Signed Iwasawa Main Conjecture

• Signed Iwasawa main conjectures refine classical Iwasawa theory by replacing unsigned arithmetic and analytic objects with signed variants tailored to handle non-ordinary and supersingular settings.
• They employ techniques such as Coleman maps, logarithmic matrix factorizations, and Euler/Kolyvagin systems to construct signed Selmer groups and p‑adic L‑functions under various local conditions.
• Different formulations—like plus/minus, #/flat, and derivative cases—lead to diverse conjectural statements with implications for μ‑invariants, BSD, and Bloch–Kato conjectures.

Searching arXiv for recent and foundational papers on signed Iwasawa main conjectures to ground the article with current literature. arxiv_search: {"^^^^^^^^17query17 Iwasawa main conjecture\"^^^^^^^^17 OR \17^^^^^^^^"plus/minus main conjecture\"^^^^^^^^17 OR \17^^^^^^^^"signed Selmer groups\"^^^^^^^^17 OR \17^^^^^^^^"Bertolini-Darmon-Prasanna p-adic L-function\"^^^^^^^^17 OR anticyclotomic non-ordinary modular forms)17^^^^^^^^", "^^^^^^^^17max_results17 ^^^^17all:(\17query17^^^^, "^^^^^^^^17sort_by17 "^^^^^^^^17submittedDate17 "^^^^^^^^17sort_order17 "^^^^^^^^17all:(\17query17 arxiv_search: {"^^^^^^^^17query17 OR \17^^^^^^^^"On the BDP Iwasawa main conjecture for modular forms\" OR ^^^^^^^^17all:(\17 OR \17^^^^^^^^"Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes\" OR ^^^^^^^^17all:(\17 OR \17^^^^^^^^"Iwasawa theory for Rankin--Selberg products of p-non-ordinary eigenforms\" OR ^^^^^^^^17all:(\17 OR \17^^^^^^^^"Iwasawa theory for Symmetric Square of non-p-ordinary eigenforms\" OR ^^^^^^^^17all:(\17 OR \17^^^^^^^^"On the Iwasawa theory of CM fields for supersingular primes\"", "^^^^^^^^17max_results17 ^^^^17all:(\17query17^^^^, "^^^^^^^^17sort_by17 "relevance", "^^^^^^^^17sort_order17 "^^^^^^^^17all:(\17query17 Signed Iwasawa main conjectures are refinements of Iwasawa main conjectures in which one replaces the classical arithmetic and analytic objects by signed variants adapted to difficult local behavior at PRESERVED_PLACEHOLDER_^^^^17query17^^^^, especially in non-ordinary and supersingular settings. In the formulations appearing for modular forms, elliptic curves, Rankin–Selberg motives, and CM fields, the analytic side is given by signed PRESERVED_PLACEHOLDER_^^^^17all:(\17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17 OR \17^^^^-functions and the algebraic side by signed Selmer groups or signed Selmer complexes; in anticyclotomic situations the conjecture may involve a square, a derivative, or a rank-one quotient rather than a single torsion characteristic ideal. The modern literature treats plus/minus, PRESERVED_PLACEHOLDER_^^^^17 OR \17^^^^, finite-slope, and sign-PRESERVED_PLACEHOLDER_^^^^17 OR \17^^^^ variants as part of one general pattern: the local condition at PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17^^^^ determines both the correct Selmer group and the correct PRESERVED_PLACEHOLDER_^^^^17max_results17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17sort_by17^^^^-function, and the main conjecture predicts equality or one-sided divisibility between the corresponding characteristic ideals (&&&^^^^17query17^^^^&&&, &&&^^^^17all:(\17^^^^&&&, &&&^^^^17 OR \17^^^^&&&, &&&^^^^17 OR \17^^^^&&&, &&&^^^^17 OR \17^^^^&&&).

