On Arithmetic of Curves over Function Fields

魏福村(Fu-Tsun Wei)

指導教授 ： 蔡孟傑 ； 于靖

國立清華大學/理學院/數學系/博士(2010年)

https://doi.org/10.6843/NTHU.2010.00571

摘要

There are two parts in the thesis.
Part One (Chapter 1 to 4) is on arithmetic of definite Shimura curves over function fields and automorphic forms.
We construct certain theta series from definite quaternion algebras over function fields which generate
the space of harmonic automorphic forms.
From the special points on definite Shimura curves and those theta series
we deduce the critical central value of $L$-series of automoprhic forms.
As applications, an analogue of Waldspurger's formula and
critical central values of Hasse-Weil $L$-function of certain elliptic curves over function fields are obtained.

Part Two (Chapter 5) is a published paper: On the independence of Heegner points over function fields.
We prove the independence of Heegner points for different "imaginary" quadratic function fields
and get a subgroup of elliptic curves with arbitrary large rank.

關鍵字

函數體 ； 四元數代數 ； 自守型 ； 橢圓曲線

並列摘要

無資料

並列關鍵字

function field ； quaternion algebra ； automorphic form ； elliptic curves

參考文獻

M. Bertolini and H. Darmon, Heegner points on Mumford-Tate curves, Inv. Math., 126 (1996) 413-456.

D. Bump, Automorphic Forms and representations, Cambridge studies in advanced mathematics 55, (1996).

F. Breuer, Higher Heegner points on elliptic curves over function fields. Journal of Number Theory, 104 (2004), 315-326.

D. A. Cardon and M. R. Murty, Exponents of Class Groups of Quadratic Function Fields over Finite Fields. Canad. Math. Bull. Vol. 44 (4), 2001 398-407.

W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201, (1973) 301-314.