Artin L-functions of almost monomial Galois groups

Abstract

<jats:title>Abstract</jats:title><jats:p>If <jats:inline-formula id="j_forum-2019-0288_ineq_9999_w2aab3b7e2669b1b6b1aab1c15b1b1Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2019-0288_eq_0054.png" /><jats:tex-math>{K/\mathbb{Q}}</jats:tex-math></jats:alternatives></jats:inline-formula> is a finite Galois extension with an almost monomial Galois group and if <jats:inline-formula id="j_forum-2019-0288_ineq_9998_w2aab3b7e2669b1b6b1aab1c15b1b3Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2019-0288_eq_0148.png" /><jats:tex-math>{s_{0}\in\mathbb{C}\setminus\{1\}}</jats:tex-math></jats:alternatives></jats:inline-formula> is not a common zero for any two Artin <jats:italic>L</jats:italic>-functions associated to distinct complex irreducible characters of the Galois group, then all Artin <jats:italic>L</jats:italic>-functions of <jats:inline-formula id="j_forum-2019-0288_ineq_9997_w2aab3b7e2669b1b6b1aab1c15b1b9Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2019-0288_eq_0054.png" /><jats:tex-math>{K/\mathbb{Q}}</jats:tex-math></jats:alternatives></jats:inline-formula> are holomorphic at <jats:inline-formula id="j_forum-2019-0288_ineq_9996_w2aab3b7e2669b1b6b1aab1c15b1c11Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2019-0288_eq_0151.png" /><jats:tex-math>{s_{0}}</jats:tex-math></jats:alternatives></jats:inline-formula>. We present examples and basic properties of almost monomial groups.</jats:p>

On supercharacter theoretic generalizations of monomial groups and Artin's conjecture

“…Theorem 1.6. ( [7]) If G is almost monomial and s 0 is not a common zero for any two distinct L-functions f k and f l then all Artin L-functions of K/Q are holomorphic at s 0 .…”

“…, f r = L(s, χ r ) the corresponding Artin L-functions. In [7] it was proved that if G is almost monomial and s 0 is not a common zero for any two distinct L-functions f k and f l then all Artin L-functions of K/Q are holomorphic at s 0 . Also in [7], some basic properties of almost monomial groups were stated.…”

On a generalization of monomial groups

“…We recall the following definition from [17]; see also [6]: Obviously, any (quasi)-monomial group G is also almost monomial; the converse being false. For instance, the already mentioned group SL 2 (F 3 ) is almost monomial, but is not monomial.…”

“…For instance, the already mentioned group SL 2 (F 3 ) is almost monomial, but is not monomial. We recall the main results from [6], regarding the almost monomial groups: Theorem 2.2. We have that:…”

“…A finite group G is called almost monomial, if for any two irreducible characters χ = φ of G, there exist a subgroup H G and a linear character λ of H, such that χ is a constituent of the induced character λ G and φ is not. Some basic properties of almost monomial groups are presented in [6]. The aim of this paper is to continue the study of almost monomial groups.…”