Kolyvagin Systems Research Articles

Refined class number formulas and Kolyvagin systems

AbstractWe use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that, for every odd prime p, each side of Darmon’s conjectured formula (indexed by positive integers n) is ‘almost’ a p-adic Kolyvagin system as n varies. Using the fact that the space of Kolyvagin systems is free of rank one over ℤp, we show that Darmon’s formula for arbitrary n follows from the case n=1, which in turn follows from classical formulas.

On Euler systems of rank $r$ and their Kolyvagin systems

In 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 Galoys representation T (that satisfies certain technical hypotheses) in terms of a certain r \times 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.

Stark units and the main conjectures for totally real fields

AbstractThe main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

Kolyvagin systems of Stark units

In this paper we construct, using Stark elements of Rubin [Ann. Inst. Fourier 46: 33–62, 1996], Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one in the sense of [Mazur and Rubin, Mem. Amer. Math. Soc. 168: viii, 96, 2004]) and prove a Gras-type conjecture, relating these Kolyvagin systems to appropriate ideal class groups, refining the results of [Rubin, J. reine angew. Math. 425: 141–154, 1992] (in a sense we explain below), and of [Perrin-Riou, Ann. Inst. Fourier 48: 1231–1307, 1998], [Rubin, Euler systems, Princeton University Press, 2000] applied to our setting.

Tamagawa defect of Euler systems

As remarked by Mazur and Rubin [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)] one does not expect the Kolyvagin system obtained from an Euler system for a p-adic Galois representation T to be primitive (in the sense of the above mentioned reference) if p divides a Tamagawa number at a prime ℓ ≠ p ; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map in terms of the local Tamagawa numbers of T, refining a result of [B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (799) (2004)]. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system.