Anticyclotomic Iwasawa theory at potentially ordinary primes

Introduction

Let $p > 3$ be a fixed prime. This article proves one divisibility in the anticyclotomic Iwasawa main conjectures when $p$ is an additive potentially ordinary prime, treating both the definite and indefinite settings. Bertolini–Darmon proved one divisibility in the definite case for elliptic curves at ordinary primes, including multiplicative reduction [5], and Howard developed the bipartite Euler-system formalism linking the definite and indefinite settings [24]. Darmon–Iovita established the corresponding results for signed Selmer groups at supersingular primes split in the imaginary quadratic field [18], while Chida–Hsieh extended the ordinary method to higher-weight modular forms [10]. More recently, signed anticyclotomic Iwasawa theories for both definite and indefinite settings, treating abelian varieties of $\mathrm{GL}_2$-type at split non-ordinary primes and elliptic curves at inert supersingular primes were studied in [1 BBL26]. In the good ordinary setting, Burungale–Castella–Kim proved Perrin-Riou’s Heegner point main conjecture under suitable hypotheses [BCK21]. Bertolini–Longo–Venerucci proved both definite and indefinite main conjectures for elliptic curves at good ordinary and supersingular primes unramified in the quadratic field under suitable hypotheses, with one divisibility in the exceptional plus-signed inert supersingular case [7]. At additive primes, Kohen–Pacetti constructed anticyclotomic $p$-adic $L$-functions in [28]. One of the motivations of the present work is to study the corresponding Iwasawa main conjecture in the additive setting. Recently, Keller–Yin proved anticyclotomic Iwasawa main conjectures for elliptic curves with potentially good ordinary reduction at Eisenstein primes [29]. Their results concern the residually reducible case, whereas we assume residual absolute irreducibility in the present article.

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $Np^2$, where $(N,p)=1$, that has potentially ordinary reduction at $p$. As explained in [De198, De102], there exists a Dirichlet character $\varepsilon$ of conductor $p$, such that the twist $f_E \otimes\varepsilon$ is ordinary at $p$, where $f_E$ is the weight two newform attached to $E$. This twisting method is utilized in [De198, De102, Mü124] to study the cyclotomic Iwasawa theory at potentially ordinary primes (see also [20] for an approach to the potentially supersingular case). In this article, we develop the corresponding theory over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$, allowing $p$ to split, remain inert or ramify in $K$. The main inputs are the bipartite Euler system machinery developed in [5, 24], together with the anticyclotomic $p$-adic $L$-function constructed by Kohen-Pacetti [28]. We impose the following (Heegner) hypothesis on $N$ for the rest of the article:

(Heeg) $N = N^{+}N^{-}$ with $N^{-}$ square-free. Every prime $q \mid N^{+}$ splits in $K$, and every prime $q \mid N^{-}$ is inert in $K$.

Our main results apply more generally to a normalized $p$-ordinary eigenform

that is new away from $p$. Here, $\epsilon$ is a non-trivial Dirichlet character of conductor $p$. If $\epsilon^{2} \ne1$, then $f$ is a newform of level $Np$. Otherwise, $f$ is the ordinary $p$-stabilization of a newform $f^{\circ} \in S_{2}(\Gamma_{0}(N))$.

Let $\widetilde{f} = \operatorname{Tw}_{\epsilon}(f)$ be its self-dual twist. Write $\operatorname{Sel}_{N^{-}}(K_{\mathrm{ac}}, A_{\widetilde{f}})$ and $\operatorname{Sel}_{N^{-}}(K_{\mathrm{ac}}, T_{\widetilde{f}})$ for the discrete and compact Selmer groups defined in § 5, respectively, and let $\mathcal{L}_{f}$ be the $p$-adic $L$-function of §5.2.2, obtained from the construction of Kohen–Pacetti. For a positive integer $M$, let $\omega(M)$ denote the number of its distinct prime divisors. The hypotheses (CR) and (irr) are stated in §3.2.1 and § 5, respectively. We prove the following results.

