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Harmonic analysis --- Shimura varieties --- Trace formulas

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This monograph is concerned with the Shimura variety attached to a quaternion algebra over a totally real number field. For any place of good (or moderately bad) reduction, the corresponding (semi-simple) local zeta function is expressed in terms of (semi-simple) local L-functions attached to automorphic representations. In an appendix a conjecture of Langlands and Rapoport on the reduction of a Shimura variety in a very general case is restated in a slightly stronger form. The reader is expected to be familiar with the basic concepts of algebraic geometry, algebraic number theory and the theory of automorphic representation.

Number theory --- Shimura varieties. --- L-functions. --- Functions, Zeta. --- Quaternions. --- Shimura varieties --- L-functions --- Functions, Zeta --- Quaternions --- Mathematical Theory --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Fonctions L. --- Fonctions zêta --- Functies L. --- Functions [Zeta ] --- Funsties [Zêta ] --- Shimura variëteiten --- Variétés de Shimura --- Zeta functions --- Zêta functies --- Number theory. --- Algebraic geometry. --- Number Theory. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Number study --- Numbers, Theory of --- Varieties, Shimura --- Arithmetical algebraic geometry

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51 <082.1> --- Mathematics--Series --- Homotopy groups. --- Shimura varieties. --- Formes automorphes --- Topologie algébrique --- Groupes d'homotopie --- Shimura, Variétés de --- Topologie algébrique --- Shimura, Variétés de --- Algebraic topology --- Automorphic forms. --- Algebraic topology. --- Automorphic forms --- Homotopy groups --- Shimura varieties --- Varieties, Shimura --- Arithmetical algebraic geometry --- Group theory --- Homotopy theory --- Automorphic functions --- Forms (Mathematics) --- Topology

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This text presents an important breakthrough in arithmetic geometry. In 2014, this work's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of 'diamonds,' which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The text follows the informal style of the original Berkeley lectures, with one chapter per lecture.

p-adic analysis. --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Adic spaces. --- Dieudonné theory. --- Drinfeld’s lemma. --- Fargues-Fontaine curve. --- Pre-adic spaces. --- Shimura varieties. --- affine Grassmannians. --- cohomology of local systems. --- flag varieties. --- formal schemes. --- integral models. --- perfectoid rings. --- torsors. --- v-sheaves. --- v-topology. --- vector bundles.

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This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action--at good places--on the G(Af)-isotypical components of the cohomology. Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups.

Shimura varieties. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Varieties, Shimura --- Arithmetical algebraic geometry --- Accuracy and precision. --- Adjoint. --- Algebraic closure. --- Archimedean property. --- Automorphism. --- Base change map. --- Base change. --- Calculation. --- Clay Mathematics Institute. --- Coefficient. --- Compact element. --- Compact space. --- Comparison theorem. --- Conjecture. --- Connected space. --- Connectedness. --- Constant term. --- Corollary. --- Duality (mathematics). --- Existential quantification. --- Exterior algebra. --- Finite field. --- Finite set. --- Fundamental lemma (Langlands program). --- Galois group. --- General linear group. --- Haar measure. --- Hecke algebra. --- Homomorphism. --- L-function. --- Logarithm. --- Mathematical induction. --- Mathematician. --- Maximal compact subgroup. --- Maximal ideal. --- Morphism. --- Neighbourhood (mathematics). --- Open set. --- Parabolic induction. --- Permutation. --- Prime number. --- Ramanujan–Petersson conjecture. --- Reductive group. --- Ring (mathematics). --- Scientific notation. --- Shimura variety. --- Simply connected space. --- Special case. --- Sub"ient. --- Subalgebra. --- Subgroup. --- Symplectic group. --- Theorem. --- Trace formula. --- Unitary group. --- Weyl group.