非同餘子群的模型式的同餘性質

Abstract

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.