Predicting root numbers with neural networks

TL;DR

The paper tackles predicting root numbers $w_E$ and analytic ranks $r_E$ of low-degree $L$-functions from Dirichlet coefficients $(a_p)$, through two experiments: interpretable small neural nets that blend murmurations with Mestre–Nagao heuristics, and large CNNs testing whether a polynomial-time statistic of $(a_p)$ exists. It finds that small models leverage a combined signal where murmurations dominate in degree $3$, while Mestre–Nagao features are more influential in degrees $2$ and $4$, and that large models require exponentially many primes to approach reliable prediction of $r_E$. The results suggest fundamental limits for standard ML in achieving polynomial-time root-number prediction and point toward deeper, perhaps non-machine-learning methods to capture the arithmetic structure. Collectively, the work links neural-network-based pattern discovery to explicit number-theoretic heuristics and clarifies the practical barriers to efficient root-number classification in low-degree $L$-functions.

Abstract

We report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree $L$-functions. The first experiment is to construct interpretable models based on murmurations, a recently discovered correlation between Dirichlet coefficients and root numbers. We show experimentally that these models achieve high accuracy by learning a combination of Mestre-Nagao type heuristics and murmurations, noting that the relative importance of these features varies with degree. The second experiment is to search for a low-complexity statistic of Dirichlet coefficients that can be used to predict root numbers in polynomial time. We give experimental evidence and provide heuristics that suggest this can not be done with standard machine learning techniques.

Figures (13)

• Figure 1: Data for small neural network experiments.
• Figure 2: Average entries of $v_{64}(L)$ over all $L$-functions in our low conductor dataset with degree $d$ and root number $w = 1$ (resp. $w = -1$) in Blue (resp. Red) for Left:$d=2$, Center:$d=3$, Right:$d=4$.
• Figure 3: Examples of $v_B(L(E_0, s)$ and $v_B(L(E_1, s)$ where $E_0:y^2=x^3+x^2-309x+1991$ is rank 0 and $E_1:y^2=x^3-704x-6960$ is rank 1.
• Figure 4: Medium conductor data for large neural network experiments.
• Figure 5: High conductor data for large neural network experiments.