Asymptotic distribution of traces of singular moduli

Asymptotic distribution of traces of singular moduli, Discrete Analysis 2022:4, 14 pp.

This paper concerns the modular $j$-function, a famous function that is fundamental to algebraic number theory and has remarkable connections to other areas of mathematics such as the representations of the monster group. Its properties are also responsible for the well-known fact that $e^{\pi\sqrt{163}}$ is extremely close to an integer.

The values taken by the $j$-function at quadratic irrationals come up naturally in class field theory and are a very important part of this story. The values of $j(z)$ for certain quadratic irrationals in the upper half plane are known as singular moduli. They are algebraic integers that have played an important role in number theory for a long time.

Given a field extension $E$ of a field $F$, one can regard $E$ as a vector space over $F$, and therefore of multiplication by an element of $E$ as a linear map on this vector space. This allows one to associate linear-algebra invariants such as the trace and determinant with elements of $E$. For instance, if $\omega$ is a non-trivial cube root of 1, then every element of $\mathbb Q(\omega)$ is a linear combination of $\omega$ and $\omega^2$. Multiplication of $a\omega+b\omega^2$ by $\omega$ is easily checked to give $-b\omega+(a-b)\omega^2$, and the trace of the linear map that takes $(a,b)$ to $(-b,a-b)$ is -1, so we define the trace of $\omega$ to be -1.

The sum of all the Galois conjugates of an element of $E$ will be an element of $F$, by symmetry, and their traces will all be the same, from which it follows that the trace of an element of $E$ is equal to the sum of its Galois conjugates. (For instance, in the case above, we have that $\omega$ has the one conjugate $\omega^2$, and $\omega+\omega^2=-1$.)

The main objects of study of this paper are the traces of the algebraic integers $j_1(z_d)$, where $j_1(z)=j(z)-744$ (this is a natural function to consider because the constant term in the Fourier expansion of $j(z)$ is 744) and for each negative fundamental discriminant $d$, $z_d=\sqrt d/2$ if $d\equiv 0$ mod 4, and $z_d=(-1+\sqrt d)/2$ if $d\equiv 1$ mod 4. The singular modulus $j_1(z_d)$ is an algebraic integer of degree $h(d)$ equal to the class number of the field $\mathbb Q(\sqrt d)$.

In 2006, the second author proved a conjecture of Brunier, Jenkins and Ono, which stated that
$$\lim_{d\to-\infty}\frac 1{h(d)}\Big(\text{Tr}j_1(z_d)-\sum_{z_Q\in\mathcal R_1}e(-z_Q)\Big)=-24,$$
where $\mathcal R_1$ is the set of complex numbers with real part in $[-1/2,1/2)$ and imaginary part greater than 1.

However, the convergence in such a limit is very slow. The basic idea of the authors in this paper is that the convergence is faster when one enlarges the window of the exponential sum by allowing the imaginary parts to be smaller. The modified series they obtain only approximate the trace, but with appropriate choices of parameters one can ensure that the error is less than 1, and since the trace is an integer this is sufficient to calculate it exactly.

They are also able to generalize the problem to traces of Faber polynomials in $j$ and to sums that are twisted by genus characters.