On Jacobi sum Hecke characters ramified only at 2

Abstract

Let K be the cyclotomic field of the mth roots of unity in some fixed algebraic closure of Q. Weil has shown in [Jacobi sums as Großencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487–495] that Jacobi sums induce Hecke characters on K modulo m 2, or equivalently homomorphisms of the idèle class group of K into K ∗ . Given such a Hecke character ϱ, we then obtain for every place p of K a continuous homomorphism ϱ p (the local component of ϱ at p), from ( K p ) ∗ into K ∗ . When p does not divide m, ϱ p is completely determined. When p divides m the determination of ϱ p is much more difficult. Coleman and McCallum in [Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41–101] have given formulas and computed the conductors of the local components of ϱ at p for p∥ m and m odd. Hasse, in [Zetafunktion und L-Funktionen zu einem arithmetischen Funktionenkorper vom Fermatschen Typus, Abh. Deut. Akad. Wiss. Berlin Kl. Math. Nat. 1954 4 (1955)] has computed the local component and the conductor when m is prime, while in [C. Jensen, Uber die Führer einer Klasse Heckescher Großencharaktere, Math. Scand. 8 (1960), 81–96; D. Rohrlich, Jacobi sums and explicit reciprocity laws, Compositio Math. 60 (1986), 97–110; C.-G. Schmidt, Uber die Führer von Gauss'schen Summen als Großencharaktere, J. Number Theory 12, No. 3 (1980), 283–310] the authors have given estimates and in some cases have determined the conductor of the local component. Here we deal with the case where m = 2 n , and we give an explicit formula for the local component in terms of Hilbert symbols, similar to the one in [R. Coleman and W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41–101]. In [D. Prapavessi, On the conductor of 2-adic Hilbert norm residue symbols, to appear in J. Algebra] we use this result to compute the conductor of ϱ when m = 2 n .