For an elliptic curve $E/\mathbb{Q}$ as above, take $f$ to be the $p$-ordinary eigenform associated with $f_{E} \otimes\epsilon$ as specified above. After extending coefficients, the representation attached to $\widetilde{f}$ identifies with that on the Tate module $T_{p}(E)$. In this setting, (CR 2) is the absolute irreducibility of $E[p]$; we discuss the interpretation of the remaining hypotheses in Remark 4.1. Theorems A and B give the following corollary.

• i) If $\omega(N^{-})$ is odd, then $\operatorname{Sel}(K_{\mathrm{ac}}, E[p^{\infty}])^{\vee}$ is $\Lambda(\Gamma_{\mathrm{ac}})$-torsion and the divisibility of Theorem A holds with $A_{\widetilde{f}}$ replaced by $E[p^{\infty}]$.

• ii) If $\omega(N^{-})$ is even, then $\operatorname{rank}_{\Lambda(\Gamma_{\mathrm{ac}})}\left(\operatorname{Sel}(K_{\mathrm{ac}}, E[p^{\infty}])^{\vee}\right) = 1$ and the divisibility of Theorem B holds with $A_{\widetilde{f}}$ replaced by $E[p^{\infty}]$.

To prove Theorems A and B, we construct a bipartite Euler system for the self-dual twist $\widetilde{f}$, relating Heegner classes to theta elements of [28]. Although we use the twisting method of [16, 17], the passage to anticyclotomic Iwasawa main conjectures is not formal. When $\epsilon^{2} \ne1$, the norm pullback of $\epsilon$ to $K$ is not anticyclotomic, so the twist cannot be absorbed into the anticyclotomic character variable. We therefore incorporate the twist into the constructions of Heegner classes and theta elements and establish their compatibility under reduction modulo powers of $\varpi$. Hida theory supplies the required integral level control and, in the case where $\epsilon^{2} \ne1$, a comparison modulo $\varpi$ with higher-weight specializations. In the indefinite case, we also prove the non-vanishing modulo $\varpi$ of the anticyclotomic Heegner class, which is the key ingredient in the proof of the rank-one statements in Theorem B.

In § 9.3, we discuss the relation between Kolyvagin’s conjecture and the criteria for equality in Theorems A and B. In § 9.4, we discuss arithmetic applications of these theorems. In the definite case, we show that $L(\bar{f}/K,1) \ne0$ implies the finiteness of the Bloch–Kato Selmer group, whose length, together with local Tamagawa terms, is bounded by the valuation of $\mathcal{L}_f$ at the trivial character. In the indefinite case, a nonzero specialized Heegner class gives a Selmer group of corank one and bounds the torsion in its Pontryagin dual, together with a finite local-condition correction, in terms of the index of the class.

Acknowledgements. The second-named author (P.N.) wishes to thank the first-named author (A.L.) for agreeing to host him at the University of Ottawa during the Summer of 2026 as part of the University of Ottawa’s ‘Doctoral Mobility Program’. This work was first started during this visit. P.N. is also supported by a NUI travelling studentship. The research of A.L. is supported by the NSERC Discovery Grants Program RGPIN-2026-07384. A.L. also thanks Daniel Delbourgo and Ariel Pacetti for interesting discussions on topics related to this article.

Preliminaries

Let $N \in\mathbb{Z}$, $p > 3$ be a fixed prime, and $f \in S_2(\Gamma_0(N) \cap\Gamma_1(p),\epsilon^2)$ all be as in the introduction.

Fix embeddings

and let $v_p$ be the induced $p$-adic valuation, normalized such that $v_p(p) = 1$. We impose the following ordinarity condition throughout the article.

(ord) $v_p(a_p(f)) = 0$.

If $\chi: (\mathbb{Z}/M\mathbb{Z})^\times\to\mathbb{C}^\times$ is a Dirichlet character, then the projection $\mathbb{A}^\times\simeq\mathbb{Q}^\times\times\widehat{\mathbb{Z}}^\times\times\mathbb{R}_{>0} \twoheadrightarrow(\mathbb{Z}/M\mathbb{Z})^\times$ allows us to pull $\chi$ back to an idelic Hecke character, $\chi_{\mathbb{A}^\times}$, satisfying