On holomorphy and special values of Artin L-functions [HBNI Th256]

Abstract

In this thesis, we investigate holomorphy and special values of Artin L-functions. Let K/F be a Galois extension of number fields with Galois group G. Artin conjectured that Artin L-functions associated to non-trivial irreducible characters of G are entire. It is known to be true for degree one irreducible characters. In an elegant work, Stark established the following; given a Galois extension K/F of number fields with Galois group G and s0 2 C \ {1}, if ords=s0 ⇣K (s)  1, then the Artin L-functions L(s, , K/F) are holomorphic at s0 for all characters of G. He established a relationship between the order of ⇣K and the orders of Artin L-functions at such a point s0 to prove the result. His proof demonstrated that (1) X 2Irr(G) n(G, , s0 )2  (ords=s0 ⇣K (s))2 , where Irr(G) denotes the set of irreducible characters of G and n(G, , s0 ) denotes the order of L(s, , K/F) at s0 . This holomorphy result was refined later by Foote and Kumar Murty [10] for solvable Galois groups. More precisely, let K/F be a Galois extension of number fields of degree n with a solvable Galois group G. Let n = p↵1 1 p↵2 2 · · · p↵k k be the prime factorisation of n with p1 < p2 < · · · < pk and ↵i > 0 for 1  i  k. Foote and Kumar Murty proved that if ords=s0 ⇣K (s)  p2 2, then L(s, , K/F) are holomorphic at s0 for all characters X of G ords=s0 ⇣K (s)  p2 2, then L(s, , K/F) are holomorphic at s0 for all will examine the behavior of ⇣⇣K+(s) at positive odd integers, where K denotes (s) K a totally imaginary field and K denotes its maximal totally real sub-extension. + For a totally real number field K, the Siegel-Klingen theorem asserts that for any integer k 0, the value of ⇣K (1 2k) is rational. Additionally, functional equation connects the value of ⇣K (1 2k) to the value of ⇣K (2k). Furthermore, Siegel and Klingen demonstrated that for an integer k>_1 and a totally real number field K,the value of ⇣K (2k) is a non zero rational multiple of ⇡ 2k[K:Q] p |dK |. In the final part of our thesis we establish some new results in this direction. We end with a summary of the main results of the thesis.