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Register a new userDistribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes
توزيع المعاملات الفوريه الجزئية لشكل مودول من الوزن النصفي الجزئي حسب الأولويات
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Dohoon Choi
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حديثا، تصنف بروينير و أونو الأشكال المنقطة $f(z):=sum_{n=0}^{infty} a_f(n)q ^n in S_{lambda+1/2}(Gamma_0(N),chi)cap mathbb{Z}[[q]]$ التي لا تلبي معيار معين للأعداد الفردية بحجم أكبر من 5. في هذا البحث، باستخدام علامة رانكين-كوهن، نمتد هذا النتيجة إلى الأشكال النصف عشوائية للأعداد الفردية بحجم أكبر من 5. كتطبيقات لنظريتنا الرئيسية، نستنتج منها عن معايير التوزيع للأعداد الفردية بحجم أكبر من 5 للأعداد المتجانسة و رقم فئة هورويتز. كما ندرس أيضا تشبيها لمقترح نيومان للتفريعات الزائدة.
Recently, Bruinier and Ono classified cusp forms $f(z) := sum_{n=0}^{infty} a_f(n)q ^n in S_{lambda+1/2}(Gamma_0(N),chi)cap mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $pgeq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newmans conjecture for overpartitions.
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We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $Gamma_0(4)$ with Kohnens plus condition and modular forms for the Weil representation associated to the discriminant form for the lattice with Gram matrix $(2)$. With such an isomorphism, we prove the Zagier duality and write down the Borcherds lifts explicitly.
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