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.
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.
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
In this paper, we consider the first negative eigenvalue of eigenforms of half-integral weight k + 1/2 and obtain an almost type bound.
We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1over 2}$ on $Gamma_0(4).$
We study central hyperplane arrangements with integral coefficients modulo positive integers $q$. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection lattices modulo $q$ are periodic except for a finite number of $q$s.