Quaternionic Artin representations of ℚ

Abstract

Isomorphism classes of dihedral Artin representations of ℚ can be counted asymptotically using Siegel's asymptotic averages of class numbers of binary quadratic forms. Here we consider the analogous problem for quaternionic representations. While an asymptotic formula is out of our reach in this case, we show that the asymptotic behaviour in the two cases is quite different.

Quaternionic Artin representations and nontraditional arithmetic statistics

“…Proof. If d is a product of primes congruent to 1 mod 4 the assertion follows from Proposition 11 of [11]. The argument in the case d ≡ 0 mod 8 is similar, but for the sake of completeness we provide the details.…”

“…x/ log x for every ε > 1/4 √ e (see [11]). A conjecture of Ambrose [1], whose work in the direction of his conjecture is the main ingredient in the lower bound in (2), would imply that the lower bound holds for all ε > 0, but our focus here is on the upper bound and the disparity in growth rates: ϑ di (x)…”