A variant of Heilbronn characters

Abstract

In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, and the non-existence of simple zeros or poles in such quotients.

Virtual characters on L-functions

“…Similarly as before, we have the following result, Theorem 5.2.2 (Arithmetic Heilbronn-Stark Lemma, [38]). For every subgroup H of G, one has…”

“…Corollary 5.1.2 (Weak arithmetic Stark Lemma, [38]). If n(G, Reg G ) 1, then n(G, χ) 0 for all irreducible characters χ of G.…”

“…Now we will see how they are matched up with Heilbronn characters in terms of Arithmetic properties. The first one of them is the following Theorem 5.1.1 (Weak arithmetic Artin-Takagi decomposition, [38]).…”

“…Theorem 5.1.2 (Weak arithmetic Heilbronn-Stark Lemma, [38]). For every cyclic subgroup H of G, we have…”

“…In this context we first have the following result. Theorem 5.1.4 (Wong, [38]). Let ρ be an arbitrary character of G. Suppose that for every cyclic subgroup H of G and irreducible character φ of H, we have n(H, ρ| H ⊗ φ) 0, then…”