ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11M41, 11M45 Diaconu, Călin Adrian, Department of Mathematics, Brown University, Providence, RI 02912, USA Some new techniques in the study of automorphic and Hecke Lfunctions Our goal is the meromorphic continuation to C^{2} of

Z(s, w) = ∑_{\begin{matrix}n ∈ Z[i], \\ n ≡ 1(1 + i)^{3} \end{matrix}} L_{λ}(s, χ_{n})L_{λ}(s, χ̄_{n})N(n)^{−w},

where χ_{n}(m) = (\frac{n}{m})_{4} , for (m, (1 + i)n) = 1 and m ≡ 1(1 + i)^{3}, is the character associated to the cyclic extension Q(i, n^{1/4})/Q(i); λ means that the Euler factor at 1 + i has been removed. Our approach is based on a functional equation relating Z(s, w) to

\begin{aligned}Ẑ(s, w) = & ζ_{Q(i),λ}(2s + 2w − 1)ζ_{Q(i),λ}(4s − 1)^{2} \\ & × ∑_{\begin{matrix}n = λ^{t}n_{0}, \\ n_{0} ≡ 1(λ^{3}) \end{matrix}} ∑_{m,m′≡1(λ^{3})} \frac{G(n, m)\overline{G(n, m′)}}{N(n)^{w}N(mm′)^{s}}. \end{aligned}

Using the theory of metaplectic Eisenstein series we prove the following Theorem. Ẑ(s, w) has a meromorphic continuation to the tube domain in C^{2} associated to the convex region bounded by w = 1, w = 2−s, w = 2−2s. It has double poles at s = \frac{1}{4} and s = \frac{3}{4}. Furthermore, up to a finite scattering matrix, it satisfies the functional equation

Ẑ(s, w) = Ẑ(1 − s, 2s + w − 1).

As a consequence, the meromorphic continuation of Z(s, w) is proved using the convexity principle in tube domains.