Explorations in math and programming
David Lowry-Duda

I am happy to announce that my collaborators — Jonathan Bober, Andy Booker, Min Lee — and I have uploaded a preprint to the arxiv on murmurations of modular forms.

See here for the arxiv preprint. See this related post about murmurations in Maass forms for related discussion.

In this note, I describe our result and some of the context of recent murmurations work. I don't go into technical details or describe computational experiments — though there are interesting things to say in both of these regards and I intend to write more about these later.

Initial Murmuration Phenomenon

The murmuration phenomenon is very new, first observed as a happy accident from attempts to apply machine learning methods to predict data associated to elliptic curves. This phenomenon was formally presented in the paper Murmurations of Elliptic Curves by He, Lee, Oliver, and Pozdnyakov [HLOP].

Since then, there have been a large number of less formal descriptions, such as

HLOP observed an apparent correlation between the root numbers of elliptic curve $L$-functions and their coefficients $a_p(E)$, varying in proportion to the conductor. It is not surprising that the coefficients of an elliptic curve carry information about the root number of the $L$-function, as of course all the coefficients of an elliptic curve contain all information about the $L$-function. But they show that coefficients in small bands have predictable correlation with root numbers, and (very surprisingly) this correlation can be both positive or negative depending on how the primes $p$ relate in size to the conductor.

This phenomenon (and motivation for the name) can be seen in the following plot from HLOP (which we re-use in our preprint as well).

Average of $a_p$ over isogeny classes of elliptic curves of conductor in $[7500, 10000]$. Points in blue come from rank $0$ curves. Points in red come from rank $1$ curves. The primes here vary from $2$ to $7919$.

Each blue point in this image comes from the average size of $a_p(E)$ over all elliptic curves of rank $0$ (which should account for $100$ percent of elliptic curves with positive root number) whose conductor is in $[7500, 10000]$. We see, for example, that coefficients $a_p(E)$ at small primes tend to positively correlate with the root number. But at certain thresholds this reverses to negative correlation.

In further experimental work, HLOP (as well as Sutherland and those at the ICERM workshop on Murmurations) have shown this to be a general phenomenon. The recent work of Zubrilina gave the first proven result for $L$-functions of $\mathrm{GL}(2)$ type.

Murmurations in the Weight Aspect

We consider murmurations with varying archimedean parameters, prompted by a suggestion of Peter Sarnak. We consider holomorphic modular cuspforms of level $1$ and weight $k \to \infty$.

For level $1$, there are no modular forms of odd weight, and the root number of the $L$-functions for even weight is exactly $(-1)^{\frac{k}{2}}$. Thus here, murmurations should be expected to take the form of biases in coefficients depending on whether $k \equiv 0 \bmod 4$ or $k \equiv 2 \bmod 4$.

The analogue of normalizing by (arithmetic) conductor for elliptic curve murmurations is to normalize by the complete analytic conductor. The analytic conductor is approximately $\left(\frac{k-1}{4 \pi} \right)^2$ for all holomorphic cusp forms of weight $k$ and level $1$.

The conductor can be thought of as a measure of complexity. As the analytic conductor varies with the square of $k$, the complexity of this family grows more rapidly than the linear growth in other murmuration phenomena.

For example, there are only approximately $X$ modular forms of level $1$ and weight $k$ with analytic conductor up to $X$; in Zubrilina's work, there are approximately $X^2$ modular forms of a fixed weight but with varying level $N$ with analytic conductor up to $X$. The relative sparseness heuristically affects what results one is able to prove.1 1Sarnak wrote about this in a letter to Drew Sutherland and Nina Zubrilina, published at https://publications.ias.edu/sarnak/.

In our paper we prove (assuming GRH for Dirichlet $L$-functions and for $L$-functions from level $1$ modular cuspforms) that there is a bias between coefficients on average in these families against the parity of $k \bmod 4$. As in Zubrilina's work, the main tool that we use is a trace formula. We use the Eichler-Selberg trace formula.

Specifically, we prove the following.

