Murmurations for elliptic curves ordered by height
Will Sawin, Andrew V. Sutherland
TL;DR
This paper investigates murmuration densities for averages of elliptic curve L-function coefficients when elliptic curves are ordered by naive height. It builds on the observed phenomenon that the pth coefficients $a_p(E)$, twisted by the root number $\epsilon(E)$, exhibit a smooth, rank-parity dependent oscillation as a function of the ratio $p/N(E)$, and extends the analysis to the height ordering by proposing an explicit murmuration density given by a convergent sum of Bessel functions. The authors derive this density by applying Voronoi summation to the modular form attached to each curve, introducing local $p$-adic factors $\\ell_{p,\nu}$ and $\hat{\ell}_{p,\nu}$ via averaging over $A,B$ in $\\mathbb Z_p^2$ describing reductions of $E_{A,B}$ modulo $p$, and then summing over $n$ with $n$ restricted to have no small prime factors to obtain a density formula. They prove a primary theorem (Theorem) for a version where the $p$-sum is replaced by $n$ with no small prime factors, expressed through $J_1$ Bessel functions and local factors, and provide analogous predictions for curves of prime conductor, along with extensive numerical evidence up to height $H(E) \,\le 2^{28}$. The computations include a detailed evaluation strategy for the left-hand side via $a_n(E)$, using the Voronoi transform and modular form averages, and for the right-hand side via discretized integrals of the predicted density, with rigorous attention to convergence and the influence of small primes. Overall, the work delivers the first explicit Murmur density formula for a family of elliptic curves under a nontrivial ordering and outlines a general framework applicable to broader $L$-function families.
Abstract
He, Lee, Oliver, and Pozdnyakov~\cite{HLOP} have empirically observed that the average of the $p$th coefficients of the $L$-functions of elliptic curves of particular ranks in a given range of conductors $N$ appears to approximate a continuous function of $p$, depending primarily on the parity of the rank. Hence the sum of $p$th coefficients against the root number also appears to approximate a continuous function, dubbed the murmuration density. However, it is not clear from this numerical data how to obtain an explicit formula for the murmuration density. Convergence of similar averages was proved by Zubrilina~\cite{Zubrilina} for modular forms of weight $2$ (of which elliptic curves form a thin subset) and analogous results for other families of automorphic forms have been obtained in further work~\cite{BBLLD,LOP}. Each of these works gives an explicit formula for the murmuration density. We consider a variant problem where the elliptic curves are ordered by naive height, and the $p$th coefficients are averaged over $p/N$ in a fixed interval. We give a conjecture for the murmuration density in this case, as an explicit but complicated sum of Bessel functions. This conjecture is motivated by a theorem about a variant problem where we sum the $n$th coefficients for $n$ with no small prime factors against a smooth weight function. We test this conjecture for elliptic curves of naive height up to $2^{28}$ and find good agreement with the data. The theorem is proved using the Voronoi summation formula, and the method should apply to many different families of $L$-functions. By a similar approach, we give a prediction murmuration density for elliptic curves of prime conductor, ordered by conductor, again matching the data but lacking a motivating theorem. This is the first work to give an explicit formula for the murmuration density of a family of elliptic curves, in any ordering.