On the structure of signed Selmer groups

Abstract

Let F be a number field unramified at an odd prime p and F∞ be the Zp-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 Gal(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 Zp[[Gal(F∞/F )]]modules, we show that they have no proper sub-Zp[[Gal(F∞/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.

On the signed Selmer groups of congruent elliptic curves with semistable reduction at all primes above $p$

“…We note that even in the case where the elliptic curves have ordinary reduction at all primes above p, our result is an improvement of the results of Greenberg-Vatsal [10] and Ahmed-Aribam-Shekhar [1,32] for we do not require the assumption that E(F )[p] = 0. We should mention that the results of Greenberg-Vatsal and Kim have also been established for modular forms of higher weight and even more general Galois representations (for instances, see [6,12,15,30]). However in these prior works, they always work with coherent reduction types above p. In other words, their Galois representations are assumed to have either ordinary reduction at all primes above p or non-ordinary reduction at all primes above p.…”

Euler characteristics and their congruences for multisigned Selmer groups

“…Their results were generalized to the p-supersingular case for the plus and minus Selmer groups by B.D. Kim in [17] and Ponsinet [32].…”

Fine Selmer groups of congruent p-adic Galois representations

“…It also appears in work of Mazur and Rubin (see [MR04, Lemma 3.5.3]) and has been used in, e.g. [BS10] and [Pon20]. This approach is of particular interest if one wants to treat Z -extensions ∞ of in which some prime above or a ramified prime is completely split.…”