On the 𝜇-invariant in Iwasawa theory

Ramdorai, Sujathai

doi:10.1090/fic/060/15

Cited by 2 publications

(1 citation statement)

References 16 publications

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“…We have (cf. Proposition 3.2 of [Ram11]): Theorem 6.8. X(E/ F ∞ ) is a finitely generated torsion Λ( G)-module and µ G (X(E/ F ∞ )) = 0.…”

Section: Iwasawa Cohomology T ∈ Repmentioning

confidence: 99%

Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

Hamidi¹,

Ray²

2020

Preprint

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Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at primes above p. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over Z 2 p -extensions of an imaginary quadratic field where p splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the µ-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when E has good supersingular reduction) which are completely analogous to the classical Selmer group (when E has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.

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“…We have (cf. Proposition 3.2 of [Ram11]): Theorem 6.8. X(E/ F ∞ ) is a finitely generated torsion Λ( G)-module and µ G (X(E/ F ∞ )) = 0.…”

Section: Iwasawa Cohomology T ∈ Repmentioning

confidence: 99%

Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

Hamidi¹,

Ray²

2020

Preprint

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On the μ-invariants of abelian varieties over function fields of positive characteristic

et al. 2021

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Let A be an abelian variety over a global function field K of characteristic p. We study the µ-invariant appearing in the Iwasawa theory of A over the unramified ‫ޚ‬ p -extension of K . Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of A and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the µ-invariant) in terms of other quantities including the Faltings height of A and Frobenius slopes of the numerator of the Hasse-Weil L-function of A/K assuming the conjectural Birch-Swinnerton-Dyer formula. Our next result is to prove this µ-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "µ = 0" locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset. 1. Introduction 863 2. The µ-invariant for abelian varieties 869 3. The µ-invariant for Jacobians 879 4. The µ-invariant for semistable abelian varieties 886 5. The µ-invariant for elliptic curves 889 6. µ = 0 for generic elliptic curves 893 7. Computations 898 Appendix: Integrality of sum of slopes less than one 902 Acknowledgments 904 References 904

show abstract

On the 𝜇-invariant in Iwasawa theory

Cited by 2 publications

References 16 publications

Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

On the μ-invariants of abelian varieties over function fields of positive characteristic

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