Non-existence of Ramanujan Congruences in Modular Forms of Level Four
نویسنده
چکیده
Abstract. Ramanujan famously found congruences for the partition function like p(5n + 4) ≡ 0 (mod 5). We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Γ1(4) which is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F -partitions.
منابع مشابه
Ramanujan Congruences for Siegel Modular Forms
We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.
متن کاملThe Partition Function and Modular Forms
1. Intro to partition function and modular forms 1 2. Partition function leading term, without modular forms 2 3. Modular form basics 5 4. First application: Rademacher’s formula 6 4.1. A Transformation Formula for the η Function 6 4.2. Rademacher’s Convergent Series 14 5. Second application: Ramanujan congruences 22 5.1. Additional Results from Modular Functions 22 5.2. Proof of Ramanujan Cong...
متن کاملArithmetic Properties of Non-harmonic Weak Maass Forms
The arithmetic behavior of the partition function has been of great interest. For example, we have the famous Ramanujan congruences p(5n+ 4) ≡ 0 (mod 5), p(7n+ 5) ≡ 0 (mod 7), p(11n+ 6) ≡ 0 (mod 11) for every n ≥ 0. In a celebrated paper Ono [13] treated this type of congruence systematically. Combining Shimura’s theory of modular forms of half-integral weight with results of Serre on modular f...
متن کاملPartition Statistics and Quasiharmonic Maass Forms
Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson’s rank statistic. He used these connections to find identities relating their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between. In this paper we show that the hypergeometric generating functions for t...
متن کاملOn Ramanujan congruences between special values of Hecke and Dirichlet L-functions
Introduction. The Fourier coefficients of modular forms of even integer weight on the full modular group SL(2,Z) can by now be considered classical in number theory. Among their intriguing properties appears a remarkable congruence of Ramanujan τ(n) ≡ σ11(n) mod 691. Plenty of excellent works, such as [9], [10], [12] and [13], that appeared in the last two decades were devoted to various genera...
متن کامل