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Modular lattices from finite projective planes

Tathagata Basak (2014)

Journal de Théorie des Nombres de Bordeaux

Using the geometry of the projective plane over the finite field 𝔽 q , we construct a Hermitian Lorentzian lattice L q of dimension ( q 2 + q + 2 ) defined over a certain number ring 𝒪 that depends on q . We show that infinitely many of these lattices are p -modular, that is, p L q ' = L q , where p is some prime in 𝒪 such that | p | 2 = q .The Lorentzian lattices L q sometimes lead to construction of interesting positive definite lattices. In particular, if q 3 mod 4 is a rational prime such that ( q 2 + q + 1 ) is norm of some element in [ - q ] , then we find a 2 q ( q + 1 ) dimensional...

Motivic cohomology and unramified cohomology of quadrics

Bruno Kahn, R. Sujatha (2000)

Journal of the European Mathematical Society

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension 4 and 11 . Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics....

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