AN APPLICATION OF BINARY QUADRATIC FORMS OF DISCRIMINANT $\boldsymbol {-31}$ TO MODULAR FORMS

Abstract

In this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant $-31$ and Fourier coefficients of a weight $16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant $-23$ .

The Ramanujan tau function $\tau $ is defined by

$$ \begin{align*} \Delta(z):=q \prod_{n=1}^{\infty} (1-q^{n})^{24} = \sum_{n=1}^{\infty} \tau(n) q^{n}, \end{align*} $$

where $q:=e^{2 \pi i z}$ ( $z \in \mathbb {C}$ , Im $(z)>0$ ). In 1930, Wilton [5] determined $\tau (n)$ modulo $23$ for all positive integers n. In 2006, Sun and Williams [3, Corollary 2.2, page 357] obtained Wilton’s congruence for $\tau (n)$ modulo $23$ as a consequence of their work on binary quadratic forms. Recently Dr. Pieter Moree of the Max Planck Institute for Mathematics in Bonn, Germany, in relation to his recent work [1] with Ciolan and Languasco, asked the second author if the analogous congruence modulo $31$ could be obtained using the ideas of [3] for the function $\tau _{16}(n)$ , where

$$ \begin{align*} \Delta(z) E_{4}(z):=\sum_{n=1}^{\infty} \tau_{16}(n) q^{n} \end{align*} $$

and $E_{4}(q)$ is the Eisenstein series

$$ \begin{align*} E_{4}(z) := 1+ 240 \sum_{n=1}^{\infty} \sigma_{3}(n) q^{n}. \end{align*} $$

Swinnerton-Dyer [4, page 34], before giving the arguments that prove the congruence relation (3), notes that ‘there seems little prospect’ of proving this congruence using Dedekind’s eta function. In this note, we show that it can be done by giving an explicit proof of the congruence relation (3) using Dedekind’s eta function. Then we combine our results with [2, Theorem 10.2, page 166] to obtain the following congruence for $\tau _{16}(n)$ modulo $31$ .

Theorem 1. For any positive integer n,

$$ \begin{align*} \tau_{16}(n) \equiv \begin{cases} 0 \pmod{31} \hspace{0.5cm} \mbox{if there is a prime }p \mid n\mbox{ with }\scriptsize{\big(\tfrac{p}{31}\big) }=-1\\ \hspace{3cm} \mbox{and }\nu_{p}(n) \equiv 1\pmod{2}\mbox{, or }\scriptsize{\big(\tfrac{p}{31}\big) }=1\mbox{, }\\ \hspace{3cm} p= 2x^{2}+xy+4y^{2}\mbox{ and }\nu_{p}(n) \equiv 2 \pmod{3}\mbox{,}\\ \hspace{2cm} \mbox{ }\\ \displaystyle (-1)^{\mu} \hspace{-15pt}\prod_{\substack{p \mid n,\\ \scriptsize{(\frac{p}{31}) }=1,\\ p= x^{2}+xy+8y^{2}}}\hspace{-15pt} (1+ \nu_{p}(n)) \pmod{31} \hspace{0.5cm} \mbox{otherwise,} \end{cases} \end{align*} $$

where

$$ \begin{align*} \mu= \hspace{-20pt}\sum_{\substack{p \mid n,\\ \scriptsize{(\frac{p}{31}) }=1,\\ p= 2x^{2}+xy+4y^{2},\\ \nu_{p}(n) \equiv 1 \pmod{3}}}\hspace{-20pt} 1. \end{align*} $$

Proof. We use the Dedekind eta function which is defined by

$$ \begin{align*} \eta(z):=q^{1/24} \prod_{n=1}^{\infty} (1-q^{n}). \end{align*} $$

