Numerical study of refined conjectures of the BSD type
Juan-Pablo Llerena-Córdova
TL;DR
The paper conducts a numerical study of Mazur-Tate refined BSD-type conjectures for elliptic curves over $Q$ with split multiplicative reduction at a prime $p$, using SageMath to compute theta_E,M and related congruences in $Q_r(R,G_M)$. It provides a multiplicative reformulation that facilitates computation and proves the equivalence between the basic conjecture and MT87's Conjecture 6 in the $M=p$ case, while highlighting the crucial dependence on the chosen ring $R$ and the torsion subgroup $E(Q)_{Tor}$. The numerical results reveal general failures of the unadjusted conjectures (with 886 and 367 counterexamples for two variants) but show that adding the extra assumption that the inverse of the torsion order lies in $R$ restores validity across the dataset, suggesting a refined conjecture that incorporates torsion. In addition, the paper documents two explicit counterexamples illustrating how Sylow-subgroup structure drives failures, and provides public code and data for reproducibility.
Abstract
In 1987, Mazur and Tate stated conjectures which, in some cases, resemble the classical Birch-Swinnerton-Dyer conjecture and its $p$-adic analog. We study experimentally three conjectures stated by Mazur and Tate using SageMath. Our findings indicate discrepancies in some of the original statements of some of the conjectures presented by Mazur and Tate. However, a slight modification on the statement of these conjectures does appear to hold.