Schneider's conjecture in rank one: non-degeneracy of the cyclotomic $p$-adic height pairing on elliptic curves of Mordell–Weil rank one

An open problem arising from Rodolfo Venerucci's proof of a $p$-converse to the Kolyvagin–Gross–Zagier theorem for elliptic curves with split multiplicative reduction at $p$. The paper's key algebraic input is the non-degeneracy of Nekovář's half-twisted weight pairing on the extended Mordell–Weil group, obtained through anticyclotomic Hida-family deformation; the author states that the cyclotomic analogue — Schneider's conjecture in rank one — seems out of reach. The paper's Sha-finiteness limitation was later resolved by independent approaches (Skinner–Zhang and Castella preprints), and the surviving open core is to prove that for a rank-one elliptic curve over $\mathbb{Q}$ the canonical cyclotomic $p$-adic height pairing (in the good ordinary case), or the extended Mazur–Tate–Teitelbaum pairing on the extended Mordell–Weil group (in the split multiplicative case), is non-degenerate. Only partial results are known, at Eisenstein primes for positive proportions of quadratic twists (Burungale–Skinner, 2023).

Background

Let $E$ be an elliptic curve over $\mathbb{Q}$. The Mordell–Weil theorem states that the group $E(\mathbb{Q})$ of rational points is finitely generated; its rank is denoted $\mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q})$. The Tate–Shafarevich group $\Sha(E/\mathbb{Q})$ measures the failure of the local-to-global principle for principal homogeneous spaces under $E$, and the Birch and Swinnerton-Dyer (BSD) conjecture predicts that $\mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q})$ equals the order of vanishing at $s=1$ of the Hasse–Weil $L$-function $L(E,s)$, that $\Sha(E/\mathbb{Q})$ is finite, and gives an exact formula for its order. Two theorems give one direction unconditionally: Gross and Zagier showed that Heegner points (points on $E$ arising from CM points on the modular curve $X_0(N)$ via a modular parametrization) have heights proportional to central $L$-derivatives, and Kolyvagin used the system of these points to show that if $L(E,s)$ vanishes to order at most one at $s=1$, then $\mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q})\leq 1$ and $\Sha(E/\mathbb{Q})$ is finite.

A “$p$-converse” to Kolyvagin–Gross–Zagier is a statement of the form: $\mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q})=1$ (together with finiteness of the $p$-primary part $\Sha(E/\mathbb{Q})[p^{\infty}]$) implies $\mathrm{ord}_{s=1}L(E,s)=1$, for a fixed prime $p$ at which $E$ has prescribed reduction. Venerucci (Comment. Math. Helv. 91, 2016) proved such a converse for curves with split multiplicative reduction at an odd prime $p$ (i.e. the reduction mod $p$ is a nodal cubic with rational tangent directions; $p$-adically such a curve is a Tate curve, uniformized by $\mathbb{Q}_p^{\times}/q^{\mathbb{Z}}$ for a Tate period $q\in p\mathbb{Z}_p$). His proof combines: the three-variable main divisibility of Skinner–Urban for the Mazur–Kitagawa $p$-adic $L$-function (a $p$-adic analytic function of the weight and cyclotomic variables, built from a Hida family $\mathbf{f}$ — a $p$-adic analytic family of modular forms containing the modular form attached to $E$); Bertolini–Darmon's exceptional-zero formula, which identifies the second weight-derivative of $L_p$ with the square of the $p$-adic logarithm of a Heegner point when $L_p$ has an exceptional zero (a forced vanishing caused by multiplicative reduction at $p$, in the sense of Mazur–Tate–Teitelbaum); and a purely algebraic input.

That algebraic input is where the open problem lies. Using Nekovář's theory of Selmer complexes (a homological framework for Selmer groups, the cohomological groups that measure rational points and $\Sha$ via Kummer theory), Venerucci relates the structure of an anticyclotomic Selmer module (a Selmer module over the anticyclotomic $\mathbb{Z}_p$-extension of $K_{\chi}$, the infinite abelian $\mathbb{Z}_p$-extension on whose Galois group the non-trivial automorphism of $K_{\chi}$ acts by inversion) to the non-degeneracy of Nekovář's half-twisted weight pairing, a skew-symmetric $\mathbb{Q}_p$-valued bilinear form on the extended Mordell–Weil group $A^{\dagger}(K_{\chi})^{\chi}$. Here $K_{\chi}$ is an auxiliary imaginary quadratic field, $\chi$ a quadratic character, and the extended Mordell–Weil group of Mazur–Tate–Teitelbaum is obtained from $A(K_{\chi})^{\chi}$ (the $\chi$-isotypic part of the Mordell–Weil group) by adjoining one formal copy of $\mathbb{Z}$ generated by the Tate period class $q_{\chi}$; in rank one, $A^{\dagger}(K_{\chi})^{\chi}\otimes\mathbb{Q}_p$ is two-dimensional, spanned by $q_{\chi}$ and a non-torsion point $P_{\chi}$. Venerucci proves the formula $\langle q_{\chi},P_{\chi}\rangle^{\mathrm{Nek},\chi}_{V_f,\pi}=c\cdot\log_A(P_{\chi})$ with $c\neq 0$, whence the pairing is non-degenerate — but crucially he needs $\Sha(A/K_{\chi})^{\chi}_{p^{\infty}}$ finite to identify the extended Selmer group with the extended Mordell–Weil group, and his non-degeneracy is deduced through the anticyclotomic (Hida-family) deformation.

