The exact anticyclotomic $p$-adic Birch–Swinnerton-Dyer leading-term conjectures for elliptic curves (Agboola–Castella, Conjectures 2.20 and 2.21)

For an elliptic curve $E/\mathbb{Q}$ of good ordinary reduction at an odd prime $p$ that splits in an imaginary quadratic field $K$ satisfying the Heegner hypothesis, Agboola–Castella formulated Birch–Swinnerton-Dyer-type conjectures for the two anticyclotomic $p$-adic $L$-functions $L_{\mathfrak p}(f)=\mathscr L_{\mathfrak p}(f)^{2}$ and $\mathscr L_{\mathfrak p}(f)$ attached to $E$: the order of vanishing along the anticyclotomic $\mathbb{Z}_p$-extension should be $2(\max\{r^+,r^-\}-1)$ and $\max\{r^+,r^-\}-1$ respectively, where $r^\pm$ are the ranks of the $\pm$-eigenspaces of $E(K)$, with leading terms expressed through derived anticyclotomic $p$-adic regulators, the order of the Tate–Shafarevich group, and Tamagawa factors. The source and later work prove only one inequality and the leading coefficient up to a $p$-adic unit, conditionally on the Iwasawa–Greenberg main conjecture. This problem asks for a proof of the two conjectures in their exact form; their derived regulators are defined in the source under the maximal non-degeneracy conjecture of Mazur–Bertolini–Darmon for the derived $p$-adic height filtration, which is itself open. As of 2026 the exact equalities, the rank-part equality, and the non-degeneracy conjecture remain open.

