* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

Modular elliptic curves and fermat's last theorem

The conjecture of Birch and Swinnerton-Dyer is one of the principal open problems of number theory today. Since it involves exact formulae rather than asymptotic questions, it has been tested numerically more extensively than any other conjecture in the history of number theory, and the numerical results obtained have always been in perfect accord with every aspect of the conjecture. The present article is aimed at the non-expert, and gives a brief account of the history of the conjecture, its precise formulation, and the partial results obtained so far in support of it.

On the conjecture of Birch and Swinnerton-Dyer

Introduction Statement of the main theorems Local character identities and the intertwining relation Trace formulas and their stabilization The Standard model Study of critical cases Local classification Nontempered representations Global classification Addendum Bibliography

Endoscopic Classification of Representations of Quasi-split Unitary Groups

The Gross-Zagier Formula on Shimura Curves:

By the relative trace formula approach of Jacquet�Rallis, we prove the global Gan�Gross�Prasad conjecture for unitary groups under some local restrictions for the automorphic representations

https://www.math.columbia.edu/~wzhang/math/online/GGP1%20final.pdf

Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups

For elliptic curves over Q, we prove the p-indivisibility of derived Heegner points for certain prime numbers p, as conjectured by Kolyvagin in 1991. Applications include the rened Birch{SwinertonDyer conjecture in the analytic rank one case, and a converse to the theorem of Gross{Zagier and Kolyvagin. A slightly dierent version of the converse is also proved earlier by Skinner.

https://www.math.columbia.edu/~wzhang/math/online/Kconj.pdf

Selmer groups and the indivisibility of Heegner points

We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of L-packets and some part of the local Gan-Gross-Prasad conjecture.

/pdf/automorphic-period-and-the-central-value-of-rankin-selberg-l-1ybakiwiye.pdf

Automorphic period and the central value of Rankin-Selberg L-function

We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura varieties. As a crucial ingredient, we formulate a conjectural arithmetic fundamental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.

https://www.math.columbia.edu/~wzhang/math/online/AFL.pdf

Wei Zhang

Papers

The Gross-Zagier Formula on Shimura Curves:

Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups

Selmer groups and the indivisibility of Heegner points

Automorphic period and the central value of Rankin-Selberg L-function

On arithmetic fundamental lemmas