On the structure of signed Selmer groups

TL;DR

This paper studies the algebraic structure of signed Selmer groups associated with certain Galois representations over number fields, proving they have no proper finite index submodules and deriving explicit formulas for related arithmetic invariants.

Contribution

It establishes the simplicity of the structure of signed Selmer groups under cotorsion assumptions and derives explicit formulas for their Euler-Poincaré characteristic and comparisons of Iwasawa invariants.

Findings

Signed Selmer groups have no proper finite index submodules.

Explicit formula for the size of the Bloch-Kato Selmer group.

Comparison of Iwasawa invariants for isomorphic modulo p representations.

Abstract

Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be the $\mathbf{Z}_p$-cyclotomic extension of $F$. Generalizing Kobayashi plus/minus Selmer groups for elliptic curves, B\"uy\"ukboduk and Lei have defined modified Selmer groups, called signed Selmer groups, for certain non-ordinary $\mathrm{Gal}(\overline{F}/F)$-representations. In particular, their construction applies to abelian varieties defined over $F$ with good supersingular reduction at primes of $F$ dividing $p$. Assuming that these Selmer groups are cotorsion $\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, we show that they have no proper sub-$\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-module of finite index. We deduce from this a number of arithmetic applications. On studying the Euler-Poincar\'e characteristic of these Selmer groups, we obtain an explicit formula on the size of the Bloch-Kato Selmer group attached to these representations. Furthermore, for two such representations that are isomorphic modulo $p$, we compare the Iwasawa-invariants of their signed Selmer groups.