Background

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ with associated newform $f$ (the weight-2 cuspidal modular form attached to $E$ by the modularity theorem), let $p$ be an odd prime of good ordinary reduction for $E$ (ordinary means $p\nmid a_p(E)$, where $a_p(E)=p+1-\#E(\mathbb{F}_p)$), and let $K$ be an imaginary quadratic field. The Heegner hypothesis $(\mathrm{Heeg})$ requires that every prime factor of $N$ splits in $K$; additionally $(\mathrm{spl})$ requires that $p=\mathfrak p\bar{\mathfrak p}$ splits in $K$. The anticyclotomic $\mathbb{Z}_p$-extension $K_\infty/K$ is the unique $\mathbb{Z}_p$-extension of $K$ — a Galois extension whose Galois group is isomorphic to the additive group of $p$-adic integers $\mathbb{Z}_p$ — on which complex conjugation (the nontrivial automorphism of $K/\mathbb{Q}$) acts by $-1$; one writes $\Gamma=\mathrm{Gal}(K_\infty/K)$, $\Lambda=\mathbb{Z}_p[[\Gamma]]\simeq\mathbb{Z}_p[[T]]$ for the Iwasawa algebra (the completed group ring of $\Gamma$), and $J=(T)$ for its augmentation ideal; the source works over $\Lambda_{\hat{\mathcal O}}:=\Lambda\hat\otimes_{\mathbb{Z}_p}\hat{\mathcal O}$, with $\hat{\mathcal O}$ the completion of the ring of integers of the maximal unramified extension of $\mathbb{Q}_p$, and denotes by $J$ the augmentation ideal of either ring. Let $r^\pm$ denote the rank of the $\pm$-eigenspace $E(K)^\pm$ under complex conjugation, so $\mathrm{rank}\,E(K)=r^++r^-$. A Heegner point $y_{\mathfrak p}$ on $E$ is the trace over its Galois orbit of the image of a CM point (a point with complex multiplication) on the modular curve $X_0(N)$, which classifies cyclic subgroups of order $N$ on elliptic curves, under a fixed modular parametrization $X_0(N)\to E$; write $\omega_E$ for the Néron differential of $E$ (the invariant differential on the Néron model, normalized over $\mathbb{Z}$) and $\log_{\omega_E}$ for the formal $p$-adic logarithm on the formal group of $E$ at $\mathfrak p$, and let $t$ denote the Heegner-point index factor, measuring the failure of $\log_{\omega_E}(y_{\mathfrak p})$ to generate the local logarithm lattice of $E(K)^+\otimes\mathbb{Z}_p$. Bertolini–Darmon–Prasanna, as strengthened by Castella–Hsieh, constructed the anticyclotomic $p$-adic $L$-function $\mathscr L_{\mathfrak p}(f)\in\Lambda_{\hat{\mathcal O}}$, interpolating square roots of the central critical values of the twists of $f/K$ by certain infinite-order characters of $\Gamma$; the trivial character lies outside the range of $p$-adic interpolation, and for it the construction yields the $p$-adic Gross–Zagier formula, expressing the value through $\log_{\omega_E}$ of a Heegner point, the period factor $\big(\frac{1-a_p(E)+p}{p}\big)$, and Tamagawa-type constants. The source sets $L_{\mathfrak p}(f):=\mathscr L_{\mathfrak p}(f)^{2}\in\Lambda_{\hat{\mathcal O}}$, the $p$-adic $L$-function interpolating the corresponding central critical values. In the cyclotomic setting the $p$-adic height pairing (the $\mathbb{Z}_p$-valued analogue of the Néron–Tate canonical height, built from the $\mathfrak p$-adic logarithm) on $E(K)\otimes\mathbb{Z}_p$ is expected to be non-degenerate; in the anticyclotomic setting it degenerates systematically whenever $|r^+-r^-|>1$. Bertolini–Darmon understood these degeneracies as giving rise to derived $p$-adic height pairings: on the pro-$p$ Selmer group $S_{\mathfrak p}(E/K)$ (the extended Mordell–Weil group in the terminology of Bertolini–Darmon), which fits into the exact sequence $0\to E(K)\otimes\mathbb{Z}_p\to S_{\mathfrak p}(E/K)\to T_p\Sha(E/K)\to 0$ with $T_p\Sha(E/K)$ the $p$-adic Tate module of the Tate–Shafarevich group $\Sha(E/K)$ of $E/K$ (so that $S_{\mathfrak p}(E/K)\simeq E(K)\otimes\mathbb{Z}_p$ under the standing finiteness of $\Sha(E/K)_{p^\infty}$), the successive radicals of the height pairings define a descending filtration $S_{\mathfrak p}^{(k)}$, and the successive derived pairings produce an enhanced (derived) $p$-adic regulator $\mathrm{Reg}_{p,\mathrm{der}}$, an element of $(J^\varrho/J^{\varrho+1})\otimes\mathbb{Q}$, and a derived square-root regulator $\mathrm{Reg}^{1/2}_{p,\mathrm{der}}$ (well-defined up to sign), generalizing the classical $p$-adic regulator $\mathrm{Reg}_p$ (the determinant of the height pairing); in the source these regulators are defined under the non-degeneracy assumption $S_{\mathfrak p}^{(p)}=US_{\mathfrak p}(E/K)$ recalled below. The Mazur–Bertolini–Darmon maximal non-degeneracy conjecture (Conjecture 2.9 of the source) predicts that the filtration has the largest possible ranks: $\mathrm{rank}_{\mathbb{Z}_p}S_{\mathfrak p}^{(2)}=|r^+-r^-|$ and $S_{\mathfrak p}^{(3)}=US_{\mathfrak p}(E/K)$, where $US_{\mathfrak p}(E/K)=\bigcap_n\mathrm{Norm}_{K_n/K}S_{\mathfrak p}(E/K_n)$ is the group of universal norms (isomorphic to $\mathbb{Z}_p$, by the theorem of Cornut–Vatsal that Heegner points become non-torsion over the anticyclotomic tower); equivalently, the graded ranks $e_k$ satisfy $e_1=2\min\{r^+,r^-\}$, $e_2=|r^+-r^-|-1$, and $e_k=0$ for $k\ge 3$. Agboola–Castella proved, under mild hypotheses, one of the inequalities predicted by the rank part of their conjectures and the predicted leading coefficient formula up to a $p$-adic unit on the algebraic side (Theorem 1.3), transferring it to the analytic side under the then-partially-proved Iwasawa–Greenberg main conjecture (an identity between the characteristic ideal of a Selmer group and a $p$-adic $L$-function) for $L_{\mathfrak p}(f)$ (Corollary 3.4). Later work sharpened the surrounding ingredients: Burungale–Castella–Kim proved Perrin-Riou's Heegner point main conjecture, which supplies the Iwasawa–Greenberg main conjecture in the ordinary split setting; Castella–Hsu–Kundu–Lee–Liu gave an independent method removing the extra hypotheses of Agboola–Castella and extending the up-to-unit leading-coefficient result to all good primes (including supersingular ones, i.e. $p\mid a_p(E)$, where they also prove the main conjecture under mild hypotheses), reducing the rank part to maximal non-degeneracy; Sano showed by the theory of derived Bockstein regulators that the Agboola–Castella conjecture follows from the Iwasawa–Greenberg main conjecture up to a $p$-adic unit. Nevertheless, in every available result the leading-coefficient formula retains the $p$-adic unit ambiguity, the order-of-vanishing statement is proved only as an inequality, and the maximal non-degeneracy conjecture itself is unproven. The natural extension of both conjectures to non-split (inert or ramified) primes $p$ and to the generalized Heegner hypothesis is left open by the source.

