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

Let K/k be a Galois extension of number fields, with Galois group G = Gal(K/k), and suppose ρ : G → GLn(C) is a nontrivial irreducible representation of G. Stark’s conjectures [Tate 84] aim to unravel the arithmetic information encoded in the leading coefficient of the Taylor series for the Artin L-function L(s, ρ) of ρ at s = 0. When G is abelian and one modifies the Artin L-function by removing the factors in the Euler product at primes in a finite set S which contains all of the infinite primes, Stark formulated an especially precise conjecture for the case of a first-order zero at 0 [Stark 80]. It states that the exact value of this coefficient may be obtained from an “L-function evaluator” element in K which is an S-unit in the typical case. Rubin [Rubin 96], Popescu [Popescu 03], Burns [Burns 01], Sands [Sands 87], and others have formulated similarly precise conjectures for abelian Lfunctions with any order of zero at s = 0. In the general nonabelian case with L(s, ρ) possessing a zero at s = 0 of order r = r(ρ), the conjecture states that the L-function coefficient equals an algebraic factor multiplied by the determinant of a regulator matrix defined in terms of a set of r special units and the representation ρ. But this algebraic factor is not fully specified and in particular may be multiplied by any nonzero rational factor without affecting the truth of the conjecture. Hence the conjecture in this generality is considered to be a conjecture “over Q,” as opposed to the more precise conjectures “over Z” mentioned above in the abelian case.