Department of Mathematics2024-11-0920100022-251810.1512/iumj.2010.59.42372-s2.0-79954429857http://dx.doi.org/10.1512/iumj.2010.59.4237https://hdl.handle.net/20.500.14288/13369In this paper we set up a general Kolyvagin system machinery for Euler systems of rank r (in the sense of Perrin-Riou) associated to a large class of Galois representations, building on our previous work on Kolyvagin systems of Rubin-Stark units and generalizing the results of Kato, Rubin and Perrin-Riou. Our machinery produces a bound on the size of the classical Selmer group attached to a Galois representation T (that satisfies certain technical hypotheses) in terms of a certain r x r determinant; a bound which remarkably goes hand in hand with Bloch-Kato conjectures. At the end, we present an application based on a conjecture of Perrin-Riou on p-adic L-functions, which lends further evidence to Bloch-Kato conjectures.MathematicsJournal Article289665400003Q11179