^^^^17all:(\17^^^^. Signed structures and the meaning of “sign”

In the context of signed Iwasawa theory, one often constructs “signed” Selmer groups and “signed” PRESERVED_PLACEHOLDER_^^^^17submittedDate17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17sort_order17^^^^-functions, for example plus/minus invariants at supersingular primes. The underlying reason is that classical unsigned objects are often not the correct bounded integral objects when PRESERVED_PLACEHOLDER_^^^^17all:(\17query17^^^^ is non-ordinary at PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17^^^^, when PRESERVED_PLACEHOLDER_^^^^17all:(\17 OR \17^^^^, or when the global root number forces vanishing at the center. In these settings, “signed analogues introduce finer projections or decompositions of the Iwasawa module based on local conditions at PRESERVED_PLACEHOLDER_^^^^17all:(\17 OR \17^^^^,” and the arithmetic side must be replaced by signed Selmer groups or complexes, while the analytic side must be replaced by signed PRESERVED_PLACEHOLDER_^^^^17all:(\17 OR \17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17all:(\17 OR anticyclotomic non-ordinary modular forms)17^^^^-functions (&&&^^^^17 OR anticyclotomic non-ordinary modular forms)17^^^^&&&, &&&^^^^17 OR \17^^^^&&&).

The sign can appear in several technically distinct ways. In Kobayashi-style theories it appears as plus/minus local conditions; in the non-ordinary literature over imaginary quadratic fields it appears as PRESERVED_PLACEHOLDER_^^^^17all:(\17max_results17^^^^ local conditions at the two PRESERVED_PLACEHOLDER_^^^^17all:(\17sort_by17^^^^-adic places; in finite-slope formulations it appears through a choice of Frobenius eigenvalue; and in anticyclotomic theories it also appears through the global root number PRESERVED_PLACEHOLDER_^^^^17all:(\17submittedDate17^^^^ or PRESERVED_PLACEHOLDER_^^^^17all:(\17sort_order17^^^^, which changes whether the conjecture concerns a torsion Selmer module, a derivative, or a rank-one quotient (&&&^^^^17sort_by17^^^^&&&, &&&^^^^17query17^^^^&&&, &&&^^^^17sort_order17^^^^&&&, &&&^^^^17all:(\17query17^^^^&&&, &&&^^^^17 OR \17^^^^&&&).

This is not merely terminological variation. A frequent misconception is that “signed” always means plus/minus in the narrow supersingular elliptic-curve sense. The literature shows a broader pattern: PRESERVED_PLACEHOLDER_^^^^17 OR \17query17^^^^, PRESERVED_PLACEHOLDER_^^^^17 OR \17all:(\17^^^^, doubly-signed, quadruply-signed, admissible finite-slope, and sign-normalized formulations are all devices for isolating the correct local or parity component of the theory. By contrast, some foundational noncommutative and equivariant main conjectures for totally real fields are explicitly unsigned and do not address signed or plus/minus structures at all (&&&^^^^17all:(\17 OR \17^^^^&&&).

^^^^17 OR \17^^^^. Local conditions at PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^: Coleman maps, logarithmic matrices, and signed Selmer groups

A central feature of signed formulations is the construction of local conditions at PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^ by means of signed Coleman maps. For elliptic modular forms over imaginary quadratic fields at non-ordinary primes, the signed Coleman maps are

PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^

and they arise from a logarithmic matrix factorization of Perrin–Riou’s big logarithm map: PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR anticyclotomic non-ordinary modular forms)17^^^^ Signed local conditions are then defined by taking kernels of these maps, and doubly-signed Selmer structures over cyclotomic, anticyclotomic, or PRESERVED_PLACEHOLDER_^^^^17 OR \17max_results17^^^^-extensions are obtained by imposing one sign at PRESERVED_PLACEHOLDER_^^^^17 OR \17sort_by17^^^^ and one sign at PRESERVED_PLACEHOLDER_^^^^17 OR \17submittedDate17^^^^ (&&&^^^^17query17^^^^&&&).

For the symmetric square of a non-PRESERVED_PLACEHOLDER_^^^^17 OR \17sort_order17^^^^-ordinary modular form, signed local conditions are written in the form

PRESERVED_PLACEHOLDER_^^^^17 OR \17query17^^^^

and doubly-signed Selmer groups are built from intersections of kernels of signed Coleman maps. The associated signed PRESERVED_PLACEHOLDER_^^^^17 OR \17all:(\17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^-functions are defined by

PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^

where the signs PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^ record the chosen local projections (&&&^^^^17 OR \17^^^^&&&).

