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, Büyükboduk 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–Poincaré 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.
Highlights
Let F be a number field and E be an elliptic curve defined over F
On the algebraic side of the Iwasawa theory for E developed by Mazur [22] is the p-Selmer group associated to E over F∞, denoted by Selp(E/F∞), which is naturally a discrete Zp[[Gal(F∞/F)]]-module
Kim [16] has extended the definition of these Selmer groups to number fields F where p is unramified and generalized Greenberg’s result and showed that if the signed Selmer groups of E over F∞ are cotorsion Zp[[Gal(F∞/F)]]modules, they have no proper submodule of finite index
Summary
Let F be a number field and E be an elliptic curve defined over F. Kim [16] has extended the definition of these Selmer groups to number fields F where p is unramified and generalized Greenberg’s result and showed that if the signed Selmer groups of E over F∞ are cotorsion Zp[[Gal(F∞/F)]]modules, they have no proper submodule of finite index (for one of the signed Selmer group, namely the plus one, he requires the additional assumption that p splits completely in F and is totally ramified in F∞). Assume that T / pT T / pT as Galois modules and that the Pontryagin dual of the signed Selmer groups associated to T , T ∗, T and T ,∗ are torsion Zp[[Gal(F∞/F)]]modules. This allows to keep track of the congruence through Büyükboduk and Lei’s construction
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