---
title: Signed Iwasawa Main Conjecture
url: https://www.emergentmind.com/topics/signed-iwasawa-main-conjecture
type: topic
---

# Signed Iwasawa Main Conjecture

Searching arXiv for recent and foundational papers on signed Iwasawa main conjectures to ground the article with current literature.
arxiv_search:
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arxiv_search:
{"query":"ti:\"On the BDP Iwasawa main conjecture for modular forms\" OR ti:\"Iwasawa theory of elliptic modular forms over imaginary quadratic fields at non-ordinary primes\" OR ti:\"Iwasawa theory for Rankin--Selberg products of p-non-ordinary eigenforms\" OR ti:\"Iwasawa theory for Symmetric Square of non-p-ordinary eigenforms\" OR ti:\"On the Iwasawa theory of CM fields for supersingular primes\"", "max_results": 10, "sort_by": "relevance", "sort_order": "descending"}
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 \(p\), 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 \(p\)-adic \(L\)-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, \(\#/\flat\), finite-slope, and sign-\(\pm1\) variants as part of one general pattern: the local condition at \(p\) determines both the correct Selmer group and the correct \(p\)-adic \(L\)-function, and the main conjecture predicts equality or one-sided divisibility between the corresponding characteristic ideals [1605.05310], [1802.04419], [1807.11517], [2211.04377], [2406.00624].

## 1. Signed structures and the meaning of “sign”

In the context of signed Iwasawa theory, one often constructs “signed” Selmer groups and “signed” \(p\)-adic \(L\)-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 \(f\) is non-ordinary at \(p\), when \(a_p(f)=0\), 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 \(p\),” and the arithmetic side must be replaced by signed Selmer groups or complexes, while the analytic side must be replaced by signed \(p\)-adic \(L\)-functions [1008.0142], [2211.04377].

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 \(\#/\flat\) local conditions at the two \(p\)-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 \(+1\) or \(-1\), which changes whether the conjecture concerns a torsion Selmer module, a derivative, or a rank-one quotient [1501.01388], [1605.05310], [1407.4371], [1408.4043], [2406.00624].

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: \(\pm\), \(\#/\flat\), 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 [1004.2578].

## 2. Local conditions at \(p\): Coleman maps, logarithmic matrices, and signed Selmer groups

A central feature of signed formulations is the construction of local conditions at \(p\) by means of signed Coleman maps. For elliptic modular forms over imaginary quadratic fields at non-ordinary primes, the signed Coleman maps are
\[
\operatorname{Col}_{\#,q}, \operatorname{Col}_{\flat, q}: H^1(K_{q}, T_{f,\chi}) \to \Lambda_{O_L}(\Gamma),
\]
and they arise from a logarithmic matrix factorization of Perrin–Riou’s big logarithm map:
\[
\begin{pmatrix} L_{\alpha, q}\\ L_{\beta, q} \end{pmatrix} = M \cdot \begin{pmatrix} \operatorname{Col}_{\#, q}\\ \operatorname{Col}_{\flat, q} \end{pmatrix}.
\]
Signed local conditions are then defined by taking kernels of these maps, and doubly-signed Selmer structures over cyclotomic, anticyclotomic, or \(\mathbf{Z}_p^2\)-extensions are obtained by imposing one sign at \(p\) and one sign at \(p^c\) [1605.05310].

For the symmetric square of a non-\(p\)-ordinary modular form, signed local conditions are written in the form
\[
H^1_{\mathcal{F}_\clubsuit}(Q, \mathbb{T}) = \ker \left( H^1(Q, \mathbb{T}) \to H^1(Q_p, \mathbb{T}) / \ker \mathrm{Col}^\clubsuit \right),
\]
and doubly-signed Selmer groups are built from intersections of kernels of signed Coleman maps. The associated signed \(p\)-adic \(L\)-functions are defined by
\[
\mathcal{L}_{\mathfrak{S}} := \mathrm{Col}^{\clubsuit} \circ \mathrm{res}_p(BF_{1,\chi}^{\spadesuit}) \in \Lambda_E(\Gamma),
\]
where the signs \((\clubsuit,\spadesuit)\) record the chosen local projections [1807.11517].

For Rankin–Selberg products of two \(p\)-non-ordinary eigenforms, Wach module theory produces an explicit \(4\times4\) logarithmic matrix \(M_{\log}\), and the Perrin–Riou regulator decomposes in terms of four signed Coleman maps \(\mathrm{Col}_{\#,\#}\), \(\mathrm{Col}_{\#,\flat}\), \(\mathrm{Col}_{\flat,\#}\), and \(\mathrm{Col}_{\flat,\flat}\). This leads to doubly-signed and quadruply-signed Selmer groups, formed using intersections of kernels of the relevant Coleman maps, and to corresponding signed \(p\)-adic \(L\)-functions defined by applying one signed Coleman map to a signed Beilinson–Flach class [1802.04419].

