Quaternionic Artin representations and nontraditional arithmetic statistics

Abstract

We classify and then attempt to count the real quadratic fields (ordered by the size of the totally positive fundamental unit, as in Sarnak [14], [15]) from which quaternionic Artin representations of minimal conductor can be induced. Some of our results can be interpreted as criteria for a real quadratic field to be contained in a Galois extension of Q with controlled ramification and Galois group isomorphic to a generalized quaternion group. Traditionally, number fields of a given degree over Q are ordered by the size of their discriminant, and Artin representations of Q of a given dimension are ordered by the size of their conductor. But in [14] and [15], Sarnak obtained asymptotic averages of ring class numbers of real quadratic fields by enumerating the corresponding orders according to the size of their fundamental totally positive unit. Sarnak’s method was subsequently used by Raulf [7], [8] to average the class number over maximal orders as well as over orders satisfying given congruence conditions. Here Sarnak’s ordering will be used to count certain Artin representations induced from real quadratic fields. In describing the results of Sarnak and Raulf, we have departed from the authors’ own formulation, for they use the language of binary quadratic forms rather than the equivalent language – more suited to Artin representations – of ring class groups. Our notation will also depart from theirs in one important respect: While Sarnak and Raulf use h and to denote the narrow class number and fundamental totally positive unit of an order, we shall instead use h and , reserving h and for the usual class number and fundamental unit. Thus if has norm −1 then + = . (Here and + are defined by the standard condition , + > 1, a condition which is meaningful because real quadratic fields will always be taken to be subfields of R.) In principle, one could use rather than + as the basis of the ordering, but in this note we shall adhere to Sarnak’s original ordering by . Our group-theoretic conventions will be as follows: A representation of a group G is a homomorphism (continuous if G is endowed with a topology) ρ : G→ GL(V ), where V is a finite-dimensional vector space over C. An irreducible ρ is dihedral if its image is isomorphic to the dihedral group D2m = 〈a, b|a = 1, a = b = 1, bab−1 = a−1〉 of order 2m for some m > 3, and quaternionic if its image is isomorphic to the generalized quaternion group Q4m = 〈a, b|a = 1, a = b, bab−1 = a−1〉 2000 Mathematics Subject Classification. Primary 11R32; Secondary 11R20.