Machine Learning and LLM-Boost Symbolic Regression for Predicting $\mathbb{Q}$-Gonality of Modular Curves

We aim to predict the $\mathbb{Q}$-gonality of modular curves, an invariant measuring the minimal degree of a nonconstant rational map to $\mathbb{P}^1$. Three machine-learning architectures—Extrem gradient-boosted trees, feedforward neural networks, and transformer-based models—achieve over 90 \% exact-match accuracy on existing curves, with more than 89 \% of predictions falling within known theoretical bounds. To improve interpretability, we employ an LLM-guided boost symbolic regression pipeline that proposes nonlinear feature combinations and uncovers concise analytic formulas. These expressions match the predictive power of our models while revealing how core arithmetic invariants interact. Our results highlight the effectiveness of combining data-driven prediction with LLM-enhanced symbolic discovery in arithmetic geometry.