In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper Arithmetic intersection theory, and this note of Müller-Stach). The height pairing is supposed to generalize the Neron-Tate height pairing. Is there a precise statement in the literature where Beilinson's height pairing for an elliptic curve over $\mathbb{Q}$ is compared to the Néron–Tate pairing?

## 1 Answer

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Beilinson's original paper on Height pairings does indicate how his height pairing generalises that of Néron–Tate: see page 10 (also, Müller-Stach and Kunneman's works expand on Remark 4.0.8 on page 14). Also, Scholl's paper provides a very nice interpretation (via Ext in mixed motives).