Anticyclotomic Indefinite Setting in Iwasawa Theory

• Anticyclotomic indefinite setting is a framework in Iwasawa theory that studies arithmetic invariants over Zₚ-extensions of imaginary quadratic fields with a sign −1 functional equation.
• The approach utilizes sophisticated Euler system techniques, p-adic L-functions interpolation, and explicit congruences linked to modular forms and non-torsion Heegner points.
• Key results include refined main conjectures, explicit Iwasawa invariants, and profound diophantine applications, extending to non-ordinary and higher weight cases.

The anticyclotomic indefinite setting encompasses a foundational class of problems and phenomena in modern Iwasawa theory, arithmetic geometry, and the paper of $L$-functions. It centers on the paper of arithmetic invariants—particularly Selmer groups and $p$-adic $L$-functions—over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields, in contexts where the sign of the functional equation is $-1$. In this regime, the arithmetic is intimately tied to the existence of non-torsion global cycles (such as Heegner points), exceptional zero phenomena, and refined (often non-cotorsion) Iwasawa-theoretic structures.

1. Structural Features of the Anticyclotomic Indefinite Setting

The anticyclotomic indefinite setting is defined by the following interlocking data:

• An imaginary quadratic field $K$;
• An anticyclotomic $\mathbb{Z}_p$-extension $K_\infty/K$ characterized by the property that complex conjugation acts by inversion on the Galois group $\Gamma = \operatorname{Gal}(K_\infty/K)$;
• Modular objects (e.g., $p$-ordinary modular forms, elliptic curves, or Hilbert modular forms) for which the root number of the relevant $L$-function over $K$ is $-1$.

The "indefinite" adjective reflects the sign of the functional equation: the functional equation's sign is $-1$, which is equivalent, via the parity conjecture, to the expectation that Mordell–Weil or Bloch–Kato Selmer groups have rank one. In this context, Selmer groups over $K_\infty$ are typically non-cotorsion as $\Lambda = \mathbb{Z}_p[[\Gamma]]$-modules; that is, they have positive (often rank one) free part and hence Iwasawa invariants demand refined interpretation (Hatley et al., 2021).

A further structural hallmark is the proliferation of nontrivial Heegner points (or their generalizations: big Heegner points, generalized Heegner cycles, diagonal cycles), whose presence reflects the rank one phenomenon and plays a decisive role in $p$-adic $L$-function constructions and Euler system arguments (Castella et al., 2015, Castella et al., 2023, Marannino, 30 Jul 2025).

2. Construction and Variation of Anticyclotomic Invariants

Analytic and Algebraic Invariants

The main analytic invariants are the anticyclotomic $p$-adic $L$-functions $L_p(f/K) \in \mathcal{O}[[T]]$, which interpolate (twisted) central values of Rankin–Selberg or triple-product $L$-functions as characters vary in the anticyclotomic direction. The fundamental algebraic invariants are attached to Selmer groups over $K_\infty$: $$\operatorname{Sel}(K_\infty, f) = \ker\left\{ H^1(K_\infty, A^f) \to \prod_{w} H^1(K_{\infty, w}, A^f/F_pA^f)\right\}$$ where $A^f$ is the $p$-adic Galois representation associated to $f$ (Castella et al., 2015).

Iwasawa $\mu$- and $\lambda$-invariants are defined on both the analytic side (as exponents and degrees in $L_p(f/K)$) and the algebraic side (from the characteristic power series of the Pontryagin dual of Selmer groups, regarded as modules over $\mathcal{O}[[T]]$).

The key feature in the indefinite setting is that, due to the positive rank of the Selmer group, the characteristic power series has zero constant term, and refined methods are required to isolate the relevant Iwasawa invariants (Hatley et al., 2021, Nguyen, 14 Oct 2025).

Variation in Hida Families

Anticyclotomic Iwasawa invariants vary in $p$-adic Hida families of modular forms, with explicit lambda-invariant comparison formulae: $$X(L_p(f_1/K)) - X(L_p(f_2/K)) = e(\alpha_2) - e(\alpha_1)$$ where $f_1, f_2$ lie on different branches (i.e., weight specializations) of the Hida family and $e(\alpha_i)$ are Euler factor invariants (Castella et al., 2015). This rigid variation mirrors the Emerton–Pollack–Weston theory for cyclotomic invariants, extended here to the anticyclotomic indefinite setting.

3. Euler Systems, Diagonal Cycles, and Selmer Group Structure

The construction of norm-compatible global cohomology classes (Euler systems) is central to bounding and sometimes determining the structure of Selmer groups in the anticyclotomic indefinite setting:

• "Big Heegner points" constructed via optimal embeddings in quaternion algebras give rise to Euler systems for modular forms and are the anticyclotomic counterpart to modular symbols (Castella et al., 2015).
• Generalized Heegner cycles, and more recently, diagonal cycles on triple products of modular curves, serve as higher rank analogues that allow treatment of modular forms of higher weight and more general Galois representations (Castella et al., 2023, Marannino, 30 Jul 2025).
• These cycles yield cohomology classes $\kappa_{f,\chi}$ whose non-vanishing directly controls (via Euler system/Kolyvagin system descent) the rank and structure of the Bloch–Kato Selmer group in the indefinite setting.

The key Euler system relation is of the form

$$\operatorname{cores}_{K[m\ell]/K[m]}(z_{f,\chi,m\ell}) = P_\ell(\operatorname{Frob}_\ell)\cdot z_{f,\chi,m}$$

where $P_\ell$ is the Euler factor at $\ell$ (Castella et al., 2023, Do, 29 Sep 2024, Alonso et al., 25 Jan 2025).

