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Register a new user$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight
حد قياسي $p$-adic للأشكال النوعية الضعيفة المودولية من الوزن النصف الصحيح
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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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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 $Gamma_0(4).$
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke modules.
In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which is related to the weight of Borcherds lifts when the weight is zero. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, and obtain divisibility results in an orthogonal direction on reduced modular forms.
Let $lambda$ be an integer, and $f(z)=sum_{ngg-infty} a(n)q^n$ be a weakly holomorphic modular form of weight $lambda+frac 12$ on $Gamma_0(4)$ with integral coefficients. Let $ellgeq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{ell^m})(z) = sum_{n=0}^infty b(n)q^n$ has the form [ (f|U_{ell^m})(z) equiv b(0) + sum_{i=1}^{t}sum_{n=1}^{infty} b(d_i n^2) q^{d_i n^2} pmod{ell}, ] where $d_1, ldots, d_t$ are square-free positive integers, and the operator $U_ell$ on formal power series is defined by [ left( sum_{n=0}^infty a(n)q^n right) bigg| U_ell = sum_{n=0}^infty a(ell n)q^n. ] Then, $lambda equiv 0 pmod{frac{ell-1}{2}}$. Moreover, if $tilde{f}$ denotes the coefficient-wise reduction of $f$ modulo $ell$, then we have [ biggl{ lim_{m rightarrow infty} tilde{f}|U_{ell^{2m}}, lim_{m rightarrow infty} tilde{f}|U_{ell^{2m+1}} biggr} = biggl{ a(0)theta(z), a(0)theta^ell(z) in mathbb{F}_{ell}[[q]] biggr}, ] where $theta(z)$ is the Jacobi theta function defined by $theta(z) = sum_{ninmathbb{Z}} q^{n^2}$. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.
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