Bloch–Kato conjectures on special values of L-functions

Prove the Bloch–Kato conjectures relating, for a motive M and its L-function L(M,s), the order of vanishing at integers to Selmer-group or K-group ranks and the leading Taylor coefficient to arithmetic invariants including regulators, periods, and Tamagawa numbers.

Background

The paper summarizes the Bloch–Kato framework that connects special values of L-functions to deep arithmetic invariants, encompassing celebrated cases such as the Birch and Swinnerton–Dyer conjecture.

These conjectures generalize and organize a wide range of conjectural relations between analytic behavior of L-functions and arithmetic geometry.