霍奇猜想是代數幾何中基本的問題。在1979 年時，Shioda 提出了一個算術條件來
驗證在費馬簇上的霍奇猜想。我們將這個條件切分為較為簡單的子問題，並且實作了
一個演算法來驗證霍奇猜想對於次數是21 的費馬簇會成立。這個次數是21 的例子在
1979 年只有部分的答案。除了費馬簇之外，另一個有關的主題是費馬型阿貝爾簇的霍
奇猜想。對於某些特定的次數，我們提出了一個霍奇類是標準元素之線性組合的必要
條件。

The Hodge conjecture is a fundamental problem in algebraic geometry. In 1979, Shioda proposed an arithmetic condition on verifying the Hodge conjecture for Fermat varieties. We divide the condition into simpler subproblems and implemented an algorithm to verify the Hodge conjecture for Fermat variety of degree 21. This case was only partially answered in 1979. Besides the Fermat varieties, a related topic is the Hodge conjecture for abelian varieties of Fermat type. For some specific degrees, we provide a necessary condition for a Hodge class to be a linear combination of “stadard elements”.

1 Introduction 1
1.1 Kähler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Hodge Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Algebraic Cycles and The Hodge Conjecture . . . . . . . . . . . . . . . . . . 4
2 The Hodge Conjecture for Fermat Varieties 7
2.1 Definition and Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Conditions (Pnm) and (Pm) . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 An Algorithm for Finding Indecomposable Elements . . . . . . . . . . . . . . 10
2.3.1 Solving a System of Linear Diophantine Equations . . . . . . . . . . . 10
2.3.2 Transforming All Coefficients Into Nonnegative Integers . . . . . . . . 11
2.3.3 Check for Indecomposability . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 The Modified Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 The Hodge Conjecture for Xn
21 . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Abelian Varieties of Fermat Type 19
3.1 The Sets Rm;Rm and the Gap Group Bm/Sm . . . . . . . . . . . . . . . . . . 19
3.1.1 Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.2 Anm, Bnm, and Dnm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.3 Bm;Dm and Sm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.4 Rm;Bm; Sm and Dm . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.5 Structure of the Gap Group Bm/Sm . . . . . . . . . . . . . . . . . . . 22
3.1.6 Rm;prim and Rm;prim . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.7 Operations of Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.8 The Map c(d) and the d-part . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Dirichlet Characters and the Map  . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Jacobian Variety of the Fermat Curve . . . . . . . . . . . . . . . . . . . . . . 25
3.4 The Case of m = pq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Bibliography 33

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