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On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

A variety X over a field K is of Hilbert type if X ( K ) is not thin. We prove that if f : X S is a dominant morphism of K -varieties and both S and all fibers f - 1 ( s ) , s S ( K ) , are of Hilbert type, then so is X . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

On volumes of arithmetic quotients of S O ( 1 , n )

Mikhail Belolipetsky (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of S O ( 1 , n ) . As a result we prove that for any even dimension  n there exists a unique compact arithmetic hyperbolic n -orbifold of the smallest volume. We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We...

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