Narrow Results

In this paper, we prove arithmetic properties modulo 5 and 7 satisfied by the function pod(n) which denotes the number of partitions of n wherein odd parts must be distinct (and even parts are unrestricted). In particular, we prove the following: for all n ≥ 0,

The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of $n$ with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for $k$-colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo $11$ by utilizing some $q$-series techniques and the theory of modular forms.

It is a classical observation of Serre that the Hecke algebra acts locally nilpotently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.

We establish the freeness of certain anticyclotomic Selmer groups of modular forms. The freeness of these Selmer groups plays a key role in the Euler system arguments introduced by Bertolini and Darmon in their work on the anticyclotomic main conjecture for modular forms. In particular, our result fills some implicit gaps which appeared in generalizations of the Bertolini-Darmon result to the case where the associated residual representation is not minimally ramified. The removal of such a minimal ramification hypothesis is essential for applications involving congruences of modular forms.

In this paper, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms $f$ and $g$ with Fourier coefficients $a_n$ and $b_n$ respectively, we consider the following questions: existence of infinitely many primes $p$ such that $a_p{b}_p\ne 0$; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.

Let b_ℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is prime and 3 ≤ ℓ ≤ 23. In this paper we prove results on the distribution of b_ℓ(n) modulo m for any odd integer m > 1 with 3 ∤ m if ℓ ≠ 3.

The rank of an overpartition pair is a generalization of Dyson's rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of , the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented.

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0{(p)}_{A_n}$, $1\le n\le 3$, and from $Ẑ$-invariants of three-manifolds associated with gauge group SU(3).

We investigate the notion of equivariant forms as functions on the upper half-plane commuting with the action of a discrete group. We put an emphasis on the rational equivariant forms for a modular subgroup that are parametrized by generalized modular forms. Furthermore, we study this parametrization when the modular subgroup is of genus zero as well as their behavior under the effect of the Schwarz derivative.

We evaluate the convolution sums ∑_{l,m∈ℕ,l+15m=n}σ(l)σ(m) and ∑_{l,m∈ℕ,3l+5m=n}σ(l)σ(m) for all n ∈ ℕ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer n by the form

The theory of quasimodular forms is used to evaluate the convolution sums

For prime levels 5 ≤ p ≤ 19, sets of theta quotients are constructed that generate graded rings of modular forms of positive integer weight for Γ₁(p). Action by Γ₀(p) is shown to cyclically permute the generators. This induces symmetric representations for modular forms. The generators are used to deduce representations for the number of t-core partitions of an integer as convolutions of L-functions. Coupled systems of differential equations of level p are constructed for each basis analogous to Ramanujan’s differential equations for Eisenstein series on the full modular group.

Let f be a normalized Hecke eigenform of weight k ≥ 4 on Γ₀(N). Let λ_f(n) denote the eigenvalue of the nth Hecke operator acting on f. We show that the number of n such that λ_f(n) takes a given value coprime to 2, is finite. We also treat the case of levels 2^aN₀ with a arbitrary and N₀ = 1, 3, 5, 15 and 17. We discuss the relationship of these results to the classical conjecture of Lang and Trotter.

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes ℓ for which their coefficients c(n) obey congruences of the form c(ℓn + a) ≡ 0 (mod ℓ). We apply this result to obtain a complete characterization of the congruences of the same form that the sequences c_N(n) satisfy, where c_N(n) is defined by . This last result answers a question of H.-C. Chan.

Our results in this paper are threefold: First, we establish the modular properties of the graded dimensions of principal subspaces of level one standard modules for , and of principal subspaces of certain higher level standard modules for . Second, we establish the modular properties of families of q-series that appear in identities due to Warnaar and Zudilin, which generalize Macdonald's identities and the Rogers–Ramanujan identities. Third, we formulate a number of conjectures regarding the modularity of series of this type related to A_N-1 root systems.

Let Γ ⊂ SL₂(ℝ) be a Fuchsian group of the first kind. In this paper, we study the non-vanishing of the spanning set for the space of cuspidal modular forms of weight m ≥ 3 constructed in [5, Corollary 2.6.11]. Our approach is based on the generalization of the non-vanishing criterion for L¹-Poincaré series defined for locally compact groups and proved in [6, Theorem 4.1]. We obtain very sharp bounds for the non-vanishing of the spaces of cusp forms for general Γ having at least one cusp. We obtain explicit results for congruence subgroups Γ(N), Γ₀(N), and Γ₁(N) (N ≥ 1).

We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K^×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.

Let ℓ ≥ 3. Using the moduli interpretation, we define certain elliptic modular forms of level Γ(ℓ) over any field k where 6ℓ is invertible and k contains the ℓth roots of unity. These forms generate a graded algebra , which, over C, is generated by the Eisenstein series of weight 1 on Γ(ℓ). The main result of this article is that, when k = C, the ring contains all modular forms on Γ(ℓ) in weights ≥ 2. The proof combines algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions. Our results give a systematic method to produce models for the modular curve X(ℓ) defined over the ℓth cyclotomic field, using only exact arithmetic in the ℓ-torsion field of a single Q-rational elliptic curve E⁰.

Let k be an odd integer ≥ 3 and N be a positive integer such that 4|N. Let χ be an even Dirichlet character modulo N. Shimura decomposes the space of half-integral weight cusp forms S_k/2(N,χ) as a direct sum

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL₂(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(p^jn))_n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.