Nonvanishing of the cyclotomic p-adic height determinant

Prove that, for a CM elliptic curve $E/\mathbb{Q}$, two independent infinite-order rational points $P_1,P_2$, and every split prime $p=\mathfrak p\bar{\mathfrak p}$ of good reduction, the determinant of their cyclotomic p-adic height-pairing matrix is nonzero.

Background

The paper explains that known transcendence results establish nonvanishing of a p-adic height for a single infinite-order point, but do not establish nonvanishing of the determinant required for a rank-two height pairing.

The stated problem asks for nondegeneracy at every split prime of good reduction. The authors emphasize that their finite computations do not resolve this universal assertion.

References

In rank at least two no theorem gives the determinant nonzero at a single prime of a single curve \S 1.1.

— Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves  (2609.08431 - Banwait, 8 Sep 2026) in Problem 1 (label prob:determinant), Section “Nonvanishing of the rank-two p-adic regulator”

For the cyclotomic logarithm, non-degeneracy is conjectured but remains unproved in general; the problem is comparable in difficulty to the Leopoldt conjecture.

— A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map  (2609.35626 - Adachi et al., 28 Sep 2026) in Section 1, Introduction

Even non-triviality, meaning that the pairing is not identically zero, remains open in general, including for elliptic curves over $Q$ with good ordinary reduction at $p$.

— A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map  (2609.35626 - Adachi et al., 28 Sep 2026) in Section 1, Introduction