Search

Let kk be a nonnegative integer. Let KK be a number field and OK{{\mathcal{O}}}_{K} be the ring of integers of KK. Let ℓ≥5\ell \ge 5 be a prime and vv be a prime ideal of OK{{\mathcal{O}}}_{K} over ℓ\ell . Let ff be a modular form of weight k+12k+\frac{1}{2} on Γ0{\Gamma }_{0}(4) such that its Fourier coefficients are in OK{{\mathcal{O}}}_{K}. In this article, we study sufficient conditions that if ff has the form f(z)≡∑n=1∞∑i=1taf(sin2)qsin2(modv)f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers si{s}_{i}, then ff is congruent to a linear combination of iterated derivatives of a single theta function modulo vv.

Dedekind sums were introduced by Dedekind to study the transformation properties of Dedekind η function under the action of SL2(ℤ). In this paper, we study properties of generalized Dedekind sums si,j(p, q). We prove an asymptotic expansion of a function on ℚ defined in terms of generalized Dedekind sums by using its modular property. We also prove an equidistribution property of generalized Dedekind sums.

Let ƒ and g be weakly holomorphic modular functions on Γ0(N) with the trivial character. For an integer d, let Trd(ƒ) denote the modular trace of ƒ of index d. Let r be a rational number equivalent to i ∞ under the action of Γ0(4N). In this paper, we prove that when z goes radially to r, the limit QH(ƒ)(r) of the sum H(ƒ)(z) = ∑d>0 Trd(ƒ)e2πidz is a special value of a regularized twisted L-function defined by Trd(ƒ) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QH(ƒ)(r) = QH(g)(r) for all r equivalent to i ∞ under the action of Γ0(4N), then Trd(ƒ) = Trd(g) for all integers d. [ABSTRACT FROM AUTHOR]

Let j ⁢ (z) {j(z)} be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane ℍ {\mathbb{H}}. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., [ ℚ (τ) : ℚ ] ≠ 2 {[\mathbb{Q}(\tau):\mathbb{Q}]\neq 2} , then j ⁢ (τ) {j(\tau)} is transcendental. Let f be a harmonic weak Maass form of weight 0 on Γ 0 ⁢ (N) {\Gamma_{0}(N)}. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m, let T m {T_{m}} denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp i ⁢ ∞ {i\infty} are algebraic, and that f has its poles only at cusps equivalent to i ⁢ ∞ {i\infty}. We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that N ≥ 23 {N\geq 23} and N ∉ { 23 , 29 , 31 , 41 , 47 , 59 , 71 } {N\notin\{23,29,31,41,47,59,71\}} , then f (T m. τ) {f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N. [ABSTRACT FROM AUTHOR]

It is known that, in many cases, the field (known as the Hecke field) generated by all Hecke eigenvalues of an elliptic modular newform f is in fact generated by the pth Hecke eigenvalue αp of f for a single prime number p. In this paper, we study the density of such primes in a more general situation in terms of Galois representations. This is applied to the study of the fields of rationality of certain automorphic representations. [ABSTRACT FROM AUTHOR]

Bringmann et al. (Math. Ann., 355(3):1085-1121, ) and Guerzhoy (Proc. Am. Math. Soc., 136:3051-3059, ) studied the multiplicity two Hecke theory for weakly holomorphic modular forms on the full modular group. In this note, we consider a generalization of these results to congruence subgroups $$\Gamma _1(N)$$ with a new form theory. Using this, we define new forms for harmonic weak Maass forms and study a level raising property modulo $$\ell $$ for them. [ABSTRACT FROM AUTHOR]

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on [ABSTRACT FROM AUTHOR]