Anticyclotomic Iwasawa Main Conjecture

• Anticyclotomic Iwasawa Main Conjecture is a framework linking Selmer groups over anticyclotomic Zₚ-extensions to p-adic L-functions via explicit reciprocity laws.
• It employs Euler systems and Heegner points to construct and control Iwasawa modules, revealing torsion and rank-one structures in various settings.
• Recent advances validate key cases with implications for BSD formulas, congruence methods in Hida families, and extensions to non-commutative Iwasawa theory.

Searching arXiv for recent and foundational papers on the anticyclotomic Iwasawa Main Conjecture. The anticyclotomic Iwasawa Main Conjecture is a family of conjectures, and in several important cases theorems, relating Selmer groups over anticyclotomic $\mathbf{Z}_p$-extensions of CM fields to anticyclotomic $p$-adic $L$-functions or to Euler-system generators such as Heegner classes. In its most classical $\mathrm{GL}_2$ form, one fixes an elliptic curve or modular form, an imaginary quadratic field $K$, and the anticyclotomic extension $K_\infty^-/K$ characterized by the action of complex conjugation by inversion on $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$. One then studies Iwasawa modules over $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$, typically Greenberg or ordinary Selmer groups, and compares their characteristic ideals with anticyclotomic $p$-adic $L$-functions such as the Bertolini–Darmon–Prasanna function or with the index of a Heegner generator. The subject now encompasses elliptic curves, modular forms of higher weight, supersingular variants with $p$0-local conditions, Hilbert modular forms, CM Hecke characters, and higher-rank Rankin–Selberg motives, together with two-variable and non-commutative refinements (Wan, 2014, Castella et al., 2015, Chida et al., 2013, Burungale et al., 2024, Liu et al., 2024).

1. Classical formulation over imaginary quadratic fields

In the standard setting one starts with an elliptic curve $p$1 or a $p$2-ordinary newform $p$3, an imaginary quadratic field $p$4, and the anticyclotomic $p$5-extension $p$6 with Galois group $p$7. The Iwasawa algebra is $p$8, or more generally $p$9 when coefficients lie in a finite extension of $L$0 (Wan, 2014, Castella et al., 2015). In the broader two-variable setting for an imaginary quadratic field $L$1, the compositum of all $L$2-extensions has Galois group $L$3, decomposing as cyclotomic and anticyclotomic directions $L$4, with $L$5 the anticyclotomic factor (Order, 2011).

For modular forms, a standard Selmer group is the minimal anticyclotomic Selmer group

$L$6

whose Pontryagin dual $L$7 is a finitely generated $L$8-module (Castella et al., 2015). In the elliptic-curve setting over an anticyclotomic tower, one also considers ordinary and Greenberg Selmer structures on $L$9, with local conditions at the two primes $\mathrm{GL}_2$0 arising from the ordinary filtration (Castella et al., 2023). These local conditions generate several related Iwasawa modules: an ordinary Selmer module of rank one in sign $\mathrm{GL}_2$1 settings, and a Greenberg Selmer module expected to be torsion and controlled by a $\mathrm{GL}_2$2-adic $\mathrm{GL}_2$3-function (Castella et al., 2023, Wan, 2014).

On the analytic side, one associates an anticyclotomic $\mathrm{GL}_2$4-adic $\mathrm{GL}_2$5-function $\mathrm{GL}_2$6 characterized by interpolation of central critical values $\mathrm{GL}_2$7 for anticyclotomic Hecke characters $\mathrm{GL}_2$8 (Castella et al., 2015). For elliptic curves over imaginary quadratic fields satisfying the Heegner hypothesis and splitting at $\mathrm{GL}_2$9, the Bertolini–Darmon–Prasanna $K$0-adic $K$1-function $K$2 plays the central role (Castella et al., 2023). In classical CM settings, one instead works with Katz-type or elliptic-unit $K$3-adic $K$4-functions attached to Hecke characters (Kezuka, 2017).

