Invariants of a quadratic form attached to a tame covering of schemes
Journal de Théorie des Nombres de Bordeaux, Volume 12 (2000) no. 2, pp. 597-660.

We build on preceeding work of Serre, Esnault-Kahn-Viehweg and Kahn to establish a relation between invariants, in modulo 2 étale cohomology, attached to a tamely ramified covering of schemes with odd ramification indices. The first type of invariant is constructed using a natural quadratic form obtained from the covering. In the case of an extension of Dedekind domains, mains, this form is the square root of the inverse different equipped with the trace form. In the case of a covering of Riemann surfaces, it arises from a theta characteristic. The second type of invariant is constructed using the representation of the tame fundamental group, which corresponds to the covering. Our formula is valid in arbitrary dimension. For unramified coverings the result was proved by the above authors. The two main contributions of our work consist in (1) showing how to eliminate ramification to reduce to the unramified case, in such a way that the reduction is possible in arbitrary dimension, and; (2) getting around the difficulties, caused by the presence of crossings in the ramification divisor, by introducing what we call “normalisation along a divisor”. Our approach relies on a detailed analysis of the local structure of tame coverings. We include a review of the relevant material from the theory of quadratic forms on schemes and of the basic simplicial techniques needed for our purposes.

Nous étendons des résultats de Serre, Esnault-Kahn-Viehweg et Kahn, et nous montrons une relation entre des invariants dans la cohomologie étale modulo 2, qui sont obtenus à partir d’un revêtement modérément ramifié de schémas à ramification impaire. Le premier type d’invariant est construit à l’aide d’une forme quadratique naturelle définie par le revêtement. Dans le cas d’un revêtement de schémas de Dedekind cette forme est donnée par la racine carrée de la codifférente avec la forme trace. Dans le cas d’un revêtement de surfaces de Riemann la forme provient de l’existence d’une caractéristique théta canonique. Le deuxième type d’invariant est défini à l’aide de la représentation du groupe fondamental modéré, qui est attachée au revêtement. Notre formule est valable sans restriction sur la dimension. Pour les revêtements non-ramifiés la formule est due aux auteurs précités. Les deux contributions essentielles de notre travail sont de montrer (1) comment ramener la démonstration de la formule au cas non-ramifié en toute dimension et (2) comment maîtriser les difficultés provenant de la présence de points singuliers dans le lieu de ramification du revêtement, en utilisant ce que nous appelons «normalisation le long d’un diviseur». Notre approche toute entière est basée sur une analyse fine de la structure locale des revêtements modérément ramifiés. Nous présentons aussi un survol des notions de la théorie des formes quadratiques sur les schémas et les techniques simpliciales de base nécessaires pour la compréhension de notre travail.

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Philippe Cassou-Noguès; Boas Erez; Martin J. Taylor. Invariants of a quadratic form attached to a tame covering of schemes. Journal de Théorie des Nombres de Bordeaux, Volume 12 (2000) no. 2, pp. 597-660. doi : 10.5802/jtnb.299. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.299/

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