Density results for automorphic forms on Hilbert modular groups

We give density results for automorphic representations of Hilbert modular groups. In particular, we show that there are infinitely many automorphic representations that have a prescribed discrete series factor at some (but not all) real places.Comment: 35 pages, LaTe

Asymptotics of the Heat Kernel on Rank 1 Locally Symmetric Spaces

We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the short-time asymptotic expansion of the heat kernel is calculated explicitly.Comment: 11 pages, LaTeX fil

Automorphic forms constructed from Whittaker vectors

AbstractLet G be a semi-simple Lie group of split rank 1 and Γ a discrete subgroup of G of cofinite volume. If P is a percuspidal parabolic of G with unipotent radical N and if χ is a non-trivial unitary character of N such that χ(Γ ∩ N) = 1 then a meromorphic family of functions M(v) on gG / G that satisfy all of the conditions in the definition of automorphic form except for the condition of moderate growth is constructed. It is shown that the principal part of M(v) at a pole v0 with Re v0 ⩾ 0 is square integrable and that “essentially” all square integrable automorphic forms with non-zero χ-Fourier coefficient can be constructed using the principal parts of the M-series. For square integrable automorphic forms that are fixed under a maximal compact subgroup the proviso “essentially” can be dropped. The Fourier coefficients of the M-series are computed. A specific term in the χ-Fourier coefficient is shown to determine the structure of the singularities of the M-series. This term is related to Selberg's “Kloosterman-Zeta function.” A functional equation for the M-series is derived. For the case of SL(2, R) the results are made more explicit and a complete family of square integrable automorphic forms is constructed. Also the paper introduces the conjecture that for semi-simple Lie groups of split rank > 1 and irreducible Γ the condition of moderate growth in the definition of automorphic form is redundant. Evidence for this conjecture is given for SO(n, 1) over a number field

Density results for automorphic forms on Hilbert modular groups II

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for \SL_2 over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the ring of integers of $F$. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips and products of prescribed small intervals for all but one of the infinite places of $F$. The main tool in the derivation is a sum formula of Kuznetsov type.Comment: Accepted for publication by the Transactions of the American Mathematical Societ

Operator Product on Locally Symmetric Spaces of Rank One and the Multiplicative Anomaly

The global multiplicative properties of Laplace type operators acting on irreducible rank one symmetric spaces are considered. The explicit form of the multiplicative anomaly is derived and its corresponding value is calculated exactly, for important classes of locally symmetric spaces and different dimensions.Comment: Int. Journal of Modern Physics A, vol. 18 (2003), 2179-218

Forms on Vector Bundles Over Compact Real Hyperbolic Manifolds

We study gauge theories based on abelian $p-$ forms on real compact hyperbolic manifolds. The tensor kernel trace formula and the spectral functions associated with free generalized gauge fields are analyzed.Comment: Int. Journ. Modern Physics A, vol. 18 (2003), 2041-205