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<p>In this article, we introduce the concepts of graded $ s $-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $ s $-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of a grading by a finite group, polycyclic-by-finite group, or by $ \mathbb{Z} $, in addition to the interesting case of crossed product grading, which includes the class of group rings.</p>
</abstract>
This paper concerns the study of the Schwartz differential equation
{\{h,\tau\}=s\operatorname{E}_{4}(\tau)}, where {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of {\operatorname{SL}_{2}({\mathbb{Z}})}. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of {\operatorname{SL}_{2}({\mathbb{Z}})}. This also leads to the solutions to the Fuchsian differential equation {y^{\prime\prime}+s\operatorname{E}_{4}y=0}.
Abstract. While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group and an arbitrary representation R of in GL(n, C), their existence, as far as we know, has been established in the literature only when restrictions are imposed on or R. In this paper, we prove the existence of n linearly independent vector-valued automorphic forms for any Fuchsian group and any n-dimensional complex representation R of . To this end, we realize these automorphic forms as global sections of a special rank n vector bundle built using solutions to the Riemann-Hilbert problem over various non-compact Riemann surfaces and Kodaira's vanishing theorem.
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