Rank-one $p$-converse for CM elliptic curves beyond good ordinary primes, and the integral anticyclotomic Heegner main conjecture

For an elliptic curve $E/\mathbb{Q}$ with complex multiplication, Burungale and Tian (Invent. Math. 2020) proved the rank-one $p$-converse to the Gross–Zagier–Kolyvagin–Rubin theorem at primes $p>3$ of good ordinary reduction: if the $p^\infty$-Selmer group has $\mathbb{Z}_p$-corank one, then $L(s,E)$ vanishes to order exactly one at $s=1$. That theorem left open the remaining primes and reduction types — $p=2$, $p=3$, supersingular (inert) primes, and primes of bad (necessarily additive) reduction — and proved the underlying anticyclotomic main conjecture only in rational form; an integral refinement of it is an open direction of the same authors' program, flagged again in their rank-zero sequel. This problem asks for the completion of that program: an unconditional rank-one $p$-converse for every CM elliptic curve over $\mathbb{Q}$ and every prime $p$, together with the integral $\Lambda^\circ$-adic form of the anticyclotomic Heegner-type main conjecture at non-ordinary primes, capturing Tamagawa factors and $\mu$-invariants. As of August 2026 the published literature contains the inert-prime case only under a finiteness-of-Šafarevič–Tate assumption (Burungale–Kobayashi–Ota), while the non-split (inert and ramified) cases exist only in unpublished preprints (Kriz; Fan–Wan), which do not establish the all-prime unconditional statement, and no integral $\Lambda^\circ$-adic Heegner-type main conjecture at non-ordinary primes is established: Castella's integral Heegner main conjecture covers split ordinary $p>3$, and the recent Burungale–Kobayashi–Nakamura–Ota preprint (arXiv:2608.06879) proves an integral main conjecture at primes $p\ge 5$ ramified in $K$ with an elliptic-unit (Rubin-type) $p$-adic $L$-function rather than a Rankin–Selberg/Heegner-type one.

Background

Let $E/\mathbb{Q}$ be an elliptic curve and $L(s,E/\mathbb{Q})$ its Hasse–Weil $L$-function. The rank part of the Birch and Swinnerton-Dyer (BSD) conjecture predicts that the Mordell–Weil rank of $E(\mathbb{Q})$ equals the analytic rank $\mathrm{ord}_{s=1}L(s,E/\mathbb{Q})$. One direction is a theorem: Gross and Zagier expressed the first derivative $L'(E,1)$ as a Néron–Tate height of a Heegner point (a special point on a modular curve cut out by CM theory), and Kolyvagin used Heegner points as an Euler system (a norm-compatible family of cohomology classes with a Kolyvagin-derivative machine) to show that analytic rank $\le 1$ forces Mordell–Weil rank $\le 1$ and finiteness of the Šafarevič–Tate group $\text{Š}(E/\mathbb{Q})$; Rubin established the CM instances via the Iwasawa theory of elliptic units (norm-compatible units in abelian extensions of an imaginary quadratic field). For a prime $p$, the $p^\infty$-Selmer group $\mathrm{Sel}_{p^\infty}(E/\mathbb{Q})$ sits in the exact sequence $0\to E(\mathbb{Q})\otimes\mathbb{Q}_p/\mathbb{Z}_p\to\mathrm{Sel}_{p^\infty}(E/\mathbb{Q})\to\text{Š}(E/\mathbb{Q})[p^\infty]\to 0$, so its $\mathbb{Z}_p$-corank is a computable arithmetic rank. The converse direction, called a $p$-converse theorem, predicts for $r\in\{0,1\}$: $\mathrm{corank}_{\mathbb{Z}_p}\mathrm{Sel}_{p^\infty}(E/\mathbb{Q})=r\Rightarrow\mathrm{ord}_{s=1}L(s,E/\mathbb{Q})=r$. Say $E$ has complex multiplication (CM) if its endomorphism ring over $\overline{\mathbb{Q}}$ is an order in an imaginary quadratic field $K$; then $p$ is split, inert, or ramified in $K$ according to the Legendre/Kronecker symbol of the discriminant of $K$, and split (resp. inert) $p$ of good reduction corresponds to ordinary (resp. supersingular) reduction, the distinction governing whether the $p$-adic Tate module is locally reducible at $p$. The natural route to $p$-converse theorems is Iwasawa theory over the anticyclotomic $\mathbb{Z}_p$-extension $K^-_\infty/K$ (the subextension of the ring-class $\mathbb{Z}_p^2$-extension on which the nontrivial automorphism of $K$ acts by inversion): a Heegner-type main conjecture asserts that over the Iwasawa algebra $\Lambda^\circ=\mathcal{O}[[\mathrm{Gal}(K^-_\infty/K)]]$, the characteristic ideal of the relevant Selmer group equals the ideal generated by a Rankin–Selberg (Waldspurger/Bertolini–Darmon–Prasanna-type) $p$-adic $L$-function. At non-ordinary primes this must be $\pm$-refined: following Rubin, the local cohomology at $p$ decomposes into $\pm$-subspaces according to the parity of interpolating characters, and the main conjecture involves a pair of $\pm$ $p$-adic $L$-functions. An integral form of the main conjecture — an equality of $\Lambda^\circ$-characteristic ideals rather than one after tensoring with the coefficient field $L$ — additionally encodes the Tamagawa factors (local volume factors entering the BSD formula) and the $\mu$-invariant (the power of $p$ dividing a $\Lambda^\circ$-adic $L$-function). Rubin's 1988–1991 main conjectures for imaginary quadratic fields gave the first CM $p$-converse results, with a finiteness assumption on $\text{Š}[p^\infty]$ in rank one; Burungale and Tian (2020) removed it for good ordinary $p>3$ by proving the Heegner main conjecture in rational form. In this circle of ideas the non-ordinary cases of the rank-one converse are known to follow from the anticyclotomic Heegner main conjecture. The extensions to the remaining primes and reduction types, together with the integral refinement of the main conjecture at non-ordinary primes — an open direction the same authors flag in their rank-zero sequel — define this problem.

