黎曼假設是一個至今未解的重要問題，對我們理解質數的性質具有深遠影響。本文旨在介紹有限域上曲線的黎曼假設，它是黎曼假設在函數域上的一個變體，已被證明，為我們深入理解一般黎曼假設提供了重要見解。我們將探討此假設的定義和重要性，並提供Enrico Bombieri提出的相關證明，以闡釋其核心思想和數學意義。

The Riemann hypothesis remains an important and unresolved problem, profoundly shaping our understanding of prime numbers. This paper aims to introduce the Riemann hypothesis for curves over finite fields, a variant of the general Riemann hypothesis that has been proven, providing deeper insights into the broader conjecture. We explore the definition and significance of this hypothesis, alongside presenting a proof proposed by Enrico Bombieri to elucidate its core concepts and mathematical implications.

致謝 i
摘要 ii
Abstract iii
1 Introduction 1
1.1 notation 1
2 Nonsingular complete curve 2
3 Zeta functions and Riemann hypothesis 8
4 Frobenius endomorphism & Frobenius element 15
5 The proof of Riemann hypothesis 18
References 23

[1] R. Hartshorne, Algebraic Geometry, vol. 52 of Graduate Texts in Mathematics. Springer, 1977.
ISBN 978-0-387-90244-9.
[2] L. Dino, An invitation to arithmetic geometry, vol. 9. American Mathematical Society, 1996.
[3] M. Rosen, Number Theory in Function Fields, vol. 210 of Graduate Texts in Mathematics.
Springer, 2002. ISBN 978-1-4757-6046-0.