For Rankin–Selberg products of two PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR anticyclotomic non-ordinary modular forms)17^^^^-non-ordinary eigenforms, Wach module theory produces an explicit PRESERVED_PLACEHOLDER_^^^^17 OR \17max_results17^^^^ logarithmic matrix PRESERVED_PLACEHOLDER_^^^^17 OR \17sort_by17^^^^, and the Perrin–Riou regulator decomposes in terms of four signed Coleman maps PRESERVED_PLACEHOLDER_^^^^17 OR \17submittedDate17^^^^, PRESERVED_PLACEHOLDER_^^^^17 OR \17sort_order17^^^^, PRESERVED_PLACEHOLDER_^^^^17 OR \17query17^^^^, and PRESERVED_PLACEHOLDER_^^^^17 OR \17all:(\17^^^^. This leads to doubly-signed and quadruply-signed Selmer groups, formed using intersections of kernels of the relevant Coleman maps, and to corresponding signed PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^-functions defined by applying one signed Coleman map to a signed Beilinson–Flach class (&&&^^^^17all:(\17^^^^&&&).

The same structural principle appears in finite-slope theories. For symmetric powers of CM modular forms at supersingular primes, the sign is encoded by the tuple PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR \17^^^^ of Frobenius eigenvalues and the local conditions are defined by choosing the filtered subspace PRESERVED_PLACEHOLDER_^^^^17 OR \17 OR anticyclotomic non-ordinary modular forms)17^^^^ attached to PRESERVED_PLACEHOLDER_^^^^17 OR \17max_results17^^^^. The resulting finite-slope Selmer modules are not naively finitely generated over the Iwasawa algebra, but they are coadmissible PRESERVED_PLACEHOLDER_^^^^17 OR \17sort_by17^^^^-modules with a characteristic ideal, and the main conjecture compares that ideal with an admissible PRESERVED_PLACEHOLDER_^^^^17 OR \17submittedDate17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17 OR \17sort_order17^^^^-function (&&&^^^^17sort_order17^^^^&&&).

^^^^17 OR \17^^^^. Anticyclotomic formulations, BDP theory, and parity PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17query17^^^^

In the anticyclotomic theory of modular forms over an imaginary quadratic field PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17all:(\17^^^^, the Bertolini–Darmon–Prasanna PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17 OR \17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17 OR \17^^^^-function PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17 OR \17^^^^ is related to the characteristic ideal of the Pontryagin dual PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17 OR anticyclotomic non-ordinary modular forms)17^^^^ of a certain anticyclotomic Selmer group. Kobayashi–Ota showed the inclusion

PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17max_results17^^^^

and the later integral refinement proves

PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17sort_by17^^^^

under explicit hypotheses. Although the main statements are formulated in terms of the full Selmer group and the BDP PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17submittedDate17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17 OR anticyclotomic non-ordinary modular forms)17sort_order17^^^^-function, the results imply properties for signed objects as well, and in the case of a PRESERVED_PLACEHOLDER_^^^^17max_results17query17^^^^-supersingular elliptic curve this recovers the vanishing of the PRESERVED_PLACEHOLDER_^^^^17max_results17all:(\17^^^^-invariants of the anticyclotomic plus and minus Selmer groups (&&&^^^^17 OR \17^^^^&&&).

A broader non-ordinary anticyclotomic framework over imaginary quadratic fields formulates integral Iwasawa main conjectures over the cyclotomic PRESERVED_PLACEHOLDER_^^^^17max_results17 OR \17^^^^-extension, the anticyclotomic PRESERVED_PLACEHOLDER_^^^^17max_results17 OR \17^^^^-extensions in both the definite and the indefinite cases, and the PRESERVED_PLACEHOLDER_^^^^17max_results17 OR \17^^^^-extension. In this theory, signed Coleman maps produce doubly-signed Selmer groups, signed Beilinson–Flach elements produce doubly-signed PRESERVED_PLACEHOLDER_^^^^17max_results17 OR anticyclotomic non-ordinary modular forms)17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17max_results17max_results17^^^^-functions, and the main conjecture relates the characteristic ideal of the signed Selmer group to the corresponding signed PRESERVED_PLACEHOLDER_^^^^17max_results17sort_by17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17max_results17submittedDate17^^^^-function, up to the factor PRESERVED_PLACEHOLDER_^^^^17max_results17sort_order17^^^^ measuring possible failure of surjectivity of the Coleman map (&&&^^^^17query17^^^^&&&).

