臺灣博碩士論文加值系統

　眾所周知的，費馬曲線　x^n+y^n=1　是一個與特殊線性群SL_2(Z)的有限指數子群Γ_n相關聯的模曲線，當n不等於1, 2, 4, 8時， Γ_n是一個非同餘子群。現在令費馬曲線的虧格為g，scholl的定理告訴我們，Γ_n上權為2的尖點型式與由此曲線相關聯的Tate模所建構出的2g維l進數伽羅瓦表現會滿足Atkin and Swinnerton-Dyer同餘。

　　在這篇論文中，我們將會分解伽羅瓦表現，然後給一個更加精確的Atkin and Swinnerton-Dyer同餘。我們將會解決n=6的情況。

It is known that each Fermat curve x^n+y^n=1 is the modular curve associated to some subgroup Γ_n of SL_2(Z) of finite index. Moreover if n≠1,2,4,8 then Γ_n is a noncongruence subgroup. Let g be the genus of the Fermat curve, by Scholl’s theorem, cuspforms of weight 2 on Γ_n, together with the 2g-dimensional l-adic Galois representations coming from the Tate module associate this curve, satisfy the Atkin and Swinnerton-Dyer congruence.
In this thesis, we decompose this Galois representation and give a more precise Atkin and Swinnerton-Dyer congruence. The case n=6 will be completely worked out.

中文提要 i
英文提要 ii
誌謝 iii
目錄 iv
1 Introduction 1
2 Review of modular forms on congruence subgroups 2
2.1 Modular forms and cusp forms 2
2.2 Hecke operators 5
2.3 Petersson inner product 9
2.4 Oldforms and Newforms 10
2.5 Hecke eigenforms 11
3 Atkin and Swinnerton-Dyer congruences for noncongruence subgroups
12
3.1 Noncongruence subgroups 13
3.2 Atkin and Swinnerton-Dyer congruence 13
4 Atkin and Swinnerton-Dyer congruences associated to Fermat curves
17
4.1 Fermat curve 17
4.2 Case x^6+y^6=1 18
5 References 24

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