The conjectural statement takes several equivalent forms depending on the sign and the chosen Selmer structure. A representative torsion-form conjecture is

$K$5

for $K$6-ordinary newforms (Castella et al., 2015). In sign $K$7 situations, Perrin–Riou’s Heegner-point formulation predicts a square relation between the torsion part of a rank-one Selmer module and the index of a Heegner generator (Wan, 2014, Castella et al., 2023).

2. Heegner-point and $K$8-adic $K$9-function forms of the conjecture

A basic distinction in the subject is between sign $K_\infty^-/K$0 formulations, where the primary analytic object is a scalar $K_\infty^-/K$1-adic $K_\infty^-/K$2-function and the relevant Selmer group is expected to be torsion, and sign $K_\infty^-/K$3 formulations, where the algebra is rank one and the conjecture is naturally expressed in terms of Heegner points or Heegner classes (Wan, 2014, Liu et al., 2024).

For elliptic curves in the ordinary sign $K_\infty^-/K$4 setting, Howard’s structure theorem gives a pseudo-isomorphism

$K_\infty^-/K$5

for the anticyclotomic dual Selmer module $K_\infty^-/K$6, together with a divisibility

$K_\infty^-/K$7

where $K_\infty^-/K$8 is the initial Heegner class in a Kolyvagin system (Wan, 2014). Perrin–Riou’s conjecture then predicts the reverse divisibility and hence equality

$K_\infty^-/K$9

equivalently

$\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$0

(Wan, 2014). Wan proved this in the ordinary sign $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$1 case, integrally under a local hypothesis and after inverting $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$2 under a weaker one, by combining Howard’s Kolyvagin-system bound with a two-variable Rankin–Selberg main conjecture and Poitou–Tate comparisons (Wan, 2014).

An analogous square formulation appears in the more recent treatment of elliptic curves at Eisenstein primes. There, Perrin–Riou’s anticyclotomic conjecture states that the compact ordinary Selmer group and its dual both have $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$3-rank $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$4, and

$\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$5

(Castella et al., 2023). The same paper proves the equivalent Greenberg formulation

$\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$6

for elliptic curves with a rational $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$7-isogeny and $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$8 (Castella et al., 2023).

In the sign $\Gamma_K^-=\mathrm{Gal}(K_\infty^-/K)$9 regime for higher-rank Rankin–Selberg motives, the expected shape is the direct torsion equality between a $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$0-adic $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$1-function and the characteristic ideal of an Iwasawa Bloch–Kato Selmer group. For $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$2 over a CM field, the proved one-sided inclusion is

$\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$3

while in the sign $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$4 case the proved statement is the Heegner-type square inclusion

$\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$5

(Liu et al., 2024). This shows that the Perrin–Riou pattern persists far beyond the classical $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$6 setting.

3. Selmer structures, Heegner systems, and Euler-system methods

The algebraic side of the conjecture is built from carefully chosen local conditions. For compact coefficients $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$7, one introduces at $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$8 the local conditions $\Lambda_K^-=\mathbf{Z}_p[[\Gamma_K^-]]$9, with

$p$0

and global Selmer groups obtained by imposing these local conditions at $p$1 and unramified conditions away from $p$2 (Castella et al., 2023). For modular forms, Greenberg local conditions at $p$3 are defined using the ordinary line $p$4, and the minimal anticyclotomic Selmer group imposes $p$5 as the local quotient at $p$6 (Castella et al., 2015, Chida et al., 2013).

Heegner points, big Heegner points, and their higher-dimensional analogues supply the primary Euler systems. In the weight-two and Hida-family setting, quaternionic Shimura curves and definite quaternion algebras provide Heegner points $p$7 satisfying horizontal and vertical norm compatibilities, from which big Heegner systems and branchwise two-variable $p$8-adic $p$9-functions are extracted (Castella et al., 2015). In the ordinary sign $L$0 elliptic-curve setting, Heegner points over ring class fields yield a Kolyvagin system $L$1, whose initial term $L$2 governs the rank-one Selmer module (Wan, 2014).

For higher-weight modular forms, direct use of higher-codimension cycles is technically difficult, so Chida–Hsieh reduce to weight-two geometry via congruences of evaluations at Gross points. They construct Euler-system classes $L$3 from Heegner points on Shimura curves attached to auxiliary weight-two quaternionic forms congruent to $L$4 modulo powers of $L$5, and prove first and second explicit reciprocity laws relating these classes to theta elements (Chida et al., 2013). This yields the one-sided divisibility

$L$6

for higher-weight ordinary modular forms (Chida et al., 2013).

The explicit reciprocity laws are decisive throughout the subject. In the Eisenstein-prime work, Beilinson–Flach classes $L$7 interpolate along two variables and satisfy regulator identities at the two $L$8-adic places: $L$9

$p$00

thereby linking cyclotomic and anticyclotomic specializations (Castella et al., 2023). In the supersingular case with $p$01, Castella–Wan construct a $p$02-Heegner Kolyvagin system and prove

$p$03

up to the stated hypotheses, so that the square of the big logarithm of the Heegner class matches the anticyclotomic Rankin–Selberg $p$04-adic $p$05-function (Castella et al., 2015).

These constructions support a recurring structural principle: anticyclotomic Euler systems naturally lead to one inclusion, while explicit reciprocity and a second main conjecture or analytic input provide the reverse inclusion.

4. Established cases and major advances

Several substantial cases of the anticyclotomic Iwasawa Main Conjecture are now theorems. For ordinary elliptic curves in the sign $p$06 setting, Wan proved Perrin–Riou’s anticyclotomic Heegner-point main conjecture, integrally under a local hypothesis and after inverting $p$07 more generally (Wan, 2014). For supersingular elliptic curves with $p$08, Castella–Wan established the $p$09 and similarly the $p$10 anticyclotomic Heegner-point main conjecture, showing that the corresponding $p$11-Selmer group has rank one and that its torsion submodule is controlled by the square of the Heegner generator (Castella et al., 2015).

A major recent development is the removal of ramification hypotheses in the ordinary $p$12 setting. Burungale–Castella–Skinner prove the Heegner-point Main Conjecture for elliptic curves $p$13 over an imaginary quadratic field $p$14 satisfying the Heegner and splitting hypotheses, assuming $p$15 and residual irreducibility. They show that both the ordinary Selmer module and the compact Heegner Selmer group have $p$16-rank one and that

$p$17

in $p$18, with integral equality under residual surjectivity (Burungale et al., 2024). Via the $p$19-adic Waldspurger formula, they obtain the equivalent $p$20-adic $p$21-function statement

$p$22

up to the same integrality refinement (Burungale et al., 2024). Their method uses base change to a quartic CM field, Wan’s divisibility toward a three-variable main conjecture, and two-variable zeta elements, rather than direct imposition of local ramification hypotheses (Burungale et al., 2024).

A parallel advance occurs for elliptic curves at Eisenstein primes. Assuming a rational $p$23-isogeny with semisimplified residual representation $p$24 and $p$25, the anticyclotomic Iwasawa–Greenberg main conjecture is proved: $p$26 and this is shown to imply Perrin–Riou’s square main conjecture in the same setting (Castella et al., 2023). The technical novelty is a Kolyvagin-system estimate uniform near the augmentation ideal $p$27, allowing control at the trivial height-one prime without the earlier auxiliary rank-one assumption $p$28 (Castella et al., 2023).

For modular forms of higher weight, Chida–Hsieh generalized Bertolini–Darmon’s weight-two theory and proved the one-sided divisibility

$p$29

under their residual hypotheses (CR) and ordinarity hypothesis (PO) (Chida et al., 2013). In the setting of Hida families, Castella–Kim–Longo showed that anticyclotomic $p$30- and $p$31-invariants vary in a controlled way across a branch; assuming equality of analytic and algebraic invariants at one specialization $p$32, the full main conjecture propagates to every specialization in the family (Castella et al., 2015).

Beyond $p$33, Li–Tian–Xiao–Zhang–Zhu establish one-sided divisibilities for Rankin–Selberg motives over CM fields along anticyclotomic directions. In the sign $p$34 case they prove that the anticyclotomic $p$35-adic $p$36-function belongs to the characteristic ideal of the Iwasawa Bloch–Kato Selmer group, and in the sign $p$37 case they prove the square inclusion predicted by the Heegner-type formulation (Liu et al., 2024). This situates the conjecture in the Gan–Gross–Prasad framework for unitary groups.

5. Variation, congruences, and specialization phenomena

A central theme in anticyclotomic Iwasawa theory is that invariants are often rigid under congruence or across $p$38-adic families. For Hida families, Castella–Kim–Longo proved anticyclotomic analogues of Emerton–Pollack–Weston’s cyclotomic variation formulas. Assuming the residual representation satisfies their ordinary hypotheses and that one specialization has vanishing analytic and algebraic $p$39-invariants, they show that these $p$40-invariants vanish for every specialization in the family, and that the analytic and algebraic $p$41-invariants vary by the same explicit local terms at split primes. Hence equality for one weight-two specialization propagates to all weights $p$42 on the branch (Castella et al., 2015).

A complementary congruence phenomenon was developed in the indefinite BDP setting. For normalized newforms $p$43 satisfying the Heegner hypothesis for the same $p$44 and with isomorphic residual Galois representations modulo $p$45, the BDP $p$46-adic $p$47-functions satisfy a congruence after multiplication by local Euler factors: $p$48 (Nguyen, 28 Feb 2025). From this, one derives equality of analytic $p$49-invariants and equality of corrected $p$50-invariants for $p$51 and $p$52 (Nguyen, 28 Feb 2025). Combined with an algebraic comparison of Selmer invariants from Lei–Müller–Xia, the paper proves a transfer principle: if the anticyclotomic main conjecture is known for one form in the congruence class, and one-sided divisibility is known for another, then the full conjecture propagates to the latter (Nguyen, 28 Feb 2025). This is an anticyclotomic analogue of Greenberg–Vatsal’s cyclotomic congruence method.

Specialization from higher-variable theories also plays a fundamental role. Van Order showed that a two-variable $p$53-adic $p$54-function $p$55 exists for non-CM elliptic curves over imaginary quadratic fields, and that the anticyclotomic conjecture is obtained by specializing along the anticyclotomic line $p$56 (Order, 2011). Burungale–Castella–Skinner use the converse direction strategically: they deduce one-variable anticyclotomic and cyclotomic main conjectures from higher-variable divisibilities via base change and factorization of Selmer groups and $p$57-adic $p$58-functions (Burungale et al., 2024). This makes anticyclotomic IMC part of a larger web of specialization and descent statements rather than an isolated one-variable conjecture.

6. Extensions: Hilbert, CM, supersingular, and non-commutative settings

The scope of the conjecture has broadened well beyond elliptic curves over $p$59. For Hilbert modular forms over a totally real field $p$60, Xie proved the one-sided divisibility

$p$61

for the anticyclotomic extension of a CM quadratic extension $p$62, removing the Ihara’s lemma hypothesis from earlier work of Longo and Wang (Xie, 2019). The argument replaces the strong second reciprocity law used previously by a valuation identity derived from global Tate duality, sufficient for the Euler-system induction (Xie, 2019).

In CM settings, the anticyclotomic main conjecture often becomes especially explicit. For the Hilbert class field $p$63 of $p$64 with $p$65, and the unique $p$66-extension $p$67 unramified outside one chosen prime $p$68, the main theorem identifies

$p$69

for each character $p$70 of $p$71, where $p$72 comes from an explicitly constructed CM $p$73-adic $p$74-function (Kezuka, 2017). This gives a full equality of characteristic ideals in a non-cyclotomic, anticyclotomic CM setting, including delicate cases with $p$75 (Kezuka, 2017).

The CM main conjecture also admits higher-codimension refinements. Bleher, Chinburg, Greenberg, Kakde, Sharifi, and Taylor study quotients of exterior powers of Iwasawa modules and show that, under the relevant CM main conjectures, their codimension-two support is controlled by finite collections of Katz $p$76-adic $p$77-functions (Bleher et al., 2019). In rank-one cases this yields identities such as

$p$78

interpretable as higher-codimension refinements of the anticyclotomic IMC (Bleher et al., 2019).

Supersingular anticyclotomic theory requires $p$79-local conditions. For $p$80, Castella–Wan define $p$81 over the anticyclotomic tower and prove a quasi-isomorphism

$p$82

together with length identities at height-one primes yielding the expected square characteristic-ideal relation with the big $p$83-Heegner class $p$84 (Castella et al., 2015). This provides the supersingular anticyclotomic counterpart of the ordinary sign $p$85 theory.

Finally, the conjecture extends to non-commutative and equivariant settings. Mejías Gil formulates a new equivariant main conjecture for arbitrary one-dimensional $p$86-adic Lie extensions containing the cyclotomic tower and proves that, in CM situations, its minus part recovers the main conjecture of Ritter and Weiss (Gil, 2022). This suggests that anticyclotomic main conjectures can be embedded into a much broader non-commutative Iwasawa-theoretic framework.

7. Arithmetic consequences and remaining directions

When established, the anticyclotomic Iwasawa Main Conjecture has strong consequences for arithmetic. Burungale–Castella–Skinner use their cyclotomic and anticyclotomic one-variable theorems to deduce the $p$87-part of the Birch–Swinnerton–Dyer formula for non-CM elliptic curves of analytic rank $p$88 or $p$89, under good ordinary hypotheses and residual irreducibility (Burungale et al., 2024). Their work also feeds into proofs of Kolyvagin’s conjecture and its cyclotomic variant (Burungale et al., 2024). In the CM setting over $p$90, the main conjecture has implications toward the $p$91-part of BSD for CM elliptic curves and weak $p$92-adic Leopoldt-type statements (Kezuka, 2017).

At the level of Heegner classes, congruence results for BDP $p$93-adic $p$94-functions imply congruences between logarithms of generalized Heegner cycles, permitting transfer of nonvanishing information across congruent modular forms (Nguyen, 28 Feb 2025). In Hida families, control of $p$95- and $p$96-invariants gives systematic propagation of main-conjecture equalities from weight two to higher weights (Castella et al., 2015). For Hilbert modular forms, one-sided divisibility already implies cotorsionness and $p$97 in substantial generality (Xie, 2019).

Several limitations remain explicit in the literature. One common issue is that many papers establish only one inclusion of characteristic ideals; the reverse inclusion may require deeper geometric input, such as Eisenstein congruences, additional reciprocity laws, or control of local deformation rings (Chida et al., 2013, Xie, 2019, Liu et al., 2024). In sign $p$98 settings, the conjecture often takes a Heegner-point form rather than a direct $p$99-adic $L$00-function equality, and passing between the two requires $L$01-adic Gross–Zagier or Waldspurger formulas (Wan, 2014, Castella et al., 2023). Supersingular theory still depends on restrictive hypotheses such as $L$02 in the cleanest formulations (Castella et al., 2015). Higher-rank generalizations prove one side of the conjecture but not yet the reverse inclusion (Liu et al., 2024).

A plausible implication is that the anticyclotomic IMC is no longer a single conjecture but a network of related statements connected by specialization, congruence, Euler systems, and base change. The most recent work shows this explicitly: the anticyclotomic theory over $L$03 can be deduced from higher-variable information over larger CM fields (Burungale et al., 2024), propagated across congruent forms (Nguyen, 28 Feb 2025), or embedded into equivariant non-commutative formulations (Gil, 2022). In that sense, the conjecture functions both as a precise statement about one-variable Iwasawa modules and as an organizing principle for modern arithmetic geometry.