L值的超幾何求值與調和馬斯形式

Abstract

布魯尼耶（Bruinier）、小野（Ono）與羅德斯（Rhoades）曾提出對於具有複數乘法的新形式所對應之良好調和馬斯形式的明確建構問題。本論文研究其中一個特殊情形：同餘子子群Γ0(N)在SL(2,R)中的正規化子為三角群。於此情形下，弱全純尖點形式的正則化L值可作超幾何求值。藉由此結果，我們構造出某些赫克特徵形式所對應之良好調和馬斯形式，且其傅立葉係數係數皆可明確計算。其中，包括一個具有複數乘法的新形式的例子。本研究為與凃芳婷教授及楊一帆教授的共同成果。

The explicit construction of good harmonic Maass forms associated with CM newforms was posed as a problem by Bruinier, Ono, and Rhoades. In this thesis, we investigate a special case where the normalizer of the congruence subgroup Γ0(N) in SL(2,R) is a triangle group. In this setting, regularized L-values of weakly holomorphic cusp forms admit hypergeometric evaluations. As a consequence, we obtain good harmonic Maass forms with explicitly computable Fourier coefficients for certain Hecke eigenforms, including a CM newform. This work is joint with Fang-Ting Tu and Yifan Yang.