Elliptic units, derived Heegner classes and the p-adic Birch and Swinnerton-Dyer quotient of CM elliptic curves of rank two
Abstract
Let E: y2=x3−Dx have rank two and let p≡1pmod4 be a prime of good reduction. We study the conjecture that the p-adic quotient qp, the leading coefficient of the cyclotomic p-adic L-function divided by the Euler factor and the p-adic regulator, equals the complex quotient q∞=L''(E,1)/(2ΩEReg∞), and that the latter is rational. In rank one the analogous statement is a theorem of Perrin-Riou and Rubin, because a Heegner point computes both sides; in rank two there is no such point. We show that qp is the constant of the quadratic term of the two-variable Katz measure; this is how the conjecture arises, and it computes qp modulo any power of p from critical Hecke L-values. We then show that qp is the coordinate of two global classes: Rubin's rank-two elliptic-unit class, whose position in E(ℚ)⊗ℚp is qp times an explicit vector, and the derived Heegner class over an auxiliary imaginary quadratic field F, for which we prove an exact identity. As consequences, whenever such a field F satisfying our hypotheses exists, qp is a square in ℚp times the product of the Tamagawa numbers of E, so that the unit qp/AE of the p-adic Birch and Swinnerton-Dyer formula is a square when Sha is finite; if moreover the derived Heegner class has non-zero formal logarithm, the conjecture of Agboola and Castella for E/F holds up to a p-adic unit, and with a theorem of Burungale, Castella and Skinner the anticyclotomic main conjecture holds integrally; and our conjecture is equivalent to the analytic form of theirs, with the products of anticyclotomic heights read sesquilinearly, which differs by a sign from its literal formulation. The conjecture is verified modulo p3 for 5520 pairs (E,p) with conductors up to 3.9·1012, including curves with analytic Sha of order 4 and 9.
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BibTeX
@misc{fabregat2026elliptic,
title = {{Elliptic units, derived Heegner classes and the p-adic Birch and Swinnerton-Dyer quotient of CM elliptic curves of rank two}},
author = {Miquel Calonge Fabregat},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09192},
eprint = {2611.09192},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09192},
}
RIS
TY - UNPB
TI - Elliptic units, derived Heegner classes and the p-adic Birch and Swinnerton-Dyer quotient of CM elliptic curves of rank two
AU - Miquel Calonge Fabregat
PY - 2026
DA - 2026/11/06
PB - arXiv
DO - 10.48550/arxiv.2611.09192
UR - https://arxiv.org/abs/2611.09192
AB - Let E: y^2=x^3−Dx have rank two and let p≡1pmod4 be a prime of good reduction. We study the conjecture that the p-adic quotient q_p, the leading coefficient of the cyclotomic p-adic L-function divided by the Euler factor and the p-adic regulator, equals the complex quotient q_∞=L''(E,1)/(2Ω_EReg_∞), and that the latter is rational. In rank one the analogous statement is a theorem of Perrin-Riou and Rubin, because a Heegner point computes both sides; in rank two there is no such point. We show that q_p is the constant of the quadratic term of the two-variable Katz measure; this is how the conjecture arises, and it computes q_p modulo any power of p from critical Hecke L-values. We then show that q_p is the coordinate of two global classes: Rubin's rank-two elliptic-unit class, whose position in E(ℚ)⊗ℚ_p is q_p times an explicit vector, and the derived Heegner class over an auxiliary imaginary quadratic field F, for which we prove an exact identity. As consequences, whenever such a field F satisfying our hypotheses exists, q_p is a square in ℚ_p times the product of the Tamagawa numbers of E, so that the unit q_p/A_E of the p-adic Birch and Swinnerton-Dyer formula is a square when Sha is finite; if moreover the derived Heegner class has non-zero formal logarithm, the conjecture of Agboola and Castella for E/F holds up to a p-adic unit, and with a theorem of Burungale, Castella and Skinner the anticyclotomic main conjecture holds integrally; and our conjecture is equivalent to the analytic form of theirs, with the products of anticyclotomic heights read sesquilinearly, which differs by a sign from its literal formulation. The conjecture is verified modulo p^3 for 5520 pairs (E,p) with conductors up to 3.9·10^12, including curves with analytic Sha of order 4 and 9.
ER -