Numerical Verification of the Stark-Chinburg Conjecture for Some Icosahedral Representations

Abstract

In this paper, we give fourteen examples of icosahedral representations for which we have numerically verified the Stark-Chinburg conjecture.

On refined Stark conjectures in the non-abelian case

“…Thus, modulo these differences (and the occurrence of the auxiliary set of places T ), the factor c S0 which occurs in Proposition 3.3 predicts that the units that occur in Stark's question and in Chinburg's conjecture should be N -th powers of real units in K × where N is the largest integer such that c S0 ⊆ N · O. We remark that the possibility of such 'extra divisibility' in the Stark-Chinburg Conjecture has already been observed numerically in both [12] and [14] (but see the following remark). …”

“…For simplicity, we also assume that O is generated as a Z-module by the set {χ(g) : g ∈ G} (as is the case, for example, for all of the examples considered by Fogel in [12] and by Jehanne, Roblot and Sands in [14] E and take S equal to the set S 0 consisting of v 1 and all rational primes which ramify in K/Q, then the question of the existence of real units in K which satisfy the conditions listed in Proposition 3.3 was first considered by Stark in [20]. Also, under our stated assumption about O, the subsequent conjecture [8, Conj.…”

“…III, §4]). Computer calculations of, amongst others, Chinburg [8], Fogel [12] and Jehanne, Roblot and Sands [14] have since given credence to this conjecture (which is referred to in loc. cit.…”

“…This conjecture is in effect a natural non-abelian [20] and also (in any case in which the Euler factors of all primes which ramify in the given Galois extension are trivial) the Stark-Chinburg Conjecture itself. At the end of §3 we shall also make an explicit correction to the paper of Jehanne, Roblot and Sands [14]. In §4 we first prove an important reduction of Conjecture 2.1 and then use this to deduce the validity of Conjecture 2.1 in the function field case from a result of Bae in [1] and to show in the number field case that Conjecture 2.1 is a consequence of the 'Strong-Stark Conjecture' that is formulated by Chinburg in [7].…”