ON THE p-ADIC BINOMIAL SERIES AND A FORMAL ANALOGUE OF HILBERT'S THEOREM 90 – ADDENDUM

The proof of Theorem 3.2 in [1] (P. Tzermias, On the p-adic binomial series and a formal analogue of Hilbert's Theorem 90, Glasgow Math. J.47 (2005), 319–326) contains two opaque claims. The necessary clarifications are provided here.

EQUIDISTRIBUTION OF ALGEBRAIC NUMBERS OF NORM ONE IN QUADRATIC NUMBER FIELDS

Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted [Formula: see text]). When K is an imaginary quadratic extension, [Formula: see text] is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in [Formula: see text] can be written as [Formula: see text] for some [Formula: see text], which yields another ordering of [Formula: see text] given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that [Formula: see text] is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that [Formula: see text] is equidistributed with respect to norm, under the map β ↦ log |β|( mod log |ϵ2|) where ϵ is a fundamental unit of [Formula: see text].

Residues of intertwining operators for SO*6 as character identities

We show that the residue at s=0 of the standard intertwining operator attached to a supercuspidal representation π⊗χ of the Levi subgroup GL2(F)×E1 of the quasisplit group SO*6(F) defined by a quadratic extension E/F of p-adic fields is proportional to the pairing of the characters of these representations considered on the graph of the norm map of Kottwitz–Shelstad. Here π is self-dual, and the norm is simply that of Hilbert’s theorem 90. The pairing can be carried over to a pairing between the character on E1 and the character on E× defining the representation of GL2(F) when the central character of the representation is quadratic, but non-trivial, through the character identities of Labesse–Langlands. If the quadratic extension defining the representation on GL2(F) is different from E the residue is then zero. On the other hand when the central character is trivial the residue is never zero. The results agree completely with the theory of twisted endoscopy and L-functions and determines fully the reducibility of corresponding induced representations for all s.