Parity changes the shape of the conjecture. When the global sign is PRESERVED_PLACEHOLDER_^^^^17sort_by17query17^^^^, the relevant Selmer group has generic corank one and the conjecture concerns the torsion quotient by a Heegner or diagonal class. For Heegner points, the anticyclotomic main conjecture in the sign PRESERVED_PLACEHOLDER_^^^^17sort_by17all:(\17^^^^ case relates the quotient of the Selmer group by the initial Heegner class PRESERVED_PLACEHOLDER_^^^^17sort_by17 OR \17^^^^ to the torsion module PRESERVED_PLACEHOLDER_^^^^17sort_by17 OR \17^^^^, while the two-variable formulation identifies the characteristic ideal of the dual Selmer group with the Rankin–Selberg PRESERVED_PLACEHOLDER_^^^^17sort_by17 OR \17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17sort_by17 OR anticyclotomic non-ordinary modular forms)17^^^^-function (&&&^^^^17all:(\17query17^^^^&&&). In the higher-rank anticyclotomic theory for PRESERVED_PLACEHOLDER_^^^^17sort_by17max_results17^^^^, when the global root number is PRESERVED_PLACEHOLDER_^^^^17sort_by17sort_by17^^^^ the result is

PRESERVED_PLACEHOLDER_^^^^17sort_by17submittedDate17^^^^

while for root number PRESERVED_PLACEHOLDER_^^^^17sort_by17sort_order17^^^^ the paper proves

PRESERVED_PLACEHOLDER_^^^^17submittedDate17query17^^^^

with PRESERVED_PLACEHOLDER_^^^^17submittedDate17all:(\17^^^^ generated by universal diagonal cycles (&&&^^^^17 OR \17^^^^&&&).

This parity dependence shows that signed main conjectures are not uniformly “torsion module equals one PRESERVED_PLACEHOLDER_^^^^17submittedDate17 OR \17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17submittedDate17 OR \17^^^^-function.” In sign PRESERVED_PLACEHOLDER_^^^^17submittedDate17 OR \17^^^^ cases the torsion formulation is often direct; in sign PRESERVED_PLACEHOLDER_^^^^17submittedDate17 OR anticyclotomic non-ordinary modular forms)17^^^^ cases the conjecture typically involves a derivative, a square containment, or a torsion quotient after removing a distinguished rank-one class (&&&^^^^17all:(\17query17^^^^&&&, &&&^^^^17 OR \17^^^^&&&).

^^^^17 OR \17^^^^. Euler systems, Kolyvagin systems, and one-sided divisibilities

The dominant method for proving one inclusion in signed main conjectures is Euler-system theory adapted to the signed local conditions. For CM fields and CM elliptic curves at supersingular primes, Rubin–Stark PRESERVED_PLACEHOLDER_^^^^17submittedDate17max_results17^^^^-restricted Kolyvagin systems are constructed by modifying the local conditions at PRESERVED_PLACEHOLDER_^^^^17submittedDate17sort_by17^^^^ via direct summands PRESERVED_PLACEHOLDER_^^^^17submittedDate17submittedDate17^^^^, so that the resulting machinery acts on “transversal” signed directions. This yields one divisibility in the two-variable main conjecture for CM fields and in the signed plus/minus main conjecture

PRESERVED_PLACEHOLDER_^^^^17submittedDate17sort_order17^^^^

with equality under the strong Rubin–Stark conjecture (&&&^^^^17sort_by17^^^^&&&).

For non-ordinary Rankin–Selberg products, the theory of Beilinson–Flach elements gives rise to four rank-one non-integral Euler systems, one for each choice of PRESERVED_PLACEHOLDER_^^^^17sort_order17query17^^^^-stabilisations. The signed theory reorganizes them via a logarithmic matrix and signed projections into bounded signed classes, and the quadruply-signed main conjecture predicts that for suitable sign data PRESERVED_PLACEHOLDER_^^^^17sort_order17all:(\17^^^^,

PRESERVED_PLACEHOLDER_^^^^17sort_order17 OR \17^^^^

up to certain cokernel errors arising from the Coleman maps. Under standard technical hypotheses and the signed-splitting conjecture, one inclusion is proved by Euler system and Poitou–Tate arguments (&&&^^^^17all:(\17^^^^&&&).