These results extend to more general settings such as Asai Galois representations of Hilbert modular forms (real quadratic base field) and to anticyclotomic extensions of biquadratic CM fields (Alonso et al., 25 Jan 2025, Do, 29 Sep 2024).

4. Main Conjectures, Fitting Ideals, and Explicit Formulae

A central achievement is the extension and proof of main conjectures identifying the algebraic and analytic invariants: $$\operatorname{Char}_\mathcal{A}(\operatorname{Sel}(K_\infty, f)^\vee) = (L_p(f/K))$$ as ideals in the anticyclotomic Iwasawa algebra $\mathcal{A}$ under explicit hypotheses (Castella et al., 2015).

In refined settings, anticyclotomic analogues of the Mazur–Tate and Kurihara conjectures are formulated: $$L_p(K(m), f) \in \operatorname{Fitt}_{\mathcal{O}[\operatorname{Gal}(K(m)/K)]}(\operatorname{Sel}(K(m), A_f)^\vee)$$ where $K(m)$ is a cyclic ring class extension and $L_p(K(m), f)$ is the square of a Bertolini–Darmon theta element (Kim, 2016, Kim, 14 May 2025). These Fitting ideal results provide not only divisibility but equalities, rendering the connection between $p$-adic $L$-functions and Selmer groups both sharp and computable.

In the context of congruent modular forms, precise formulae relate the difference of $\lambda$-invariants to explicit sums of local correction terms: $$\lambda(f) - \lambda(g) = \sum_{\ell | N} (\delta_\ell(f) - \delta_\ell(g))$$ where $\delta_\ell(-)$ are local invariants at bad primes, and both sides are directly computable even in the positive rank (indefinite) case (Nguyen, 14 Oct 2025).

5. Exceptional Zero Phenomena and $p$-adic Heights

The anticyclotomic indefinite setting admits exceptional zero phenomena: the vanishing of $p$-adic $L$-functions at special characters due to sign considerations in the functional equation. In these cases, the first non-vanishing derivative is directly related to $p$-adic height pairings: $L_p'(E/K, x, 1) = [H: H_p]\cdot\log_E(P_x - P_x^c)$ with $P_x$ a twisted sum of Heegner points (Longo et al., 2017), extending the classical Mazur–Tate–Teitelbaum formula to the anticyclotomic, ramified, or multivariable setting (Xie, 2021). The leading term of $p$-adic $L$-functions is then explicitly expressed in terms of arithmetic $L$-invariants defined by $p$-adic logarithms, cup products in group cohomology, and explicit period integrals.

6. Statistical, Diophantine, and Broader Arithmetic Consequences

Statistical Results and Non-Cotorsion Phenomena

In the indefinite case, the non-cotorsion nature of Selmer groups dramatically alters the landscape: Selmer groups may have rank one, and their torsion submodules (measured by Iwasawa invariants) are controlled by $p$-adic logarithms of Heegner points. For elliptic curves $E$ over $K$: $$\mu_p(E/K_\infty)=0,\; \lambda_p(E/K_\infty)=0 \iff \frac{\log_{\omega_E}(P)}{p} \in \mathbb{Z}_p^\times$$ where $P$ is a Heegner point (Hatley et al., 2021).

The statistical frequency of vanishing invariants (over families of elliptic curves or primes) is then reducible to the distribution of $p$-adic heights.

Diophantine Applications: Hilbert's Tenth Problem

Results in the indefinite anticyclotomic setting have been used to deduce undecidability (in the sense of Hilbert's Tenth Problem) for rings of integers in all finite layers of anticyclotomic towers, by leveraging rank stability of Mordell–Weil groups and congruence properties of elliptic curves (Ray et al., 2023).

Extension to Non-Ordinary and Non-Elliptic Settings

Recent developments include the extension of the entire Iwasawa–theoretic framework, including plus/minus Selmer groups and signed $p$-adic $L$-functions, to the non-ordinary (supersingular) case and to abelian varieties of GL$_2$-type, even when $p$ is inert in $K$ (Burungale et al., 2022, Longo et al., 2 Apr 2025).

7. Axiomatic and Generalized Frameworks

There now exists an axiomatic theory of anticyclotomic Euler systems and their associated Kolyvagin systems, suitable for any $p$-adic Galois representation and compatible with anticyclotomic twists (Mastella et al., 13 May 2025). This formalism clarifies the descent constructions from Euler systems, their compatibility with local conditions, and provides a flexible language for analyzing deep structural results like the anticyclotomic main conjecture.

The core structure is:

• Euler system: Families $\{c(n)\}$ of global cohomology classes satisfying precise norm and local compatibility;
• Universal Kolyvagin system: Derived classes $\{\kappa(n)\}$ controlling Selmer groups and their torsion submodules;
• Twisting formalism: Identifies Selmer groups over $K_\infty$ with the Selmer group for the anticyclotomic twist.

This framework covers classical settings (Heegner points), higher weight cases (generalized cycles), and Hida families, and allows the generalization of core results to much broader contexts.

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The anticyclotomic indefinite setting thus acts as a crucible for the interplay between global and local arithmetic (via Euler systems and Selmer group theory), $p$-adic analytic invariants, refined main conjectures, explicit congruences, and applications ranging from exceptional zero formulas to the undecidability of diophantine equations. Its paper integrates geometric cycles, p-adic analysis, modular forms, and advanced Galois cohomology, with increasing generality to encompass non-ordinary forms, Hilbert and Siegel modular varieties, and arbitrary (possibly non-commutative) extensions.