Applications to Arithmetic Geometry Values of L-functions Let X be a smooth projective variety over a number field. The Bloch-Kato conjecture on values of L-functions predicts that the order of vanishing of an L-function of X at an integer point is equal to the rank of a suitable motivic cohomology group. This is one of the central problems of number theory, incorporating earlier conjectures by Deligne and Beilinson. The Birch–Swinnerton-Dyer conjecture is a special case. More precisely, the conjecture predicts the leading coefficient of the L-function at an integer point in terms of regulators and a height pairing on motivic cohomology. History The first clear sign of a possible generalization from Chow groups to a more general motivic cohomology theory for algebraic varieties was Quillen's definition and development of algebraic K-theory (1973), generalizing the Grothendieck group K0 of vector bundles. In the early 1980s, Beilinson and Soulé observed that Adams operations gave a splitting of algebraic K-theory tensored with the rationals; the summands are now called motivic cohomology (with rational coefficients). Beilinson and Lichtenbaum made influential conjectures predicting the existence and properties of motivic cohomology. Most but not all of their conjectures have now been proved. Bloch's definition of higher Chow groups (1986) was the first integral (as opposed to rational) definition of motivic homology for schemes over a field k (and hence motivic cohomology, in the case of smooth schemes). The definition of higher Chow groups of X is a natural generalization of the definition of Chow groups, involving algebraic cycles on the product of X with affine space which meet a set of hyperplanes (viewed as the faces of a simplex) in the expected dimension. Finally, Voevodsky (building on his work with Suslin) defined the four types of motivic homology and motivic cohomology in 2000, along with the derived category of motives. Related categories were also defined by Hanamura and Levine. Notes References See also Presheaf with transfers A¹ homotopy theory External links Harrer Daniel, Comparison of the Categories of Motives defined by Voevodsky and Nori Wiesława Nizioł, p-adic motivic cohomology in arithmetic Cohomology theories Homotopical algebra Topological methods of algebraic geometry