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Let F be a newform of weight 2k on Γ 0 (N) with an odd integer N and a positive integer k, and ℓ be a prime larger than or equal to 5 with (ℓ , N) = 1 . For each fundamental discriminant D, let χ D be a quadratic character associated with quadratic field Q (D) . Assume that for each D, the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is non-negative. Let W ℓ + (resp. W ℓ - ) be the set of positive (resp. negative) fundamental discriminants D with (D , N) = 1 such that the ℓ -adic valuation of the algebraic part of L (F ⊗ χ D , k) is zero. We prove that for each sign ϵ , if W ℓ ϵ is a non-empty finite set, then W ℓ ϵ ⊂ 1 , (- 1) ℓ - 1 2 ℓ. By this result, we prove that if ϵ is the sign of (- 1) k , then k ≥ ℓ - 1 or k = ℓ - 1 2. These are applied to obtain a lower bound for # { D ∈ W ℓ ϵ : | D | ≤ X } and the indivisibility of the order of the Shafarevich–Tate group of an elliptic curve over Q . To prove these results, first we refine Waldspurger's formula on the Shimura correspondence for general odd levels N. Next we study mod ℓ modular forms of half-integral weight with few non-vanishing coefficients. To do this, we use the filtration of mod ℓ modular forms and mod ℓ Galois representations. [ABSTRACT FROM AUTHOR]

Let λ be an integer, and f (z) = ∑ n ≫ - ∞ a (n) q n be a weakly holomorphic modular form of weight λ + 1 2 on Γ 0 (4) with integral coefficients. Let ℓ ≥ 5 be a prime. Assume that the constant term a(0) is not zero modulo ℓ . Further, assume that, for some positive integer m, the Fourier expansion of (f | U ℓ m) (z) = ∑ n = 0 ∞ b (n) q n has the form (f | U ℓ m) (z) ≡ b (0) + ∑ i = 1 t ∑ n = 1 ∞ b (d i n 2) q d i n 2 (mod ℓ) , where d 1 , ... , d t are square-free positive integers, and the operator U ℓ on formal power series is defined by ∑ n = 0 ∞ a (n) q n | U ℓ = ∑ n = 0 ∞ a (ℓ n) q n. Then, λ ≡ 0 (mod ℓ - 1 2) . Moreover, if f ~ denotes the coefficient-wise reduction of f modulo ℓ , then we have { lim m → ∞ f ~ | U ℓ 2 m , lim m → ∞ f ~ | U ℓ 2 m + 1 } = { a (0) θ (z) , a (0) θ ℓ (z) ∈ F ℓ [ [ q ] ] } , where θ (z) is the Jacobi theta function defined by θ (z) = ∑ n ∈ 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. [ABSTRACT FROM AUTHOR]

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]

Let $$\Gamma $$ be a subgroup of finite index in $$\mathrm {SL}(2,\mathbb {Z})$$ . Eichler defined the first cohomology group of $$\Gamma $$ with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on $$\Gamma $$ with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups $$\Gamma ^{(1,j)}$$ and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if $$j=1$$ and the weights of Jacobi cusp forms are in $$\frac{1}{2}\mathbb {Z}-\mathbb {Z}$$ , then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms. [ABSTRACT FROM AUTHOR]

Let k be a positive integer and P ⊂ C[ X] the set of polynomials of degree less than or equal to k. There exists an isomorphism, called the Eichler-Shimura isomorphism, between the space of cusp forms of integral weight k and a first parabolic cohomology group with coefficient module P . Moreover, P contains period functions of cusp forms of weight k. The Eichler-Shimura isomorphism was extended to the space of cusp forms of real weight with coefficient module P. Here, in contrast to the case of integral weight P is an infinite-dimensional vector space consisting of holomorphic functions on the complex upper half plane with a certain growth condition. However, period functions of cusp forms of real weight have not been described in terms of a finite-dimensional space even for the case of half-integral weight. In this paper, we construct a new isomorphism between the space of cusp forms of real weight and an Eichler-Shimura cohomology group with coefficient module P so that we obtain a finite-dimensional space of period functions containing those of cusp forms up to coboundaries. As applications, we prove analogues of the Haberland formula for the case of real weight, and we construct injective linear maps from the spaces of mixed mock modular forms to the spaces of quantum modular forms and classify Zariski closures in A(C) of images of mixed mock modular forms under these linear maps in terms of their weights. [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]

In his last letter to Hardy, Ramanujan introduced mock theta functions. For each of his examples f(q), Ramanujan claimed that there is a collection {Gj} of modular forms such that for each root of unity ζ, there is a j such that.....Moreover, Ramanujan claimed that this collection must have size larger than 1. In his 2001 PhD thesis, Zwegers showed that the mock theta functions are the holomorphic parts of harmonic weak Maass forms. In this paper, we prove that there must exist such a collection by establishing a more general result for all holomorphic parts of harmonic Maass forms. This complements the result of Griffin, Ono and Rolen that shows such a collection cannot have size 1. These results arise within the context of Zagier's theory of quantum modular forms. A linear injective map is given from the space of mock modular forms to quantum modular forms. Additionally, we provide expressions for "Ramanujan's radial limits" as L-values. [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]