Science Mathematics

Topics in complex function theory. Abelian and modular by Carl Ludwig Siegel

By Carl Ludwig Siegel

Develops the better elements of functionality thought in a unified presentation. begins with elliptic integrals and capabilities and uniformization conception, keeps with automorphic features and the idea of abelian integrals and ends with the speculation of abelian capabilities and modular services in different variables. The final subject originates with the writer and looks right here for the 1st time in booklet shape.

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Extra resources for Topics in complex function theory. Abelian and modular functions of several variables

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7) Nous admettrons ici ce crit`ere, qui est cens´e figurer au paragraphe des plats. Son principal avantage est qu’il ne suppose pas a priori A r´eduit, ni a fortiori int`egre. Ici, on peut d´ej` a supposer dim A = dim B = 0. D’apr`es les rappels du d´ebut du num´ero, les id´eaux premiers p de A qui sont de rang 1 (resp. de rang 2) sont exactement les traces sur A des id´eaux premiers q de B qui sont de rang 1 (resp. de rang 2). 1). Appliquant le crit`ere de Serre, on trouve que A est normal si et seulement si B l’est.

La fi´e (resp. ´etale) alors f ∗ ΩY1 /S → ΩX/S r´eciproque est vraie dans le cas « non ramifi´e », si f est suppos´e localement de type fini. (5) Pour ces formules, cf. 3. ´ ES ´ DIFFERENTIELLES ´ 4. 1), mais peut aussi se voir directement comme le cas ´etale. Consid´erons le diagramme X ∆X/Y / X ×Y X  Y / X ×S X ∆Y /S  / Y ×S Y dans lequel X ×Y X s’identifie au produit fibr´e de Y et X ×S X sur Y ×S Y . Comme f est non ramifi´e, X → X ×Y X est une immersion ouverte, donc le faisceau « conormal » de l’immersion compos´ee ∆X/S de cette derni`ere avec X ×Y X → X ×S X est isomorphe ` a l’image inverse sur X du faisceau conormal pour l’immersion X ×Y X → X ×S X.

Tn ] et X ´etale sur Y [s1 , . . , sm ], la premi`ere hypoth`ese implique donc que Y [s1 , . . , sm ] est ´etale sur Z[t1 , . . , tn ][s1 , . . , sm ] = Z[t1 , . . , sm ], donc X est ´etale sur Z[t1 , . . , sm ], cqfd. 5. — L’entier n qui figure dans d´ef. 1 est bien d´etermin´e, car on constate aussitˆot que c’est la dimension de l’anneau local de x dans sa fibre f −1 f (x) . On l’appelle « dimension relative » de X sur Y . Elle se comporte additivement pour la composition des morphismes.

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