By Günter Harder
ISBN-10: 3540574085
ISBN-13: 9783540574088
The purpose of this booklet is to teach that Shimura types supply a device to build definite fascinating items in mathematics algebraic geometry. those gadgets are the so-called combined causes: those are of significant mathematics curiosity. they are often seen as quasiprojective algebraic forms over Q that have a few managed ramification and the place we all know what we need to upload at infinity to compactify them. The life of yes of those combined factors is expounded to zeroes of L-functions hooked up to yes natural factors. this can be the content material of the Beilinson-Deligne conjectures that are defined in a few element within the first bankruptcy of the booklet. the remainder of the booklet is dedicated to the outline of the overall rules of building (Chapter II) and the dialogue of numerous examples in bankruptcy II-IV. In an appendix we clarify how the (topological) hint formulation can be utilized to get a few figuring out of the issues mentioned within the ebook. just some of this fabric is basically proved: the booklet additionally includes speculative concerns, which offer a few tricks as to how the issues should be tackled. consequently the e-book can be seen because the define of a programme and it bargains a few fascinating difficulties that are of significance and will be pursued by means of the reader. within the widest experience the topic of the paper is quantity thought and belongs to what's known as mathematics algebraic geometry. therefore the reader could be acquainted with a few algebraic geometry, quantity conception, the speculation of Liegroups and their mathematics subgroups. a few difficulties pointed out require simply a part of this history wisdom.
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Extra info for Eisensteinkohomologie und die Konstruktion gemischter Motive
Example text
Vari´et´es presque rationnelles, leurs points rationnels et leurs d´eg´en´erescences 33 On dispose alors d’un morphisme d’´evaluation M 0,2 (X, e) → X × X. La fibre g´en´erale de ce morphisme est un analogue de l’espace des chemins a` points base en topologie. La vari´et´e (projective et lisse) X est dite rationnellement simplement connexe si pour e ≥ 1 suffisamment grand il existe une composante M de M 0,2 (X, e) dominant X × X telle que la fibre g´en´erique de M → X × X soit une vari´et´e rationnellement connexe.
6 (d) de [17] et les r´esultats de [20]. 11 pour les surfaces (projectives et lisses) g´eom´etriquement rationnelles d´efinies sur C(t) impliquerait l’unirationalit´e des vari´et´es de dimension 3 sur C qui admettent une fibration en coniques sur le plan projectif. Il s’agit l`a d’une question largement ouverte. 12 Soient K un corps de nombres et X une K-vari´et´e rationnellement connexe. Le quotient X (K)/R est-il fini ? C’est connu dans les cas suivants : (i) La vari´et´e X est une compactification lisse d’un groupe lin´eaire connexe G.
Supposons p ≡ 1 mod 3, et soit a ∈ Z× p non cube. Qu’en est-il pour l’hypersurface x3 + y3 + z3 + p(u31 + au32) + p2 (v31 + av32) = 0 dans P6Q p ? Vari´et´es presque rationnelles, leurs points rationnels et leurs d´eg´en´erescences 37 Sur le corps K = C((a))((b)), en utilisant la th´eorie de l’intersection sur un mod`ele au-dessus de C((a))[[b]], Madore [58] a montr´e que pour l’hypersurface cubique lisse X ⊂ P4K d’´equation x3 + y3 + az3 + bu3 + abv3 = 0, on a A0 (X) = 0. 3 Intersections lisses de deux quadriques Soit K un corps p-adique, et soit X ⊂ PnK , avec n ≥ 4, une intersection compl`ete lisse de deux quadriques poss´edant un K-point.
Eisensteinkohomologie und die Konstruktion gemischter Motive by Günter Harder
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