Congruences for a mock modular form on SL2(Z) and the smallest parts function

Scott Ahlgren; Byungchan Kim

doi:10.1016/j.jnt.2017.11.013

Congruences for a mock modular form on SL₂(Z) and the smallest parts function

Scott Ahlgren, Byungchan Kim

School of Natural Sciences

University of Illinois

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight 3/2 on SL₂(Z) modulo primes ℓ≥5. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo ℓ. As another application, we show that congruences modulo ℓ for the smallest parts function are rare in a precise sense.

Original language	English
Pages (from-to)	81-89
Number of pages	9
Journal	Journal of Number Theory
Volume	189
DOIs	https://doi.org/10.1016/j.jnt.2017.11.013
State	Published - Aug 2018

Keywords

Mock modular forms
Modular forms modulo ℓ
Smallest parts function

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10.1016/j.jnt.2017.11.013

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abstract = "Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight 3/2 on SL2(Z) modulo primes ℓ≥5. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo ℓ. As another application, we show that congruences modulo ℓ for the smallest parts function are rare in a precise sense.",

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AB - Using a family of mock modular forms constructed by Zagier, we study the coefficients of a mock modular form of weight 3/2 on SL2(Z) modulo primes ℓ≥5. These coefficients are related to the smallest parts function of Andrews. As an application, we reprove a theorem of Garvan regarding the properties of this function modulo ℓ. As another application, we show that congruences modulo ℓ for the smallest parts function are rare in a precise sense.

KW - Mock modular forms

KW - Modular forms modulo ℓ

KW - Smallest parts function

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JF - Journal of Number Theory

ER -

Congruences for a mock modular form on SL₂(Z) and the smallest parts function

Abstract

Keywords

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