For the symmetric square motive of a non-PRESERVED_PLACEHOLDER_^^^^17sort_order17 OR \17^^^^-ordinary eigenform, Beilinson–Flach elements factorize into integral signed Beilinson–Flach elements, giving evidence toward the existence of a rank-two Euler system predicted by Perrin–Riou. The resulting signed Euler systems are used to prove the inclusion

PRESERVED_PLACEHOLDER_^^^^17sort_order17 OR \17^^^^

on isotypic components, and an analytic analogue is established for Pottharst-style Selmer groups (&&&^^^^17 OR \17^^^^&&&).

Heegner-point and diagonal-cycle theories supply the rank-one analogues in parity PRESERVED_PLACEHOLDER_^^^^17sort_order17 OR anticyclotomic non-ordinary modular forms)17^^^^. The Heegner point Kolyvagin system controls the rank-one part of the anticyclotomic Selmer group in the global sign PRESERVED_PLACEHOLDER_^^^^17sort_order17max_results17^^^^ case, and the higher-rank anticyclotomic PRESERVED_PLACEHOLDER_^^^^17sort_order17sort_by17^^^^ theory uses a bipartite Euler system and explicit reciprocity laws to produce the square containment involving PRESERVED_PLACEHOLDER_^^^^17sort_order17submittedDate17^^^^ (&&&^^^^17all:(\17query17^^^^&&&, &&&^^^^17 OR \17^^^^&&&).

A related local input is the epsilon-isomorphism formalism. For rank one Iwasawa modules, the epsilon-isomorphism construction corrects a sign ambiguity in Kato’s unpublished construction by using minus the classical Coleman map, and this clarification is directly relevant to local main conjectures for CM elliptic curves and to signed theories where the precise normalization of local maps matters (&&&^^^^17 OR \17submittedDate17^^^^&&&).

^^^^17 OR anticyclotomic non-ordinary modular forms)17^^^^. Major variants of the conjecture

The literature now contains several parallel signed formulations. They differ in local conditions, in the analytic object, and in whether the conjecture is stated as equality or as one-sided divisibility.

Setting	Signed data	Typical statement
CM elliptic curves at supersingular primes	PRESERVED_PLACEHOLDER_^^^^17sort_order17sort_order17^^^^ Selmer groups and PRESERVED_PLACEHOLDER_^^^^17all:(\17query17query17^^^^	PRESERVED_PLACEHOLDER_^^^^17all:(\17query17all:(\17^^^^ (&&&^^^^17sort_by17^^^^&&&)
Modular forms over imaginary quadratic fields at non-ordinary primes	doubly-signed PRESERVED_PLACEHOLDER_^^^^17all:(\17query17 OR \17^^^^ local conditions at PRESERVED_PLACEHOLDER_^^^^17all:(\17query17 OR \17^^^^	PRESERVED_PLACEHOLDER_^^^^17all:(\17query17 OR \17^^^^ (&&&^^^^17query17^^^^&&&)
Rankin–Selberg products of non-ordinary eigenforms	doubly-signed and quadruply-signed Coleman maps	PRESERVED_PLACEHOLDER_^^^^17all:(\17query17 OR anticyclotomic non-ordinary modular forms)17^^^^ (&&&^^^^17all:(\17^^^^&&&)
Symmetric square of a non-PRESERVED_PLACEHOLDER_^^^^17all:(\17query17max_results17^^^^-ordinary eigenform	signed and doubly-signed Selmer groups	PRESERVED_PLACEHOLDER_^^^^17all:(\17query17sort_by17^^^^ (&&&^^^^17 OR \17^^^^&&&)
Anticyclotomic BDP theory for modular forms	full Selmer group with signed consequences	PRESERVED_PLACEHOLDER_^^^^17all:(\17query17submittedDate17^^^^ (&&&^^^^17 OR \17^^^^&&&)
Anticyclotomic Rankin–Selberg motives	parity PRESERVED_PLACEHOLDER_^^^^17all:(\17query17sort_order17^^^^, diagonal cycles in sign PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17query17^^^^	PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17all:(\17^^^^ or PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17 OR \17^^^^ (&&&^^^^17 OR \17^^^^&&&)

A further variant appears in the function-field literature. For rank one, sign-normalized Drinfeld modular Iwasawa towers split at infinity, sign-normalization selects the “real” piece of the theory, and this is described as analogous to the “plus” part in signed main conjectures over number fields. The resulting equivariant main conjecture is expressed by Fitting ideals rather than the Selmer-group characteristic ideals more common in the number-field modular-form literature (&&&^^^^17 OR \17 OR anticyclotomic non-ordinary modular forms)17^^^^&&&).

