On Arithmetic Over Function Fields

李亭芳(Ting-Fang Lee)

指導教授 ： 于靖 ； Solomon Friedberg ； 潘戍衍

國立清華大學/理學院/數學系/博士(2013年)

https://doi.org/10.6843/NTHU.2013.00475

摘要

There are two parts in this dissertation.
The first part (Chapter 1) is trace formula of a Brandt matrix. Given a division quaternion algebra over a global function field which is definite with respect a fixed chosen infinity place. A family of Brandt matrices is then introduced to encode information from the arithmetic of the division quaternion algebra. Adapting Eichler’s method from the rational field case, we build up a fine formula expressing the trace of these Brandt matrices in terms of class numbers of specific orders inside imaginary quadratic extensions of the global function field embeddable into the division quaternion. The proof is base on a detailed study of the so-called optimal embeddings of quadratic orders into the division quaternion.
The second part (Chapter 2 and 3) is multiple Dirichlet series over a global function field. Fisher and Friedberg [c.f. FF and FF2] constructed and studied a family of multiple (2 and 3 variables) Dirichlet series over general function fields. They proved that these multiple Dirichlet series satisfy finite, non-abelian groups of functional equations and are rational functions with specific denominator. We then are interested in explicitly finding these multiple Dirichlet series over a curve. In the 2-variable case, we work on in particular the elliptic curve C : y^2 = x^3 + 2x over F_5. For the 3-variable case, we compute the curve P^1 over F_q with q odd which the function field is rational function field over F_q.

關鍵字

function field ； automorphic form ； multiple Dirichlet series ； Brandt matrix

參考文獻

1. G. Chinta and P. Gunnells, Constructing Weyl Group Multiple Dirichlet Series, Journal of The American Mathematical Society, Volume 23, Number 1, January 2010, Pages 189-215

3. M. Eichler, Lectures on Modular Correspondences, Tata Institute of Fundamental Research, Bombay 1957

4. B. Fisher and S. Friedberg, Double Dirichlet Series Over Function Fields, Compositio Math. 140 (2004) 613-630

5. B. Fisher and S. Friedberg, Sums of Twisted GL(2) $L$-functions over function fields, Duke Mathematical Journal Vol. 117, No. 3, 2003

6. H. Friedlander, Ph.D Dissertation, University of Massachusetts, 2013