The cyclotomic counterpart is the theory of $p$-adic heights. For an elliptic curve with good ordinary reduction at $p$ (good reduction whose formal group has height one, equivalently $p\nmid a_p$, where $a_p=p+1-\#E(\mathbb{F}_p)$ is the $p$th Fourier coefficient of the modular form attached to $E$), Schneider and Mazur–Tate constructed a canonical cyclotomic $p$-adic height pairing $\langle\cdot,\cdot\rangle_p:E(\mathbb{Q})\times E(\mathbb{Q})\to\mathbb{Q}_p$, a $p$-adic analogue of the archimedean Néron–Tate height built from local $p$-adic heights. Schneider conjectured that this pairing is non-degenerate, i.e. its kernel is exactly the torsion subgroup. In rank one this is the statement $\langle P,P\rangle_p\neq 0$ for a non-torsion point; equivalently the $p$-adic regulator is non-zero. For split multiplicative reduction at $p$ the corresponding object is the extended $p$-adic height pairing of Mazur–Tate–Teitelbaum on the extended Mordell–Weil group, and the rank-one conjecture is the non-vanishing of $\langle q,P\rangle_p$. This non-vanishing is the missing algebraic input for rank-one instances of the algebraic $p$-adic BSD-type formulae of Schneider and Perrin-Riou, which express the leading term of an algebraic $p$-adic $L$-function (the Mellin transform of a Selmer-module characteristic ideal) through the determinant of the height pairing. Venerucci states explicitly that the analogue of his non-degeneracy input in cyclotomic Iwasawa theory — Schneider's conjecture in rank one — seems out of reach at present, and this is precisely the problem treated here. It is known how to compute these pairings numerically (Mazur–Stein–Tate), and numerical evidence supports non-degeneracy, but no general proof is known; the one general partial result (Burungale–Skinner 2023, at Eisenstein primes — primes at which the Galois representation on the torsion of the curve admits a congruence with an Eisenstein representation, enabling Ribet–Wiles-style congruence methods) covers only specific congruence primes and families of quadratic twists.

Problem Statement

Let $E$ be an elliptic curve over $\mathbb{Q}$ with $\mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q})=1$, let $P\in E(\mathbb{Q})$ be a point of infinite order, and let $p$ be an odd prime. Prove the rank-one non-degeneracy of the cyclotomic $p$-adic height pairing — Schneider's conjecture in rank one — in the two linked settings:

(a) (Good ordinary case.) If $E$ has good ordinary reduction at $p$, let $\langle\cdot,\cdot\rangle_p$ be the canonical cyclotomic $p$-adic height pairing of Schneider and Mazur–Tate on $E(\mathbb{Q})$ with values in $\mathbb{Q}_p$. Prove that $\langle P,P\rangle_p\neq 0$, i.e. that the kernel of the pairing is exactly the torsion subgroup.

(b) (Split multiplicative case, the cyclotomic analogue of Venerucci's input.) If $E$ has split multiplicative reduction at $p$, let $E^{\dagger}(\mathbb{Q})$ be the extended Mordell–Weil group of Mazur–Tate–Teitelbaum (the Mordell–Weil group with one formal copy of $\mathbb{Z}$ generated by the class $q$ of the Tate period adjoined), equipped with the extended $p$-adic height pairing $\langle\cdot,\cdot\rangle_p^{\dagger}$. Prove that $\langle q,P\rangle_p^{\dagger}\neq 0$, i.e. that the extended pairing is non-degenerate on $E^{\dagger}(\mathbb{Q})\otimes\mathbb{Q}_p$ (under the standard Mazur–Tate–Teitelbaum normalization $\langle q,q\rangle_p^{\dagger}=0$).

A complete resolution establishes the non-vanishing of the $p$-adic regulator in rank one in both settings; the common scientific objective is the direct algebraic input needed for rank-one instances of algebraic $p$-adic Birch–Swinnerton-Dyer type formulae in cyclotomic Iwasawa theory. A certified counterexample in either setting (a rank-one curve with an exactly vanishing pairing value, or a degenerate extended pairing) would refute the non-degeneracy statement of that setting, and hence the problem as posed. The problem is the rank-one instance, over $\mathbb{Q}$, of Schneider's classical non-degeneracy conjecture.

References

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