Another distinction is between finite-slope and plus/minus theories. For symmetric powers of CM modular forms at supersingular primes, the main conjecture is stated for “admissible” PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17 OR \17^^^^-adic PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17 OR \17^^^^-functions and “finite-slope” Selmer modules: PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17 OR anticyclotomic non-ordinary modular forms)17^^^^ and the plus/minus theory is recovered as a special case rather than being the primary formulation (&&&^^^^17sort_order17^^^^&&&).

^^^^17max_results17^^^^. Consequences, limitations, and unresolved aspects

A major arithmetic consequence of integral signed or partially signed divisibilities is the vanishing of Iwasawa PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17max_results17^^^^-invariants. In the BDP anticyclotomic setting, the integral inclusion together with Hsieh’s result on the zero PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17sort_by17^^^^-invariant of PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17submittedDate17^^^^ implies

PRESERVED_PLACEHOLDER_^^^^17all:(\17all:(\17sort_order17^^^^

and these vanishing results propagate to other anticyclotomic Selmer groups and to signed Selmer groups, including the anticyclotomic plus and minus Selmer groups for PRESERVED_PLACEHOLDER_^^^^17all:(\17 OR \17query17^^^^-supersingular elliptic curves (&&&^^^^17 OR \17^^^^&&&).

The same circle of ideas has consequences toward the Birch and Swinnerton-Dyer conjecture and the Bloch–Kato conjecture. For CM elliptic curves at supersingular primes, the signed main conjectures obtained from Rubin–Stark PRESERVED_PLACEHOLDER_^^^^17all:(\17 OR \17all:(\17^^^^-restricted Kolyvagin systems are used to derive conditional consequences toward BSD (&&&^^^^17sort_by17^^^^&&&). For quadratic Hilbert modular forms, a cyclotomic main conjecture proved by comparing “Kato divisibility” with the opposite divisibility due to Wan gives new cases of the Bloch–Kato conjecture and of the equivariant BSD conjecture, although this theory is ordinary rather than a signed non-ordinary theory in the narrow sense (&&&^^^^17 OR \17sort_order17^^^^&&&).

Several limitations are explicit in the current literature. Many of the strongest statements are one-sided divisibilities rather than equalities. In non-ordinary settings the signed classes are often obtained only after factorization of unbounded objects, and extra hypotheses such as non-vanishing, big image, admissibility, or strong Rubin–Stark are frequently required for the passage from divisibility to equality (&&&^^^^17sort_by17^^^^&&&, &&&^^^^17all:(\17^^^^&&&, &&&^^^^17 OR \17^^^^&&&). A plausible implication is that the signed theory is currently more robust on the “upper-bound” side than on the full equality side.

It is also important to distinguish signed theories from unsigned noncommutative main conjectures. The main conjecture of equivariant Iwasawa theory for totally real extensions and Kakde’s noncommutative main conjecture for totally real fields are formulated in an unsigned framework; the papers explicitly state that signed or plus/minus main conjectures are not treated there, even though some of the reduction techniques and congruence ideas provide a blueprint for later signed generalizations (&&&^^^^17all:(\17 OR \17^^^^&&&, &&&^^^^17 OR anticyclotomic non-ordinary modular forms)17^^^^&&&).

Taken together, these works show that the signed Iwasawa main conjecture is best understood not as a single conjecture but as a family of conjectural correspondences adapted to local non-ordinarity and global parity. The recurring template is stable across settings: define bounded signed analytic classes, define local signed Selmer conditions via Coleman maps or analogous regulators, and compare the resulting characteristic or Fitting ideals by Euler-system, Kolyvagin-system, or control-theoretic arguments. This suggests that future progress will continue to depend on improving integrality, control at height-one primes, and reciprocity laws for the signed local factors that determine the correct arithmetic side of the theory (&&&^^^^17 OR \17^^^^&&&, &&&^^^^17 OR \17^^^^&&&).