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تسجيل مستخدم جديدحد قياسي $p$-adic للأشكال النوعية الضعيفة المودولية من الوزن النصف الصحيح
$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight
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تحصل سير على حدود بادية لمعامل الفوريه للأشكال المودولية على $SL_2(mathbb{Z})$ ل $p=2,3,5,7$. في هذا البحث، نمتد إلى نتيجة سير للأشكال المودولية الضعيفة النصف الوزنية على $Gamma_{0}(4N)$ ل $N=1,2,4$. ويستند الإثبات إلى العلاقات الخطية بين معاملات الفوريه للأشكال النصف الوزنية. كتطبيقات، نحصل على توافقات من الأسباب البورشردز، وتوافقات من كسور السلسلة الإيزنشتاين، وتوافقات من قيم الدوال $L$ في نقطة معينة. بالإضافة إلى ذلك، يتم الحصول على توافقات معاملات الفوريه للأشكال المودولية سيجل على الفضاء المااس باستخدام رفع إكيدا.
Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Gamma_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coefficients of Siegel modular forms on Maass Space are obtained using Ikeda lifting.
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