arXiv subjects

Ryotaro Sakamoto

Publications and source records attributed to Ryotaro Sakamoto.

21 records · Page 2Linked to original sources

arXiv2019

On the theory of higher rank Euler, Kolyvagin and Stark systems, III: applications

In an earlier article we proved the existence of a canonical Kolyvagin derivative homomorphism between the modules of Euler and Kolyvagin systems (in any given rank) that are associated to $p$-adic representations over number fields. We now explain how the existence of such a homomorphism leads to new results on the structure of the Selmer modules of Galois representations over Gorenstein orders and to a strategy for verifying (refinements of) the Tamagawa number conjecture of Bloch and Kato. We describe concrete applications relating to the multiplicative group over arbitrary number fields and to elliptic curves over abelian extensions of the rational numbers.

math.NT↗

arXiv2018

On the theory of higher rank Euler, Kolyvagin and Stark systems, II

We prove the existence of a canonical `higher Kolyvagin derivative' homomorphism between the modules of higher rank Euler systems and higher rank Kolyvagin systems, as has been conjectured to exist by Mazur and Rubin. This homomorphism exists in the setting of $p$-adic representations that are free with respect to the action of a Gorenstein order $\mathcal{R}$ and, in particular, implies that higher rank Euler systems control the $\mathcal{R}$-module structures of Selmer modules attached to the representation. We give a first application of this theory by considering the (conjectural) Euler system of Rubin-Stark elements.

math.NT↗

arXiv2017

Stark Systems over Gorenstein Local Rings

In this paper, we define a module of Stark systems over a complete Gorenstein local ring with a finite residue field of odd characteristic. Under some mild assumptions, we show that it is free of rank one and that these systems control all the Fitting ideals of the Pontryagin dual of the dual Selmer group. These results generalize the work of B. Mazur and K. Rubin on Stark systems over a principal ideal ring.

math.NT↗