Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCongruences involving the $U_{ell}$ operator for weakly holomorphic modular forms
170
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
In this note, we generalize the isomorphisms to the case when the discriminant form is not necessarily induced from real quadratic fields. In particular, this general setting includes all the subspaces with epsilon-conditions, only two spacial cases of which were treated before. With this established, we shall prove the Zagier duality for canonical bases. Finally, we raise a question on the integrality of the Fourier coefficients of these bases elements, or equivalently we concern the existence of a Miller-like basis for vector valued modular forms.
Let $M_k^!(Gamma_0^+(3))$ be the space of weakly holomorphic modular forms of weight $k$ for the Fricke group of level $3$. We introduce a natural basis for $M_k^!(Gamma_0^+(3))$ and prove that for almost all basis elements, all of their zeros in a fundamental domain lie on the circle centered at 0 with radius $frac{1}{sqrt{3}}$.
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 this paper we apply results from the theory of congruences of modular forms (control of reducible primes, level-lowering), the modularity of elliptic curves and Q-curves, and a couple of Frey curves of Fermat-Goldbach type, to show the existence of newforms of weight 2 and trivial nebentypus with coefficient fields of arbitrarily large degree and square-free or almost square-free level. More precisely, we prove that for any given numbers t and B, there exists a newform f of weight 2 and trivial nebentypus whose level N is square-free (almost square-free), N has exactly t prime divisors (t odd prime divisors and a small power of 2 dividing it, respectively), and the degree of the field of coefficients of f is greater than B.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
