Learning Fricke signs from Maass form Coefficients
This work addresses a specific challenge in number theory by applying machine learning to classify Maass forms, representing an incremental advancement in computational methods for mathematical data analysis.
The paper tackles the problem of predicting Fricke signs for Maass forms, achieving 96% accuracy for even parity and 94% for odd parity using Linear Discriminant Analysis, and verifies predictions with a 95% match against heuristic methods.
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.