Shimura varieties and the Selberg trace formula

Geometric Zeta Functions, L-Theory, and Compact Shimura Manifolds

INTRODUCTION 4 Introduction Zeta functions encoding geometric information such as zeta functions of algebraic varieties over finite fields or zeta functions of finite graphs will loosely be called geometric zeta functions in the sequel. Sometimes the geometric situation gives one tools at hand to prove analytical continuation, functional equation and an adapted form of the Riemann hypothesis. T...

Rankin-selberg L-functions and Cycles on Unitary Shimura Varieties

Gross–Zagier formula and arithmetic fundamental lemma

The recent work [YZZ] completes a general Gross–Zagier formula for Shimura curves. Meanwhile an arithmetic version of Gan–Gross–Prasad conjecture proposes a vast generalization to higher dimensional Shimura varieties. We will discuss the arithmetic fundamental lemma arising from the author’s approach using relative trace formulae to this conjecture.

The basic stratum of some simple Shimura varieties

Under simplifying hypotheses we prove a relation between the l-adic cohomology of the basic stratum of a Shimura variety of PEL-type modulo a prime of good reduction of the reflex field and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields. We obtain our results using the trace formula and tr...

Points on Some Shimura Varieties over Finite Fields

The Hasse-Weil zeta functions of varieties over number fields are conjecturally products (and quotients) of automorphic L-functions. For a Shimura variety S associated to a connected reductive group Gover Q one can hope to be more specific about which automorphic L-functions appear in the zeta function. In fact Eichler, Shimura, Kuga, Sato, and Ihara, who studied GL2 and its inner forms, found ...