G2 and some exceptional Witt vector identities

We find some new one-parameter families of exponential sums in every odd characteristic whose geometric and arithmetic monodromy groups are G2. 1. The exceptional identities Fix a prime p, and consider the p-Witt vectors of length 2 as a ring scheme over Z. The addtion law is given by (x, a) + (y, b) := (x+ y, a+ b+ (x + y − (x+ y))/p). The multiplication law is given by (x, a)(y, b) := (xy, xb+ ya+ pab). For an odd prime p, we have (x, 0) + (y, 0) + (−x− y, 0) = (0, (x + y − (x+ y))/p). Let us define, for odd p, the integer polynomial Fp(x, y) := (x p + y − (x+ y))/p ∈ Z[x, y]. For p = 2, we have (x, 0) + (y, 0) + (−x− y, 0) = (0, x + xy + y), and we define F2(x, y) := x 2 + xy + y ∈ Z[x, y]. Thus F3 = −xy(x+ y). The exceptional identities we have in mind are F5 = F3F2, F7 = F3(F2) . 1