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Bilinear forms for SL(2,q), An and similar groups.

Alexandre Turull (1992)

Publicacions Matemàtiques

The set of invariant symmetric bilinear forms on irreducible modules over fields of characteristic zero for certain groups is studied. Results are obtained under the presence in a finite group of elements of order four whose square is central. In particular, we find that the relevant modules for the groups mentioned in the title always accept an invariant symmetric bilinear form under which the module admits an orthonormal basis.

Binary quadratic forms and Eichler orders

Montserrat Alsina (2005)

Journal de Théorie des Nombres de Bordeaux

For any Eichler order 𝒪 ( D , N ) of level N in an indefinite quaternion algebra of discriminant D there is a Fuchsian group Γ ( D , N ) SL ( 2 , ) and a Shimura curve X ( D , N ) . We associate to 𝒪 ( D , N ) a set ( 𝒪 ( D , N ) ) of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to Γ ( D , N ) , for primitive forms contained in ( 𝒪 ( D , N ) ) . In particular, the classification theory of primitive integral binary quadratic forms by SL ( 2 , ) is recovered. Explicit fundamental domains for Γ ( D , N ) allow the characterization...

Birational geometry of quadrics

Burt Totaro (2009)

Bulletin de la Société Mathématique de France

We construct new birational maps between quadrics over a field. The maps apply to several types of quadratic forms, including Pfister neighbors, neighbors of multiples of a Pfister form, and half-neighbors. One application is to determine which quadrics over a field are ruled (that is, birational to the projective line times some variety) in a larger range of dimensions. We describe ruledness completely for quadratic forms of odd dimension at most 17, even dimension at most 10, or dimension 14....

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