The Ramanujan Journal ( IF 0.7 ) Pub Date : 2021-05-11 , DOI: 10.1007/s11139-021-00413-y Jaban Meher , Sujeet Kumar Singh
In this article, we first give a complete description of the algebra of integer weight modular forms on the congruence subgroup \(\Gamma _0(2)\) modulo a prime \(p\ge 3\). This result parallels results of Swinnerton-Dyer in the \(SL_2(\mathbb {Z})\) case, Katz on the subgroup \(\Gamma (N)\) for \(N\ge 3\), Gross on the subgroup \(\Gamma _1(N)\) for \(N\ge 4\) and Tupan on modular forms of half-integral weight on \(\Gamma _1(4)\). Next, we use the theory of mod p modular forms on \(\Gamma _0(2)\) to prove the non-existence of simple congruences for Fourier coefficients of quotients of certain integer weight Eisenstein series on \(\Gamma _0(2)\). The non-existence of simple congruences for coefficients of quotients of Eisenstein series on \(SL_2(\mathbb {Z})\) has been shown by Dewar.
中文翻译:
Mod p模块化形式和简单的全等
在本文中,我们首先对模子素\(p \ ge 3 \)的同余子群\(\ Gamma _0(2)\)上的整数加权模形式的代数进行完整描述。此结果与\(SL_2(\ mathbb {Z})\)情况下Swinnerton-Dyer的结果相似,子组\(\ Gamma(N)\)上的Katz为\(N \ ge 3 \),Gross上的结果与Swinnerton-Dyer的结果相似。子组\(\伽玛_1(N)\)为\(N \ GE 4 \)和Tupan上模形式半整重量对\(\伽玛_1(4)\) 。接下来,我们使用\(\ Gamma _0(2)\)上的mod p模块化形式的理论证明\(\ Gamma _0(2)\)上某个整数权重爱森斯坦级数的商的傅立叶系数的简单同余不存在。杜瓦尔已经证明了\(SL_2(\ mathbb {Z})\)上爱森斯坦级数的商系数的简单同余不存在。