Problem Statement

Under the standing hypotheses of Agboola–Castella (2022) — $E/\mathbb{Q}$ an elliptic curve of conductor $N$, $p$ an odd prime of good ordinary reduction for $E$, $K$ an imaginary quadratic field of discriminant prime to $Np$ satisfying $(\mathrm{Heeg})$ (every prime factor of $N$ splits in $K$) and $(\mathrm{spl})$ ($p=\mathfrak p\bar{\mathfrak p}$ splits in $K$), the residual representation $\bar\rho_{E,p}$ of $G_\mathbb{Q}=\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the $p$-torsion $E[p]$ surjective, $p\nmid\#E(\mathbb{F}_p)$, the mod-$p$ representation $\bar\rho_{E,p}$ (equivalently $E[p]$) ramified at every prime $\ell|N$, and $\#\Sha(E/K)_{p^\infty}<\infty$ — prove the anticyclotomic $p$-adic Birch–Swinnerton-Dyer conjectures for the two $p$-adic $L$-functions over $K_\infty$ in their exact form, where $J$ is the augmentation ideal of $\Lambda_{\hat{\mathcal O}}=\mathbb{Z}_p[[\mathrm{Gal}(K_\infty/K)]]\hat\otimes_{\mathbb{Z}_p}\hat{\mathcal O}$, in which $\mathscr L_{\mathfrak p}(f)$ and $L_{\mathfrak p}(f)=\mathscr L_{\mathfrak p}(f)^{2}$ live, $r^\pm=\mathrm{rank}\,E(K)^\pm$, $\rho=\max\{r^+,r^-\}-1$, $c_\ell$ is the Tamagawa number of $E/\mathbb{Q}_\ell$ (the index of $E(\mathbb{Q}_\ell)$ in its subgroup of points with nonsingular reduction), $\mathrm{Reg}_{p,\mathrm{der}}$ is the derived $p$-adic regulator and $\mathrm{Reg}^{1/2}_{p,\mathrm{der}}$ the derived square-root regulator of Bertolini–Darmon type: (i) Conjecture 2.20: $\mathrm{ord}_J L_{\mathfrak p}(f)=2(\max\{r^+,r^-\}-1)$, and the image of $L_{\mathfrak p}(f)$ in $J^{2\rho}/J^{2\rho+1}$ equals $\big(\frac{1-a_p(E)+p}{p}\big)^{2}\cdot \mathrm{Reg}_{p,\mathrm{der}}\cdot \#\Sha(E/K)\cdot \prod_{\ell|N}c_\ell^{2}$; (ii) Conjecture 2.21: $\mathrm{ord}_J \mathscr L_{\mathfrak p}(f)=\max\{r^+,r^-\}-1$, and the image of $\mathscr L_{\mathfrak p}(f)$ in $J^{\rho}/J^{\rho+1}$ equals $\pm\big(\frac{1-a_p(E)+p}{p}\big)\cdot \mathrm{Reg}^{1/2}_{p,\mathrm{der}}\cdot \sqrt{\#\Sha(E/K)}\cdot \prod_{\ell|N}c_\ell$. In the source these two conjectures are formulated using the derived regulators of its Definitions 2.10 and 2.16, whose definition presupposes the maximal non-degeneracy conjecture of Mazur–Bertolini–Darmon (Conjecture 2.9 of the source: $\mathrm{rank}_{\mathbb{Z}_p}S^{(2)}_{\mathfrak p}=|r^+-r^-|$ and $S^{(3)}_{\mathfrak p}=US_{\mathfrak p}(E/K)$, equivalently the graded ranks $e_1=2\min\{r^+,r^-\}$, $e_2=|r^+-r^-|-1$, $e_k=0$ for $k\ge 3$), which remains open; the exact equalities above are the target. Note that the currently known results establish only the inequality $\mathrm{ord}_J L_{\mathfrak p}(f)\ge 2\rho$ and the above leading-term formulas up to a $p$-adic unit, so removing that unit ambiguity is an integral part of the problem.

References

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