Search

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 k be an even integer and S k be the space of cusp forms of weight k on SL 2 (Z). Let S = ⊕ k ∈ 2 Z S k. For f , g ∈ S , we let R (f , g) be the set of ratios of the Fourier coefficients of f and g defined by R (f , g) : = { x ∈ P 1 (C) | x = [ a f (p) : a g (p) ] for some prime p } , where a f (n) (resp. a g (n)) denotes the n th Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R (f , g) is finite, then f = c g for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL 2 (Z). We apply it to study the number of representations of a positive integer by a quadratic form. [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]

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]

Let D be the differential operator defined by D : = 1 2 π i d d z. This induces a map D k + 1 : M − k ! (Γ 0 (N)) → M k + 2 ! (Γ 0 (N)) , where M k ! (Γ 0 (N)) is the space of weakly holomorphic modular forms of weight k on Γ 0 (N). The operator D k + 1 plays important roles in the theory of Eichler-Shimura cohomology and harmonic weak Maass forms. On the other hand, eta-quotients are fundamental objects in the theory of modular forms and partition functions. In this paper, we show that the structure of eta-quotients is very rarely preserved under the map D k + 1 between dual spaces M − k ! (Γ 0 (N)) and M k + 2 ! (Γ 0 (N)). More precisely, we classify dual pairs (f , D k + 1 f) under the map D k + 1 such that f is an eta-quotient and D k + 1 f is a non-zero constant multiple of an eta-quotient. When the levels are square-free, we give the complete classification of such pairs. In general, we find a necessary condition for such pairs: the weight of the primitive eta-quotient f (z) = η (d i 1 z) b 1 ⋯ η (d i t z) b t is less than or equal to 4 and every prime divisor of each d i is less than 11. We also give various applications of these classifications. In particular, we find all eta-quotients of weight 2 and square-free level N such that they are in the Eisenstein space for Γ 0 (N). To prove our main theorems, we use various combinatorial properties of a Latin square matrix whose rows and columns are exactly divisors of N. [ABSTRACT FROM AUTHOR]