We have

$$ \begin{align*} \Delta(z) E_{4}(z) &= \frac{\eta^{32}(z) \eta^{32}(4z)}{\eta^{32}(2z)} + \frac{1}{2}\bigg( \frac{\eta^{64}(2z)}{\eta^{32}(4z)} - \frac{\eta^{64}(z)}{\eta^{32}(2z)} \bigg) \\[6pt] &\quad + 31 \bigg( 69271552 \frac{\eta^{64}(4z)}{\eta^{32}(2z)} -34095104 \frac{\eta^{48}(4z)}{\eta^{8}(z)\eta^{8}(2z)} + 7915008 \frac{\eta^{16}(2z)\eta^{32}(4z)}{\eta^{16}(z)} \\[6pt] &\quad -1050688 \frac{\eta^{40}(2z)\eta^{16}(4z)}{\eta^{24}(z)} + 82977 \frac{\eta^{64}(2z)}{\eta^{32}(z)} -3840 \frac{\eta^{88}(2z)}{\eta^{40}(z)\eta^{16}(4z)}\\[6pt] &\quad + 96 \frac{\eta^{112}(2z)}{\eta^{48}(z)\eta^{32}(4z)} - \frac{\eta^{136}(2z)}{\eta^{56}(z)\eta^{48}(4z)} \bigg). \end{align*} $$

All of the functions in this modular equation are in $M_{16}(\Gamma _{0}(4))$ and the identity can be proved using Sturm’s theorem. Thus, we have

(1) $$ \begin{align} \Delta(z) E_{4}(z) \equiv \frac{\eta^{32}(z) \eta^{32}(4z)}{\eta^{32}(2z)} + \frac{1}{2}\bigg( \frac{\eta^{64}(2z)}{\eta^{32}(4z)} - \frac{\eta^{64}(z)}{\eta^{32}(2z)} \bigg) \ \pmod{31}. \end{align} $$

If $f:=ax^{2}+bxy+cy^{2}$ is a positive-definite integral binary quadratic form, we denote by $r(f;n)$ the number of representations of a nonnegative integer n by f. We set

$$ \begin{align*} \phi_{2}(z)&:=\frac{1}{2}\sum_{n \geq 0} (r(x^{2}+xy+8 y^{2};n)-r(2x^{2}+xy+4 y^{2};n)) q^{n}\\[6pt] &\ = \frac{1}{2} \sum_{x,y=-\infty}^{\infty} (q^{x^{2}+xy+8 y^{2}} - q^{2x^{2}+xy+4 y^{2}}). \end{align*} $$

The theta functions $\sum _{x,y=-\infty }^{\infty } q^{x^{2}+xy+8 y^{2}}$ and $\sum _{x,y=-\infty }^{\infty } q^{2x^{2}+xy+4 y^{2}}$ belong to the space $M_{1}(\Gamma _{0}(124),\scriptsize {(\frac{-31}{*}) })$ as do the eta quotients

$$ \begin{align*} \frac{\eta(z) \eta(4z) \eta(31z)\eta(124z)}{\eta(2z)\eta(62z)}, \quad \frac{\eta^{2}(2z)\eta^{2}(62z)}{\eta(4z)\eta(124z)} \quad\mbox{and}\quad \frac{\eta^{2}(z)\eta^{2}(31z)}{\eta(2z)\eta(62z)}. \end{align*} $$

Then it is straightforward to prove the modular identity

$$ \begin{align*} \phi_{2}(z) = \frac{\eta(z) \eta(4z) \eta(31z)\eta(124z)}{\eta(2z)\eta(62z)} + \frac{1}{2}\bigg( \frac{\eta^{2}(2z)\eta^{2}(62z)}{\eta(4z)\eta(124z)} - \frac{\eta^{2}(z)\eta^{2}(31z)}{\eta(2z)\eta(62z)} \bigg) \end{align*} $$

using Sturm’s theorem. We have $1-A^{31} \equiv (1-A)^{31}$ (mod $31$ ) by the binomial theorem, so that

(2) $$ \begin{align} \phi_{2}(z) \equiv \frac{\eta^{32}(z) \eta^{32}(4z)}{\eta^{32}(2z)} + \frac{1}{2}\bigg( \frac{\eta^{64}(2z)}{\eta^{32}(4z)} - \frac{\eta^{64}(z)}{\eta^{32}(2z)} \bigg) \ \pmod{31}. \end{align} $$

From (1) and (2), we deduce that

(3) $$ \begin{align} \Delta(z) E_{4}(z) \equiv \phi_{2}(z) \ \pmod{31}. \end{align} $$

Appealing to the formula for $\tfrac 12(r(x^{2}+xy+8 y^{2};n)-r(2x^{2}+xy+4 y^{2};n))$ given in [2, Theorem 10.2, page 166], we obtain from (3) the congruence for $\tau _{16}(n)$ stated in the theorem.