Machine learning the arithmetic of Boyd's Mahler measure conjectures

Technical Summary: Machine Learning the Arithmetic of Boyd's Mahler Measure Conjectures

Problem Statement
The paper investigates the arithmetic nature of the rational factors $r_k$ appearing in Boyd's conjectures regarding the Mahler measure of the two-variable Laurent polynomial $P_k(x, y) = x + y + \frac{1}{x} + \frac{1}{y} + k$. Boyd conjectured that for integer $k \notin \{-4, 0, 4\}$, the Mahler measure $\mu_k$ satisfies:
$$\mu_k = r_k L'(E_k, 0)$$
where $E_k$ is the elliptic curve associated with the zero locus of $P_k$, and $r_k$ is a non-zero rational number. While the sign of $r_k$ is determined by the root number of $E_k$, the magnitude and $p$-adic properties of $r_k$ (specifically its reciprocal $n_k = 1/r_k$) remain largely mysterious. The authors aim to understand the arithmetic structure of $n_k$ through large-scale statistical analysis and machine learning, addressing the question of how $r_k$ relates to $k$ and the elliptic curve $E_k$.

Methodology
The study employs a dual approach combining massive numerical computation with transformer-based machine learning:

1. Data Generation: The authors computed the first 250,000 values of $\mu_k$ and $L'(E_k, 0)$ for $1 \le k \le 250,000$ (excluding $k=4$).

   • $\mu_k$ was calculated using explicit hypergeometric formulas.
   • $L'(E_k, 0)$ was computed to 19 decimal digits of precision using PARI/GP.
   • The rational quotient $r_k$ was identified numerically using the LLL algorithm.
   • This process required approximately 9.6 $\times$ 10$^7$ seconds of single-core computation time.

2. