Anticyclotomic Euler System

• Anticyclotomic Euler system is a framework that constructs cohomology classes over anticyclotomic towers using norm compatibility and reciprocity laws.
• It employs constructions from Heegner points, diagonal cycles, and harmonic analysis to bridge geometric elements with p-adic L-functions.
• The system provides explicit tools for controlling Selmer groups and refining main conjectures in Iwasawa theory.

An anticyclotomic Euler system is a collection of global cohomology classes, or in definite settings companion scalar-valued elements such as theta elements or modified $L$-values, indexed by ring class conductors or auxiliary squarefree products of inert or split primes in an anticyclotomic tower over an imaginary quadratic field or, more generally, a CM extension. Its defining feature is a system of norm-compatibility and reciprocity laws that ties the classes to local Euler factors and thereby connects special points, diagonal cycles, or special values on Shimura varieties to Selmer groups, Shafarevich–Tate groups, and Iwasawa modules (Sweeting, 2020). In modern usage the term encompasses the classical Heegner-point setting, Howard’s bipartite formalism for definite and indefinite signs, and a broader family of constructions from diagonal cycles, Hirzebruch–Zagier cycles, adjoint or Asai representations, and split anticyclotomic towers on unitary or orthogonal Shimura varieties (Castella et al., 2023).

1. Anticyclotomic framework and ambient Galois representations

Let $K/\mathbf Q$ be an imaginary quadratic field. The anticyclotomic $\mathbf Z_p$-extension $K_\infty/K$ is the unique $\mathbf Z_p$-extension on which complex conjugation acts by inversion, with Galois group

$$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$

In the standard Iwasawa-theoretic formulation there is a tautological character $\Psi:G_K\to \Lambda^\times$, and for a $p$-adic Galois representation $T$ one forms the anticyclotomic deformation $\mathbf T=T\otimes \Lambda(\Psi)$ (Sweeting, 2020). Over more general CM fields $K/\mathbf Q$0, the anticyclotomic direction is often realized by the norm-one torus $K/\mathbf Q$1, and the relevant tower is the maximal pro-$K/\mathbf Q$2 anticyclotomic extension or a split anticyclotomic quotient cut out by norm-one ideles (Lai et al., 2024).

The arithmetic input is usually a rank-two representation attached to an elliptic curve or modular form, but the range is now wider. The literature includes representations attached to elliptic curves and weight-two newforms (Sweeting, 2020), twists $K/\mathbf Q$3 of modular Galois representations by anticyclotomic Hecke characters (Castella et al., 2023), adjoint representations $K/\mathbf Q$4 over $K/\mathbf Q$5 (Alonso et al., 2022), Asai representations of Hilbert modular forms (Alonso et al., 25 Jan 2025), and Rankin–Selberg-type representations for $K/\mathbf Q$6 arising from RACSDC automorphic representations (Lai et al., 2024). The Selmer structures imposed on these representations are typically Greenberg ordinary conditions at primes above $K/\mathbf Q$7, together with unramified or ordinary conditions away from $K/\mathbf Q$8, though relaxed–strict, balanced, and signed variants also occur (Castella et al., 2023).

A persistent structural feature is the dependence on the sign of the functional equation. In the classical anticyclotomic theory of elliptic curves, the decomposition $K/\mathbf Q$9 with $\mathbf Z_p$0 supported on inert primes governs whether the arithmetic is “indefinite” or “definite.” When $\mathbf Z_p$1 is even, CM points on Shimura curves are available and produce global cohomology classes. When $\mathbf Z_p$2 is odd, CM points on curves are unavailable, and the Euler-system role is played instead by theta elements or modified $\mathbf Z_p$3-values on definite Shimura sets (Sweeting, 2020).

2. Classical Heegner-point systems and the bipartite formalism

The prototypical anticyclotomic Euler system is Kolyvagin’s system of Heegner points. Under a Heegner hypothesis, for each squarefree product $\mathbf Z_p$4 of inert primes one has a ring class field $\mathbf Z_p$5 and CM points $\mathbf Z_p$6. Their trace relations give the basic Euler-system norm compatibilities. Kolyvagin then applies derivative operators

$\mathbf Z_p$7

to define derived points $\mathbf Z_p$8, whose Kummer images yield classes $\mathbf Z_p$9. Gross’s relations control their local behavior and underpin finite–singular compatibility (Sweeting, 2020).

Howard’s decisive abstraction was the notion of a bipartite Euler system. In this framework one has, for one parity of auxiliary sets, cohomology classes $K_\infty/K$0 in modified Selmer groups, and for the opposite parity scalar-valued elements $K_\infty/K$1 in the coefficient ring, linked by reciprocity laws

$K_\infty/K$2

This packages the indefinite and definite cases in a single object and makes the anticyclotomic main conjecture accessible to Kolyvagin-system methods (1202.05181). The same paradigm underlies later treatments of elliptic curves with weaker residual hypotheses, where one-sided divisibility in both sign cases is proved assuming only $K_\infty/K$3 rather than residual irreducibility (Aribam et al., 2023).

Recent work has pushed the abstraction further. A $K_\infty/K$4-complete anticyclotomic Euler system is axiomatized as a family $K_\infty/K$5 over the full anticyclotomic tower, satisfying vertical compatibility in the $K_\infty/K$6-direction together with the horizontal Euler relations at admissible inert primes. From such data one constructs a universal anticyclotomic Kolyvagin system for $K_\infty/K$7 and for its anticyclotomic twist $K_\infty/K$8, recovering in a uniform way the Heegner-point, generalized Heegner-cycle, and Hida-family examples (Mastella et al., 13 May 2025). This axiomatization isolates the minimal hypotheses needed for Selmer bounds and clarifies that the anticyclotomic theory is not tied to any single geometric construction.

3. Geometric constructions beyond Heegner points

A major development has been the replacement of CM points by higher-dimensional algebraic cycles. For modular forms twisted by anticyclotomic Hecke characters, one source is the étale Abel–Jacobi image of generalized Gross–Kudla–Schoen diagonal cycles on triple products of modular curves. After projection to suitable isotypic components and patching over ring class towers, this produces classes

$K_\infty/K$9

satisfying tame Euler-system relations

$\mathbf Z_p$0

for split primes $\mathbf Z_p$1 (Castella et al., 2023). The same circle of ideas yields an anticyclotomic Euler system for the Rankin–Selberg convolution of two modular forms, constructed from $\mathbf Z_p$2-adic families of generalized diagonal cycles and used to study Bloch–Kato in analytic ranks zero and one (Alonso et al., 2021).

The geometric range now extends well beyond $\mathbf Z_p$3. For $\mathbf Z_p$4, split anticyclotomic Euler systems are built from diagonal cycles on unitary Shimura varieties. The classes

$\mathbf Z_p$5

satisfy

$\mathbf Z_p$6

with $\mathbf Z_p$7 the local Euler factor of the corresponding Rankin–Selberg Galois representation (Lai et al., 2024). In the Asai setting, an anticyclotomic Euler system is constructed from Hirzebruch–Zagier cycles on products of modular curves and Hilbert modular surfaces, and the resulting classes vary $\mathbf Z_p$8-adically in Hida families (Alonso et al., 25 Jan 2025). For adjoint modular Galois representations, diagonal-cycle constructions similarly produce an anticyclotomic Euler system related to unitary-group $\mathbf Z_p$9-adic $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$0-functions (Alonso et al., 2022).

A conceptual refinement is the realization that tame norm relations can often be recovered formally from harmonic analysis on spherical varieties. In the relative-Langlands approach, the inverse relative Satake transform and refined Cartan decomposition imply that, for suitable integral Hecke operators, the trace from level $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$1 to $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$2 matches the action of the local Euler polynomial on the basic vector. This yields a uniform derivation of tame Euler-system relations and recovers many known examples, including a new split anticyclotomic Euler system in a case studied by Cornut (Cai et al., 2024). This suggests that a substantial part of the Euler-system mechanism is representation-theoretic rather than ad hoc.

4. Reciprocity laws, local conditions, and analytic avatars

Anticyclotomic Euler systems are governed not only by norm relations but also by local reciprocity laws. In the Heegner-point and bipartite settings, the crucial maps are the localization map $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$3 and the residue map $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$4, relating global classes to ordinary or unramified local conditions. In patched anticyclotomic Iwasawa theory one has schematic laws

$$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$5

and these identities are the engine behind Selmer-rank bounds and characteristic-ideal divisibilities (Sweeting, 2020).

At $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$6, the local condition is usually ordinary in the Greenberg sense. For $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$7 and $$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$8,

$$\Gamma_{\mathrm{ac}}:=\mathrm{Gal}(K_\infty/K)\simeq \mathbf Z_p, \qquad \Lambda:=\mathbf Z_p[[\Gamma_{\mathrm{ac}}]].$$9

with unramified conditions away from a finite set (Sweeting, 2020). In higher-weight diagonal-cycle constructions one also encounters relaxed–strict and ordinary–ordinary Selmer structures, designed to place the classes in the expected Bloch–Kato group for a given sign and Hodge range (Castella et al., 2023). In non-ordinary settings, signed local conditions become necessary: plus/minus Heegner point theories and Sprung-type $\Psi:G_K\to \Lambda^\times$0 constructions define signed Coleman maps and signed Selmer groups over the anticyclotomic tower (Burungale et al., 2023).

The analytic side is supplied by anticyclotomic $\Psi:G_K\to \Lambda^\times$1-adic $\Psi:G_K\to \Lambda^\times$2-functions and regulators. In the definite elliptic-curve case, a distinguished element $\Psi:G_K\to \Lambda^\times$3 has square interpolating central values

$\Psi:G_K\to \Lambda^\times$4

up to $\Psi:G_K\to \Lambda^\times$5-adic units (Sweeting, 2020). For diagonal cycles, triple-product $\Psi:G_K\to \Lambda^\times$6-adic $\Psi:G_K\to \Lambda^\times$7-functions factor through anticyclotomic $\Psi:G_K\to \Lambda^\times$8-adic $\Psi:G_K\to \Lambda^\times$9-functions of Bertolini–Darmon or Bertolini–Darmon–Prasanna type, and the explicit reciprocity law identifies the image of the global cohomology class under a regulator with the relevant $p$0-adic $p$1-function (Castella et al., 2023). In the adjoint setting, a Perrin–Riou style logarithm satisfies

$p$2

up to a nonzero scalar (Alonso et al., 2022).

A compact comparison is useful.

Setting	Euler-system datum	Analytic counterpart
Indefinite elliptic-curve case	Heegner or Kolyvagin classes $p$3, $p$4	$p$5, Heegner-point regulators
Definite elliptic-curve case	Theta or modified $p$6-value elements $p$7, $p$8	Anticyclotomic $p$9-adic $T$0-function
Non-ordinary signed theory	Signed classes $T$1 and signed theta elements	Signed anticyclotomic main conjectures

The reciprocity picture is not uniform in every higher-rank construction. Some works obtain a full explicit reciprocity law, while others establish only the Euler-system side and leave the analytic comparison to future work. For example, the split anticyclotomic system for $T$2 proves tame and wild norm relations but explicitly does not assert a full reciprocity law to a $T$3-adic $T$4-function (Lai et al., 2024). This marks a genuine boundary of the present theory rather than a notational omission.

5. Arithmetic applications

The primary use of anticyclotomic Euler systems is the control of Selmer groups. In the Heegner-point setting, new nonvanishing results toward Kolyvagin’s conjecture imply explicit Selmer-rank formulas. If $T$5, then in the indefinite case

$T$6

with parity and eigenspace sign determined by $T$7 (Sweeting, 2020). In the definite case, with $T$8 replacing $T$9, one has $\mathbf T=T\otimes \Lambda(\Psi)$0 finite and equal to $\mathbf T=T\otimes \Lambda(\Psi)$1 (Sweeting, 2020). These statements convert the vanishing pattern of the Euler system into a precise description of Selmer rank.

Anticyclotomic Euler systems also furnish results toward Bloch–Kato. Diagonal-cycle constructions for $\mathbf T=T\otimes \Lambda(\Psi)$2 prove that, in the definite case with $\mathbf T=T\otimes \Lambda(\Psi)$3, nonvanishing of the central value forces $\mathbf T=T\otimes \Lambda(\Psi)$4, while in the higher-weight rank-one range, nontriviality of the distinguished class $\mathbf T=T\otimes \Lambda(\Psi)$5 implies that $\mathbf T=T\otimes \Lambda(\Psi)$6 is one-dimensional (Castella et al., 2023). In higher rank, the split anticyclotomic diagonal-cycle system gives the implication

$\mathbf T=T\otimes \Lambda(\Psi)$7

for certain $\mathbf T=T\otimes \Lambda(\Psi)$8 representations (Lai et al., 2024). The Asai Euler system of Hirzebruch–Zagier cycles similarly yields rank-one Bloch–Kato statements and a divisibility toward the anticyclotomic Iwasawa main conjecture (Alonso et al., 25 Jan 2025).

