The method for computing degrees of freedom that lme() uses is laid out in Pinheiro & Bates 2000, p. 91, but the most comprehensive discussion of the related controversies and pitfalls that I know of (including a case where lme() calculates the degrees of freedom incorrectly!) is at Ben Bolker's GLMM FAQ:
https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#why-doesnt-lme4-display-denominator-degrees-of-freedomp-values-what-other-options-do-i-have
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