Lower bounds for the Artin conductor

Abstract

Abstract. In this paper we improve on Odlyzko's lower bounds for the Artin conductor.

Artin L-functions of small conductor

“…The derivation of (5.2) uses the simple formula in The character φ S is used prominently in [20,24]. When χ = n − 2, the lower bound m(G, c, χ, φ S , α) coincides with that of Lemma 3.2.…”

“…However, as we explain in Sect. 10.3, we also expect that the asymptotic lower bound of √ ≈ 6.7 [24] is not optimal, and that this bound is more likely to be at least ≈ 44.8. Our second result unconditionally identifies initial segments:…”

“…It appears in [20,24] In this section we establish Theorem 4.2, which is a family of lower bounds on the root conductor δ X of a given Artin character, dependent on the choice of an auxiliary character φ.…”

“…There are 84 such groups, and we consider all associated Galois types (G, χ) with χ a rationally irreducible faithful character. Our first result gives conditional lower bounds: The bounds in Tables 3, 4, 5, 6 and 7 are graphed with the best previously known bounds from [24] in Fig. 3.…”

“…The vertical axis corresponds to lower bounds on root conductors. The piecewise-linear curve connects bounds from [24], and there is one dot at height d(G, χ) for each (G, χ) from Tables 3, 4, 5, 6 and 7 with χ 1 (e) ≤ 20. Here we are freely passing back and forth between a rational character χ and an irreducible constituent χ 1 via δ 1 (G, χ) = δ 1 (G, χ 1 ), which is a direct consequence of (2.4).…”