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Register a new userOn Modular Signs Of Half Integral Weights
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In this paper, we consider the first negative eigenvalue of eigenforms of half-integral weight k + 1/2 and obtain an almost type bound.
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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.
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 $Gamma_0(4).$
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-C
ohen 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.
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.
In the proposed work, we establish a total of six new $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 7 and 21.These equations can be regarded as modular identities in the alternate theory of signature 3. As a consequence, several values of quotients of theta--function are evaluated.
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