On Selmer complexes, Stark systems and derived $p$-adic heights

TL;DR

This paper advances the theory of Selmer complexes by establishing their equivalence to Poitou-Tate complexes, linking determinants to Stark systems, and comparing $p$-adic height pairings, with applications to controlling Selmer groups.

Contribution

It develops a canonical isomorphism between Selmer complexes and Poitou-Tate complexes, introduces a Heegner point Stark system, and compares different $p$-adic height pairings.

Findings

Selmer complexes are quasi-isomorphic to Poitou-Tate complexes.

Determinants of Selmer complexes are isomorphic to Stark systems.

Derived $p$-adic height pairings coincide with those of Nekovár.

Abstract

We develop the theory of Nekov\'a\v{r}'s Selmer complexes. We prove that, under mild hypotheses, Nekov\'a\v{r}'s Selmer complexes are canonically quasi-isomorphic to ``Poitou-Tate complexes", which arise from Poitou-Tate global duality exact sequences. We give two applications. Firstly, we prove that the determinant of a Selmer complex is canonically isomorphic to the module of Stark systems and, by using this result, we construct a canonical ``Heegner point Stark system" which controls Selmer groups. Secondly, we prove that the derived $p$-adic height pairing of Bertolini-Darmon concides with that of Nekov\'a\v{r}.