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 \(t=(t_0,\dots,t_{r-1})\) of Frobenius eigenvalues and the local conditions are defined by choosing the filtered subspace \(F_i\subset D_{\mathrm{cris}}\) attached to \(t_i\). The resulting finite-slope Selmer modules are not naively finitely generated over the Iwasawa algebra, but they are coadmissible \(H_E(G_\infty)\)-modules with a characteristic ideal, and the main conjecture compares that ideal with an admissible \(p\)-adic \(L\)-function [1407.4371].

## 3. Anticyclotomic formulations, BDP theory, and parity \(\pm1\)

In the anticyclotomic theory of modular forms over an imaginary quadratic field \(K\), the Bertolini–Darmon–Prasanna \(p\)-adic \(L\)-function \(\mathscr{L}(f/K)\) is related to the characteristic ideal of the Pontryagin dual \(\mathcal{X}_{(\emptyset,0)}(f)\) of a certain anticyclotomic Selmer group. Kobayashi–Ota showed the inclusion
\[
\mathscr{L}(f/K)^2 \in \operatorname{char}_\Lambda\left(\mathcal{X}_{(\emptyset,0)}(f)\right) \otimes_\Lambda \Lambda^{\mathrm{ur}} \otimes_\mathbb{Z} \mathbb{Q},
\]
and the later integral refinement proves
\[
\mathscr{L}(f/K)^2 \in \operatorname{char}_\Lambda\left(\mathcal{X}_{(\emptyset,0)}(f)\right) \otimes_\Lambda \Lambda^{\mathrm{ur}}
\]
under explicit hypotheses. Although the main statements are formulated in terms of the full Selmer group and the BDP \(p\)-adic \(L\)-function, the results imply properties for signed objects as well, and in the case of a \(p\)-supersingular elliptic curve this recovers the vanishing of the \(\mu\)-invariants of the anticyclotomic plus and minus Selmer groups [2211.04377].

A broader non-ordinary anticyclotomic framework over imaginary quadratic fields formulates integral Iwasawa main conjectures over the cyclotomic \(\mathbf{Z}_p\)-extension, the anticyclotomic \(\mathbf{Z}_p\)-extensions in both the definite and the indefinite cases, and the \(\mathbf{Z}_p^2\)-extension. In this theory, signed Coleman maps produce doubly-signed Selmer groups, signed Beilinson–Flach elements produce doubly-signed \(p\)-adic \(L\)-functions, and the main conjecture relates the characteristic ideal of the signed Selmer group to the corresponding signed \(p\)-adic \(L\)-function, up to the factor \(\xi_\star^?\) measuring possible failure of surjectivity of the Coleman map [1605.05310].

Parity changes the shape of the conjecture. When the global sign is \(-1\), 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 \(-1\) case relates the quotient of the Selmer group by the initial Heegner class \(\kappa_1\) to the torsion module \(M\), while the two-variable formulation identifies the characteristic ideal of the dual Selmer group with the Rankin–Selberg \(p\)-adic \(L\)-function [1408.4043]. In the higher-rank anticyclotomic theory for \(\mathrm{GL}(n)\times\mathrm{GL}(n+1)\), when the global root number is \(+1\) the result is
\[
L_{F_\infty}(\Pi_0 \times \Pi_1) \in \operatorname{char}_\Lambda X(F_\infty, V),
\]
while for root number \(-1\) the paper proves
\[
\operatorname{char}_\Lambda \mathscr{K}^2 \subset \operatorname{char}_\Lambda \left( X(F_\infty, V)_{\mathrm{tor}} \right),
\]
with \(\mathscr{K}\) generated by universal diagonal cycles [2406.00624].

This parity dependence shows that signed main conjectures are not uniformly “torsion module equals one \(p\)-adic \(L\)-function.” In sign \(+1\) cases the torsion formulation is often direct; in sign \(-1\) cases the conjecture typically involves a derivative, a square containment, or a torsion quotient after removing a distinguished rank-one class [1408.4043], [2406.00624].

## 4. 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 \(\mathcal{L}\)-restricted Kolyvagin systems are constructed by modifying the local conditions at \(p\) via direct summands \(\mathcal{L}\), 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
\[
\operatorname{char}(\mathrm{Sel}^\pm_p(E/F^\mathrm{cyc})^\vee) \mid (L_p^\pm(E/F^+)),
\]
with equality under the strong Rubin–Stark conjecture [1501.01388].

