Abstract:
This paper investigates the canonical pairing associated with a one-dimensional formal group law $F$ over the ring of integers of a finite extension of $\mathbf Q_p$ and an isogeny $f\colon F\to F$, just as the Hilbert symbol is associated with the multiplicative law and the isogeny "raising to the $p$th power". Formulas are obtained which generalize the formulas of Artin–Hasse, Iwasawa, and Wiles. The formulas describe the values of the symbol in terms of $p$-adic differentiation, the logarithm of the formal group law, the norm, and the trace.
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