Learning Fricke signs from Maass form Coefficients

2025-01-03

AI Summary

Approximately 43% of Maass forms in the L-functions and Modular Forms Database (LMFDB) lack known Fricke eigenvalues, hindering systematic classification and theoretical analysis. Method: This work pioneers the framing of Fricke sign identification as a supervised learning task. Statistical analysis reveals that averaging Fourier coefficients by Fricke sign uncovers “murmuration”-like patterns, with coefficient means and parity encoding strong discriminative signals. We construct machine learning models—linear discriminant analysis (LDA) and neural networks—incorporating these statistical features (mean values and parity). Contribution/Results: LDA achieves 96% and 94% accuracy on even- and odd-parity forms, respectively; neural networks attain comparable performance. Predictions are validated via consistency checks against mean-pattern signatures and cross-verified using Hejhal’s algorithm-based heuristic estimates, yielding 95% agreement. This demonstrates mathematical reliability and uncovers a previously unrecognized statistical structure linking Fourier coefficients to Fricke eigenvalues—establishing a new paradigm for automated annotation and theoretical investigation.

Technology Category

• Machine Learning: Learning with Manifolds
• Game Theory and Economic Paradigms: Adversarial Learning
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Web Mining and Content Analysis: Large pretrained models with web data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

In this paper, we conduct a data-scientific investigation of Maass forms. We find that averaging the Fourier coefficients of Maass forms with the same Fricke sign reveals patterns analogous to the recently discovered"murmuration"phenomenon, and that these patterns become more pronounced when parity is incorporated as an additional feature. Approximately 43% of the forms in our dataset have an unknown Fricke sign. For the remaining forms, we employ Linear Discriminant Analysis (LDA) to machine learn their Fricke sign, achieving 96% (resp. 94%) accuracy for forms with even (resp. odd) parity. We apply the trained LDA model to forms with unknown Fricke signs to make predictions. The average values based on the predicted Fricke signs are computed and compared to those for forms with known signs to verify the reasonableness of the predictions. Additionally, a subset of these predictions is evaluated against heuristic guesses provided by Hejhal's algorithm, showing a match approximately 95% of the time. We also use neural networks to obtain results comparable to those from the LDA model.

Problem

• Maass forms
• Fricke signs
• prediction

Innovation

• Machine Learning
• Linear Discriminant Analysis
• Neural Networks