Artin L-functions of small conductor

Abstract

We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and obtain much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.

Twist‐minimal trace formulas and the Selberg eigenvalue conjecture

“…This partially explains why our result for is within a factor of 4 of that for , despite the conductors being much larger. By the Langlands–Tunnell theorem, the Artin conjecture is true for tetrahedral and octahedral representations, so the conclusion of Theorem 1.2 holds unconditionally for those types. In the icosahedral case, by [3] it is enough to assume the Artin conjecture for all representations in a given Galois conjugacy class; that is, if there is a twist‐minimal, even icosahedral representation of conductor that does not appear in Table 1, then Artin's conjecture is false for at least one of its Galois conjugates.The entries of Table 1 were computed by Jones and Roberts [20] by a thorough search of number fields with prescribed ramification behavior, and we verified the completeness of the list via the trace formula. In principle, the number field search by Jones and Roberts [19] is exhaustive, so it should be possible to prove Theorem 1.2 unconditionally with a further computation, but that has not yet been carried out to our knowledge.For comparison, we note that Buzzard and Lauder [7] have characterized the odd 2‐dimensional representations of conductor by computing bases of the associated spaces of weight 1 holomorphic modular forms. Theorem 1.1 for improves on the result from [6] by extending to nonsquarefree and increasing the upper bound from 854 to 880.…”

“…The entries of Table 1 were computed by Jones and Roberts [20] by a thorough search of number fields with prescribed ramification behavior, and we verified the completeness of the list via the trace formula. In principle, the number field search by Jones and Roberts [19] is exhaustive, so it should be possible to prove Theorem 1.2 unconditionally with a further computation, but that has not yet been carried out to our knowledge.…”

Varieties via their L-functions

“…This same general method has been applied in many contexts. For example, simply taking (λ 1 , λ 2 ) = (0, 0) one is saying that certain L-functions coming from number fields can only exist if conductors are large enough [13]. The method, and the more powerful exclusionary techniques of [6], can be extended to establish L-point-free regions of other landscapes.…”