Gauss Sums and Binomial Coefficients

AbstractSuppose p=tn+r is a prime and splits as p1p2 in Q(−t). Let q=pf where f is the order of r modulo t, χ=ω(q−1)/t where ω is the Teichmüller character on Fq, and g(χ) is the Gauss sum. For suitable τi∈Gal(Q(ζt, ζp)/Q) (i=1, …, g), we show that ∏gi=1τi(g(χ))=pα((a+b−t)/2) such that 4ph=a2+tb2 for some integers a and b where h is the class number of Q(−t). We explicitly compute amod(t/gcd(8, t)) and amodp, in particular, a is congruent to a product of binomial coefficients modulo p