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 \(p\)-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 \(S\),
\[
\operatorname{char}_{\mathcal{O}[[\Gamma_1]]} \left( e_\eta \operatorname{Sel}_S(T^\vee(1)/\mathbb{Q}(\mu_{p^\infty}))^{\vee} \right) \mid (e_\eta L_S),
\]
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 [1802.04419].

For the symmetric square motive of a non-\(p\)-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
\[
\mathrm{char}\left( Sel_{\mathfrak{S}}(T^\vee(1)/Q(\mu_{p^\infty}))^\vee \right)\mid (\mathcal{L}_{\mathfrak{S}})
\]
on isotypic components, and an analytic analogue is established for Pottharst-style Selmer groups [1807.11517].

Heegner-point and diagonal-cycle theories supply the rank-one analogues in parity \(-1\). The Heegner point Kolyvagin system controls the rank-one part of the anticyclotomic Selmer group in the global sign \(-1\) case, and the higher-rank anticyclotomic \(\mathrm{GL}(n)\times\mathrm{GL}(n+1)\) theory uses a bipartite Euler system and explicit reciprocity laws to produce the square containment involving \(\mathscr{K}\) [1408.4043], [2406.00624].

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 [1204.4269].

## 5. 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 | \(\pm\) Selmer groups and \(L_p^\pm\) | \(\operatorname{char}(\mathrm{Sel}^\pm_p(E/F^\mathrm{cyc})^\vee) \mid (L_p^\pm(E/F^+))\) [1501.01388] |
| Modular forms over imaginary quadratic fields at non-ordinary primes | doubly-signed \(\#/\flat\) local conditions at \(p,p^c\) | \(char(\mathfrak{X}^?_{\star,\bullet}) \sim \mathfrak{L}_{\star,\bullet}^?/\xi_\star^?\) [1605.05310] |
| Rankin–Selberg products of non-ordinary eigenforms | doubly-signed and quadruply-signed Coleman maps | \(\operatorname{char}(\Sel_S^\vee)\mid(L_S)\) [1802.04419] |
| Symmetric square of a non-\(p\)-ordinary eigenform | signed and doubly-signed Selmer groups | \(\mathrm{char}(Sel_{\mathfrak S}^\vee)\mid(\mathcal L_{\mathfrak S})\) [1807.11517] |
| Anticyclotomic BDP theory for modular forms | full Selmer group with signed consequences | \(\mathscr{L}(f/K)^2 \in \operatorname{char}_\Lambda(\mathcal X_{(\emptyset,0)}(f))\otimes_\Lambda\Lambda^{\mathrm{ur}}\) [2211.04377] |
| Anticyclotomic Rankin–Selberg motives | parity \(+1/-1\), diagonal cycles in sign \(-1\) | \(L_{F_\infty}\in \operatorname{char}_\Lambda X\) or \(\operatorname{char}_\Lambda \mathscr K^2\subset \operatorname{char}_\Lambda(X_{\mathrm{tor}})\) [2406.00624] |

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 [2209.02440].

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” \(p\)-adic \(L\)-functions and “finite-slope” Selmer modules:
\[
\operatorname{char}_{H_E(G_\infty)} \mathrm{Sel}_{k_\infty}^t(V_m^*)^\vee = (\mathrm{Tw}_1\, L_{V_m, t}),
\]
and the plus/minus theory is recovered as a special case rather than being the primary formulation [1407.4371].

## 6. Consequences, limitations, and unresolved aspects

A major arithmetic consequence of integral signed or partially signed divisibilities is the vanishing of Iwasawa \(\mu\)-invariants. In the BDP anticyclotomic setting, the integral inclusion together with Hsieh’s result on the zero \(\mu\)-invariant of \(\mathscr{L}(f/K)\) implies
\[
\mu(\mathscr{L}(f/K)) = \mu(\mathcal{X}_{(\emptyset,0)}(f)) = 0,
\]
and these vanishing results propagate to other anticyclotomic Selmer groups and to signed Selmer groups, including the anticyclotomic plus and minus Selmer groups for \(p\)-supersingular elliptic curves [2211.04377].

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 \(\mathcal{L}\)-restricted Kolyvagin systems are used to derive conditional consequences toward BSD [1501.01388]. 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 [2006.14491].

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 [1501.01388], [1802.04419], [1807.11517]. 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 [1004.2578], [1008.0142].

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 [2211.04377], [2406.00624].

Source: https://www.emergentmind.com/topics/signed-iwasawa-main-conjecture