Kavli Affiliate: Hsiao-Mei (Sherry) Cho
| First 5 Authors: [#item_custom_name[1, [#item_custom_name[2, [#item_custom_name[3, [#item_custom_name[4, [#item_custom_name[5| Summary:Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C subseteq mathrmGal(K/k) simeq G$ be a conjugacy invariant subset. It is well known that there exists an unramified prime ideal $mathfrakp$ of $k$ with Frobenius element lying in $C$ and norm satisfying $mathrmNmathfrakp ll |mathrmDisc(K)|^α$ for some constant $α= α(G,C)$. There is a rich literature establishing unconditional admissible values for $α$, with most approaches proceeding by studying the zeros of $L$-functions. We give an alternative approach, not relying on zeros, that often substantially improves this exponent $α$ for any fixed finite group $G$, provided $C$ is a union of rational equivalence classes. As a particularly striking example, we prove that there exist absolute constants $c_1,c_2 > 0$ such that for any $ngeq 2$ and any conjugacy class $C subset S_n$, one may take $α(S_n,C) = c_1 exp(-c_2n)$. Our approach reduces the core problem to a question in character theory.| Search Query: arXiv Query: search_query=au:Cho OR all:Hsiao-Mei&id_list=&start=0&max_results=3Read More