motive construction recovers classical CM field structures and connects to work by Shimura and Taniyama on complex multiplication and modularity.
The André motive framework interacts strongly with central topics in algebraic geometry: it refines the theory of Hodge structures studied by Wilfried Schmid and Phillip Griffiths, links to the Tate conjecture via motivic Galois groups as in work by Tate and Serre, and informs the study of algebraic cycles initiated by Spencer Bloch and Beilinson. André motives are used in research on periods and transcendence involving contributors such as Goncharov and Pierre Deligne, and they provide tools for understanding the motivic underpinnings of results by André Weil, Grothendieck, and Deligne on comparison isomorphisms between cohomology theories. In the context of Shimura varietys and automorphic forms, André motives connect to conjectures tying Langlands program objects to motivic Galois representations, following paths traced by Michael Harris, Richard Taylor, and Jean-Pierre Serre.
Computational approaches to André motives leverage explicit descriptions of motivated cycles, algorithmic techniques for Hodge decomposition computation, and methods from l-adic cohomology calculations used by researchers like Andrew Wiles in modularity contexts and by Faltings in the proof of finiteness theorems. Practical computations often focus on explicit families such as elliptic curves, K3 surfaces, and modular curve cohomology where period matrices, Galois representation traces, and motivic Galois groups can be approximated numerically or symbolically, drawing on software inspired by computational projects in algebraic number theory and arithmetic geometry led by teams around John Cremona and William Stein. Applications appear in the study of period relations, transcendence questions, and the arithmetic of special values of L-functions, connecting André motives to conjectures of Beilinson, Bloch–Kato, and the broader Langlands program.
Category:Motives (algebraic geometry)