\(p\)-adic gamma function and the trace of Frobenius of elliptic curves
In [Pac. J. Math. 261, No. 1, 219--236 (2013; Zbl 1296.11079)], \textit{D. McCarthy} defined a function \(_n G_n[\cdots]\) using the Teichmüller character of finite fields and quotients of the \(p\)-adic gamma function and could express the trace of Frobenius of elliptic curves over a prime field in terms of special values of \(_2 G_2[\cdots]\). In the present article, the authors find two new formulas for the trace of elliptic curves over \(\mathbb{F}_q\), for an arbitrary \(q\), in terms of \(_2 G_2[\cdots]\) with different parameters. Hence, this can be seen as transformation formulas for McCarthy's result (which actually also holds more generally over \(\mathbb{F}_q\)).
scientific article; zbMATH DE number 6334897
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