International Journal of Data Science in the Mathematical Sciences

We outline how murmurations were first observed in the context of data scientific exploration. Specifically, we highlight the relationship between murmurations and PCA. Subsequently, we apply PCA to datasets of real quadratic fields, and speculate on an emergent perspective on arithmetic suggested by these experiments.

The moments of the coefficients of elliptic curve $L$-functions are related to numerous important arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate’s conjecture to the rank of the corresponding elliptic surface over $\mathbb{Q}(T)$; one can also construct families of moderate rank by finding families with large first moments. Michel proved that if $j(T)$ is not constant, then the second moment of the family is of size $p^2+O(p^{3/2})$; these two moments show that for suitably small support the behavior of zeros near the central point agree with that of eigenvalues from random matrix ensembles, with the higher moments impacting the rate of convergence.

In his thesis, Miller noticed a negative bias in the second moment of every one-parameter family of elliptic curves over $\mathbb{Q}$ whose second moment had a (by him) calculable closed-form expression, specifically the first lower-order term which does not average to zero is on average negative. This Bias Conjecture has now been confirmed for many families; however, these are highly non-generic families as they are specially chosen so that the resulting Legendre sums can be determined. For cohomological reasons, each subsequent term in the second moment expansion is smaller than the previous by a factor on the order of $\sqrt{p}$, and thus numerically, it is hard to see a term of size $p$ with a small negative average as it can be masked by a term of size $p^{3/2}$ which averages to zero.

Inspired by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov and others in investigations of murmurations of elliptic curve coefficients with machine learning techniques, we pose a similar problem for trying to understand the Bias Conjecture. As a start to this program, we numerically investigate the bias conjecture and provide a visual representation of the bias for the second moment. We find a one-parameter family of elliptic curves whose bias is positive for half the primes. However, the numerics do not offer conclusive evidence that negative bias for the other half is enough to overwhelm the positive bias. Without an explicit expansion for the second moment, we are not able to extract potential negative bias of the order $p$ term.

This note summarizes my talk at the ICERM workshop on Murmurations and Arithmetic, 6 June 2023 to 8 June 2023. The talk was based on the paper [14], to which the reader is referred for details and proofs. The two addenda at the end appeared neither in the talk nor in [14] but are complementary to both. It is a pleasure to thank ICERM and the organizers of the workshop for inviting me to participate in a meeting that was both enjoyable and productive.