Contou-Carrere symbols and Riemann-Roch theorems

The Contou-Carrere symbol was introduced by C. Contou-Carrere and P. Deligne. This symbol generalizes the residue of differential form and the tame symbol at a point on an algebraic curve, and can be considered as the deformation of the tame symbol. By two invertible elements of the algebra of Laurent series over a commutative ring this symbol defines an invertible element of the base ring. The Contou-Carrere symbol is connected with the class field theory for an algebraic curve over a finite field, and it satisfies the reciprocity laws. In my talk I will speak about these classical results and also about my recent results on the connection of the Contou-Carrere symbol with the Grothendieck-Riemann-Roch theorem for the family of projective curves. This connection is via the central extension of the group that is the semidirect product of the group of invertible elements of the algebra of Laurent series over a ring and the group of continuous automorphisms of this algebra.