Constantin-Nicolae Beli
Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania

The cohomology with multiliear differentials is defined same as the usual cohomology, but with the group of cocycles, which are given by the relation $da=0$, replaced by the larger group of those cochains $a$ such that $da$ is multilinear, i.e. linear in each variable. This notion allows us to produce an exact sequence involving the $2$-torsion of the Brauer group of a number field. With the help of this sequence, we define a function with many arithmetic properties, which is useful in the theory of the spinor genus of integral quadratic forms. In the case when $F=\mathbb Q$ our map is given in terms of Legendre symbols of the type $(a+b\sqrt m,p)$ and by using its properties we were able to recover all existing biquadratic reciprocity laws, as well as produce new ones.