Ternary quadratic forms with rational zeros

Counting dihedral and quaternionic extensions

“…We say that m is a square modulo n if and only if the equation x 2 ≡ m mod n is solvable. This is equivalent to the fact that for every odd prime p dividing n, we have ( A recent result due to Friedlander and Iwaniec [4,Theorem 1] states that for any fixed δ > 0 and uniformly for A, B ≥ exp (log AB) δ , we have the estimate…”

“…Results such as the aforementioned asymptotic formula (3) due to Friedlander and Iwaniec are not new in the literature. Let us mention here the work of Guo [5], where the solvability of the ternary equation (4) It is natural to notice that the analytic tools appearing in the proofs of the main results in [2], [4] and [5] are all of the same nature: the use of Jacobi symbols as characters, the Siegel-Walfisz theorem for these characters, and the double oscillations theorem. We review these tools in Lemmas 1 and 2 below.…”

“…The proofs of Propositions 2 and 3 are quite similar and are based on estimates of some auxiliary sums. We extract from [4] two technical results which we use throughout this section. The first one is a variant of Siegel's theorem concerning the distribution of primes in arithmetic progressions.…”

“…The first one is a variant of Siegel's theorem concerning the distribution of primes in arithmetic progressions. This appears as [4,Corollary 2]. Let us define (7) c(r) = π…”

“…where C m stands for the cyclic group of order m, D 4 is the group of the symmetries of the square, and H denotes the quaternionic group.…”