Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms – Part II

Hatley, Jeffrey; Lei, Antonio

doi:10.1016/j.jnt.2021.05.004

Cited by 7 publications

(7 citation statements)

References 24 publications

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“…This result is in the vein of work by ) and extends to the supersingular setting results of Hatley-Lei in the p-ordinary case ( [16,17]). The reader is referred to the introduction to Sect.…”

Section: Theorem 12 Assume (Heeg) Holds and Let Esupporting

confidence: 76%

“…Remark 1.3. Our arguments for obtaining these results have some significant differences from the strategy developed by Greenberg in [11], which has been generalized to many different settings (see, e.g., [15,16,20,22,24,35,37]). In [12], Greenberg showed that his strategy works for very general Selmer groups that can be defined by a surjective global-to-local map in cohomology.…”

Section: Theorem 12 Assume (Heeg) Holds and Let Ementioning

confidence: 84%

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$$\Lambda $$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

2021

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Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic Z p -extension of K does not admit any proper -submodule of finite index, where is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of p-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have -corank one, so they are not -cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg-Vatsal on Iwasawa invariants of p-congruent elliptic curves, extending to the supersingular case results for p-ordinary elliptic curves due to Hatley-Lei.

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Section: Theorem 12 Assume (Heeg) Holds and Let Esupporting

confidence: 76%

Section: Theorem 12 Assume (Heeg) Holds and Let Ementioning

confidence: 84%

$$\Lambda $$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

2021

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“…Moreover, over -extensions one can then often prove equality of -invariants (we refer to Section 2 for the definition of the Iwasawa invariants). Analogous results have been obtained for Selmer groups of Galois representations over the anticyclotomic -extension of an imaginary quadratic base field K (see [HL19]) and for signed Selmer groups of Galois representations over the cyclotomic -extension of a number field in the non-ordinary setting (see, e.g., [Pon20, Section 3]). Moreover, there exist vast generalisations to Selmer groups attached to families of modular forms (see, e.g., [EPW06, Sha09, Bar13]).…”

Section: Introductionsupporting

confidence: 68%

Fine Selmer groups of congruent p-adic Galois representations

Kleine

Müller²

2021

Can. Math. Bull.

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We compare the Pontryagin duals of fine Selmer groups of two congruent -adic Galois representations over admissible pro-, -adic Lie extensions ∞ of number fields . We prove that in several natural settings the -primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the -invariants. In the special case of a Z -extension ∞ / , we also compare the Iwasawa -invariants of the fine Selmer groups, even in situations where the -invariants are non-zero. Finally, we prove similar results for certain abelian non--extensions.

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“…Remark 1.3. Our arguments for obtaining these results have some significant differences from the strategy developed by Greenberg in [10], which has been generalized to many different settings (see, e.g., [14,15,17,19,21,33,32]). In [11], Greenberg showed that his strategy works for very general Selmer groups that can be defined by a surjective global-to-local map in cohomology.…”

Section: Introductionmentioning

confidence: 82%

“…In [13], Greenberg and Vatsal showed that the Iwasawa invariants (defined over the cyclotomic Z p -extension of Q) of p-congruent, p-ordinary elliptic curves are related by an explicit formula, and they used this to prove many cases of the Iwasawa Main Conjecture. These results were later extended to Hida families by Emerton, Pollack and Weston ([9]), while results in the non-ordinary case were established for elliptic curves by B. D. Kim ([19]) and for more general modular forms by the first two named authors ( [14]).…”

Section: Proposition 52mentioning

confidence: 99%

$Λ$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

Hatley¹,

Lei²,

Vigni³

2020

Preprint

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Let p be an odd prime and K an imaginary quadratic field where p splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a p-ordinary elliptic curve over the anticyclotomic Zp-extension of K does not admit any proper Λ-submodule of finite index, where Λ is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of p-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have Λ-corank one, so they are not Λ-cotorsion. As an application of our main theorem, we prove results in the vein of Greenberg-Vatsal on Iwasawa invariants of p-congruent elliptic curves, extending to the supersingular case results for p-ordinary elliptic curves due to Hatley-Lei.

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Comparing anticyclotomic Selmer groups of positive coranks for congruent modular forms – Part II

Cited by 7 publications

References 24 publications

$$\Lambda $$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

$$\Lambda $$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

Fine Selmer groups of congruent p-adic Galois representations

$Λ$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

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