Some observations on the arithmetic of Eisenstein series for the modular group SL(2, \Bbb Z)

Abstract.

For even integers $k\geqq4$, let $\varphi_k(X)$ be the separable rational polynomial that encodes the j-invariants of non-elliptic zeroes of the Eisenstein series E _k for the modular group SL$(2,{Bbb Z})$. We prove Kummer-type congruence properties for the $\varphi_k$ and, based on extensive calculations, make observations about the Galois group, the discriminant, and the distribution of zeroes of $\varphi_k(X)$.