Assume GRH. Fix $\varepsilon > 0$, small. Fix $\delta \in \{ 0, 1\}$ and consider weights $k \equiv 2\delta \bmod 4$. Fix also a compact interval $E \subset \mathbb{R}_{>0}$ with $\lvert E \rvert > 0$. Let $K, H \in \mathbb{R}_{>0}$ with $K^{\frac{5}{6} + \varepsilon} < H < K^{1 - \varepsilon}$. Write $N$ to mean the analytic conductor for forms of level $1$ and weight $K$. Then as $K \to \infty$, we have \begin{align*} &\frac{ \sum_{p/N \in E} \log p \sum_{\substack{k \equiv 2 \delta \bmod 4 \\ \lvert k - K \rvert \leq H}} \sum_{f \in S_k} \lambda_f (p) }{ \sum_{p/N \in E} \log p \sum_{\substack{k \equiv 2 \delta \bmod 4 \\ \lvert k - K \rvert \leq H}} \sum_{f \in S_k} 1 } \\ &\qquad \qquad \qquad = \frac{(-1)^\delta}{\sqrt{N}} \left( \frac{\nu(E)}{\lvert E \rvert} + o_{E, \varepsilon}(1) \right), \end{align*} where $\nu(E)$ is given explicitly by \begin{equation*} \frac{1}{\zeta(2)} \sum_{\substack{a, q \in \mathbb{Z}_{>0} \\ \gcd(a, q) = 1 \\ (a/q)^{-2} \in E}} \frac{\mu(q)^2}{\varphi(q)^2 \sigma(q)} \left( \frac{q}{a} \right)^3. \end{equation*}

In this theorem, $f \in S_k$ range over a basis of Hecke eigenforms of level $1$ and weight $k$, and $\lambda_f(p)$ denotes the $p$th Hecke eigenvalue of the form $f$.

Digesting this Theorem

This looks complicated. Let's make sense of it.

This left-hand-side of the theorem has a complicated ratio of terms. Looking closely, you'll see that the only difference between the numerator and the denominator is that the numerator has $\lambda_f(p)$ and the denominator has $1$. The denominator serves merely to count the number of terms appearing; this ultimately has to do with average behavior, and this is how we take an average.

The murmuration is formed from taking the average across all forms of a given weight $k$, for all $k$ of the correct parity that is "within $H$ of $K$", across all primes in a certain interval (relative to the analytic conductor of $K$). For technical reasons we weight the primes by $\log p$.

From a high level, we can say that the left hand side is a (weighted) average of Hecke eigenvalues $\lambda_f(p)$ across all forms with the right root number and weight near $K$, and across primes $p$ in a certain size relative to $K$.

The right-hand-side of the theorem states essentially states that there is a measure $\nu$ on compact intervals $E$ that governs the murmuration behavior. All bias comes from the effect of the root number, which comes exclusively from $\delta \in \{0, 1 \}$. Note that $\nu$ is always nonnegative, and so there is a strong, consistent bias here.

The effect of the measure is interesting. Any point $(a/q)$ with $\gcd(a, q) = 1$, $q$ squarefree, and $(a/q)^{-2} \in E$ will have a positive contribution to the murmuration function. If one chooses a sequence of compact sets $E'$ with $\lvert E' \rvert \to 0$ but such that $(a/q)^{-2} \in E'$ always, then the contribution of this point to the murmuration function won't change — but the RHS of the theorem also divides by $\lvert E' \rvert$. The effect is that on arbitrarily small intervals, the murmuration function actually behaves like Kronecker delta functions (and are positive in the root number $1$ case and negative in the root number $-1$ case)!

Here is a concrete example. Take $E$ to be a very small interval containing the point $1$. Then the RHS will have a term looking like \begin{equation*} \frac{(-1)^\delta}{\sqrt{N}} \frac{1}{\lvert E \rvert} + \text{error} \approx (-1)^\delta\frac{4 \pi}{K} \frac{1}{\lvert E \rvert}. \end{equation*} This suggests that for primes $p$ very close to $N \approx \frac{K}{4 \pi}$, we have a relatively strong bias in the values of the coefficients $\lambda_f(p)$ towards the root number.

In intervals without any large contributions from rationals $a/q$, we should expect much weaker biases. In practice, there are relatively sparse rationals with very large individual contributions to the murmuration function — though in a different sense, for small enough intervals they're all infinite delta function spikes anyway!

Experimentally, the rate of convergence isn't particularly fast. If you were trying to become rich by making a betting game of guessing signs or biases of weight $1$ forms based on the weights $k$ and primes $p$, it would take a very long time for you to win big... but you would eventually (almost surely) win!

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