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Multiplicity estimates for certain pairs of functions involving Drinfeld modular forms. (English) Zbl 1231.11083
This paper deals with multiplicity estimates for certain pairs of functions involving Drinfeld modular forms. First, we establish such an estimate for the two functions $$t$$ and $$\widehat{j}(t)$$, where $$\widehat{j}(t)$$ is the expansion at infinity of the Drinfeld modular invariant. To this end, we introduce a new method, which is based on an induction on the degree of the polynomial occurring in the estimate, and on Newton’s method to approximate the roots of a polynomial in non-Archimedean, complete fields. We then apply the method to get multiplicity estimates for some various pairs of Drinfeld modular functions.
##### MSC:
 11J93 Transcendence theory of Drinfel’d and $$t$$-modules 11F52 Modular forms associated to Drinfel’d modules
##### Keywords:
Drinfeld modular forms; multiplicity estimates
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