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As you probably know, you are also asking for a particular Gaussian prime ideal $(\pi )$ with norm $p$. Which is to say that such a choice of square root of $-1$ is the same as giving a homomorphism from the Gaussian integers to the field with $p$ elements, which is the same as giving its kernel. This accounts for the formulae in $a$ and $b$. The theory of complex multiplication gives a little bit more, namely by identifying the Frobenius automorphism which is subject to some congruence conditions depending on choice of curve with complex multiplication by the Gaussian integers. The formulae with binomial coefficients come out of formulae for the Hasse invariant.