QuarterSumLadder: the quarter-sum ladder and the weight-two congruences
For a prime p = 1 (mod 8) and a quartic character chi modulo p (even, with w = chi(2) = +-1), the development proves that the coordinates of B_(2,chi) = c0 + i c1 are integers with c0 = 4(E_1 + E_-1), c1 = 4(E_i + E_-i) and c0 + c1 = p - 1 modulo 8, and that 8 divides both exactly when w = 1, i.e. when 2 is a fourth power. A Gauss-Lehmer product argument gives the quartic Gauss lemma w = (-1)^(R_1 + R_3) for the quarter counts R_r, the parity of the four quarter counts when w = 1, and the parity of R_2 through the order of 2. For the quarter sum Omega = x + iy (2N(Omega) = h^-(K) for the…
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