Problem Statement

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an order in an imaginary quadratic field $K$ (equivalently, let $\psi$ be the self-dual CM Hecke character of $K$ of Archimedean type $(-1,0)$ attached to $E$). Establish, unconditionally:

(i) Rank-one $p$-converse at every prime. For every prime $p$ — including $p=2$ and $p=3$, and in complete generality as to whether $p$ is split, inert, or ramified in $K$ and whether $E$ has good ordinary, good supersingular, or bad (necessarily additive) reduction at $p$ — $\mathrm{corank}_{\mathbb{Z}_p}\,\mathrm{Sel}_{p^\infty}(E/\mathbb{Q})=1\ \Longrightarrow\ \mathrm{ord}_{s=1}\,L(s,E/\mathbb{Q})=1.$

(ii) Integral anticyclotomic Heegner main conjecture. Over the anticyclotomic $\mathbb{Z}_p$-extension $K^-_\infty/K$ and the integral Iwasawa algebra $\Lambda^\circ=\mathcal{O}[[\mathrm{Gal}(K^-_\infty/K)]]$ (with $\pm$-refinements of the local condition at $p$ at non-ordinary primes), prove that, for each sign $\star\in\{+,-\}$, the characteristic ideal of the $\star$-Selmer module equals the ideal generated by the corresponding $\star$ anticyclotomic Rankin–Selberg (Waldspurger/Bertolini–Darmon–Prasanna-type) $p$-adic $L$-function — an equality in $\Lambda^\circ$, not merely after tensoring with the coefficient field — so that the Tamagawa factors and the $\mu$-invariant are captured integrally, at least for all primes $p$ at which $E$ does not have good ordinary reduction — that is, all supersingular primes and all primes of bad, additive reduction, whether $p$ is split, inert, or ramified in $K$, including $p=2$. The intended conventions: the coefficient ring $\mathcal{O}$, and the requirement of an integral equality of characteristic ideals in $\Lambda^\circ=\mathcal{O}[[\mathrm{Gal}(K^-_\infty/K)]]$ (rather than after tensoring with the coefficient field), are those of the integral ($\Lambda_{\mathcal{O}}$-characteristic-ideal) anticyclotomic Heegner main conjecture established by Castella at split ordinary primes $p>3$ (reference 14); the $\pm$ local conditions on the Selmer lattice and the normalization of the pair of $\pm$ anticyclotomic Rankin–Selberg $p$-adic $L$-functions at non-ordinary primes are the standard signed objects of the non-ordinary anticyclotomic theory (Rubin's $\pm$-decomposition of the local cohomology at $p$), as in the non-ordinary main-conjecture literature cited above; and fixing the corresponding integral normalization — the $\mu$-invariant and the Tamagawa factors at non-ordinary primes — is itself part of what is to be proved.

These two components serve one objective: completing, for complex-multiplication elliptic curves over $\mathbb{Q}$, the program left open by Burungale–Tian (2020), who settled the rank-one converse at good ordinary $p>3$ and proved the underlying main conjecture only in rational form. The non-ordinary cases of component (i) are covered by component (ii), and the good ordinary case $p>3$ is already published; the unpublished preprints of references 11 and 12 do not by themselves establish component (i) unconditionally. Both components are required, and the cases of component (i) covered by neither component (ii) nor the published good ordinary theorem are the good ordinary primes $p=2$ and $p=3$. Further directions sometimes associated with this program — non-CM curves, higher-weight CM forms, and CM fields of degree greater than two — are excluded from the required core because they are either resolved in the published literature (non-CM; higher-weight ordinary CM) or constitute a distinct program without a comparable completion boundary (higher-degree CM fields).

References

1. A. A. Burungale and Y. Tian, 'p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin', Inventiones mathematicae 220 (2020), no. 1, 211–253. DOI: 10.1007/s00222-019-00929-7, https://doi.org/10.1007/s00222-019-00929-7
2. B. H. Gross and D. B. Zagier, 'Heegner points and derivatives of L-series', Inventiones mathematicae 84 (1986), no. 2, 225–320. DOI: 10.1007/BF01388809, https://doi.org/10.1007/BF01388809
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