At the $\mathbf T=T\otimes \Lambda(\Psi)$9-adic level, the main conjectural pattern is that the characteristic ideal of a torsion Selmer module is generated by an Euler-system regulator or $K/\mathbf Q$00-adic $K/\mathbf Q$01-function. For Heegner-point Iwasawa theory, one obtains pseudo-isomorphisms

$K/\mathbf Q$02

in the indefinite case, with

$K/\mathbf Q$03

and the definite analogue

$K/\mathbf Q$04

up to the stated integrality hypotheses (Sweeting, 2020). In Howard’s bipartite formulation and its generalizations, one-sided divisibility toward the anticyclotomic main conjecture is proved for both signs, and equality is reduced to nonvanishing of sufficiently many $K/\mathbf Q$05-adic $K/\mathbf Q$06-functions attached to congruent modular forms (1202.05181).

A refined development is the determination of higher Fitting ideals. For anticyclotomic Selmer and Shafarevich–Tate groups over $K/\mathbf Q$07-extensions, all higher Fitting ideals of the Pontryagin dual are described in terms of the bipartite Euler systems of Bertolini–Darmon. In the definite case,

$K/\mathbf Q$08

and in the indefinite case

$K/\mathbf Q$09

thereby refining the characteristic-ideal formulation of the main conjecture to all higher Fitting levels (Ronche et al., 12 Mar 2026).

6. Variants, generalizations, and current limitations

The modern theory includes several genuinely different variants. One is the signed anticyclotomic theory at non-ordinary primes. For supersingular or more general non-ordinary $K/\mathbf Q$10, one constructs plus/minus or $K/\mathbf Q$11 Heegner point Euler systems together with signed anticyclotomic theta elements. These satisfy signed norm relations, define signed Selmer groups, and yield signed main conjectures; in the inert supersingular case the minus main conjecture is proved integrally for semistable elliptic curves (Burungale et al., 2023). Another variant occurs when $K/\mathbf Q$12 is inert in $K/\mathbf Q$13 for diagonal-cycle Euler systems: in that setting vertical norm relations along the $K/\mathbf Q$14-power anticyclotomic tower may fail because denominators obstruct the norm maps, so one works with a purely tame system indexed by split auxiliary primes and extracts arithmetic information from the bottom class (Marannino, 30 Jul 2025).

The base field can also be enlarged. Anticyclotomic Euler systems have been constructed over imaginary biquadratic fields $K/\mathbf Q$15, where the anticyclotomic tower is $K/\mathbf Q$16-adic and the Euler-system classes are built from balanced diagonal cycles together with two CM inputs. The resulting classes satisfy tame and wild norm relations and lead to Bloch–Kato vanishing in the rank-zero range and a divisibility toward the corresponding Iwasawa–Greenberg main conjecture (Do, 2024). For general CM fields $K/\mathbf Q$17, Urban’s congruence method on definite unitary groups produces anticyclotomic Euler systems for finite-order anticyclotomic Hecke characters, together with one side of the divisibility in the anticyclotomic main conjecture after inverting $K/\mathbf Q$18 (Lee, 7 Aug 2025).

Several limitations remain explicit in the literature. Some constructions are only “split anticyclotomic,” meaning that tame norm relations are proved only at primes split in the CM extension, because the local harmonic analysis or Satake formalism is presently available only there (Cai et al., 2024). Higher-rank diagonal-cycle constructions may depend on conjectural cohomological realizations of automorphic Galois representations, such as Hypothesis conj:Coh in the $K/\mathbf Q$19 setting (Lai et al., 2024). The inverse relative Satake formula in mixed characteristic is still conjectural in the general strongly tempered case (Cai et al., 2024). Even when a full Euler system is available, explicit reciprocity laws to $K/\mathbf Q$20-adic $K/\mathbf Q$21-functions are not yet universal.

These limitations should not be confused with a lack of structural coherence. On the contrary, the current body of work shows a remarkably stable template: anticyclotomic towers, carefully chosen local conditions, geometric classes or definite-side theta elements, Euler-factor norm relations, and reciprocity laws tying the global objects to $K/\mathbf Q$22-adic $K/\mathbf Q$23-functions or regulators. What varies is the geometry that manufactures the classes and the analytic input needed to convert divisibilities into equalities. In that sense, the anticyclotomic Euler system has become less a single construction than a unifying mechanism for anticyclotomic Iwasawa theory across $K/\mathbf Q$24, unitary groups, adjoint and Asai motives, and higher-rank Rankin–Selberg problems (Mastella et al., 13 May 2025).