General study of modular forms and arithmetic varieties

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The purpose of head investigator was to study properties of Hilbert modular cusps. We joined some workshops and seminars to get information and discuss about modular forms. We organized a workshop on Hilbert modular forms. Thanks to this workshop, we understood fundamental and recent results.Our results are as follows :1.We established some results on the structure of rings of modular forms and infinite products of modular forms.2.We established some results on the existence of perfect numbers and their analogue.3.We described Selmer groups of elliptic curves in terms of graphs.