Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds

Geometric zeta functions with non-unitary twists

It is shown that the zeta functions of Ruelle and Selberg admit analytic continuation to meromorphic functions on the plane for every compact locally-symmetric space and every non-unitary twist.

Geometric zeta-functions on p-adic groups

Geometric zeta functions of Ihara and Hashimoto are generalized to higher rank. The $p$-adic version of the Patterson conjecture is proven.

Geometric zeta-functions of locally symmetric spaces

The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes the Patterson conjecture. We also extend the range of zeta functions in considering higher dimensional flats.

典范高度、周期与赫尔维茨 zeta 函数

Canonical heights, periods and the Hurwitz zeta function

本文为相对丰富的对数典范线丛引入典范高度，证明其与 K-稳定性及 Kähler-Einstein 度量高度的关系，并用周期极限公式显式计算算术对数曲面等情形，应用于志村曲线整模型和 Arakelov 度量界。

Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean and Mueller to groups which are not necessarily cofinite.

Holomorphic torsion and geometric zeta functions for certain Hermitian locally symmetric manifolds

We give a dynamical description, in terms of a Weil-type zeta function, to the holomorphic torsion with coefficients for certain compact Hermitian locally symmetric manifolds, whose connected group G of isometries of the universal cover has only one conjugacy class of cuspidal maximal parabolic subgroup and satisfies a technical Ansatz relative to the given coefficients. A distinguishing feature of our zeta function is that its construction involves in an essential way the geometry of a standard compactification of the universal cover. The two senior authors are indebted to their junior collea…