일시: 10월24일(금) 11시-12시
장소: 27동325호

연사: 조재현(유니스트)
제목: The least prime with a given factorization type
초록: Let G be a finite group. Let K/k be a Galois extension of number fields, and let
C ⊆ Gal(K/k) ≃ G be a conjugacy invariant subset. It is well known that there exists a
prime ideal p of k with Frobenius element lying in C and norm satisfying Np ≪ |Disc(K)| α for
some constant α = α(G, C). There is a rich literature establishing unconditional admissible
values for α and, aside from a narrow subset of cases, they follow the traditional approach
with the zeros of L-functions. We generalize an alternative approach via sign changes to
substantially improve 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 exists
absolute constant c 1 , c 2 > 0 such that α(S n , C) ≤ c 1 e −c 2 n for any conjugacy class C ⊆ S n .
Our approach reduces the core problem to a question in character theory.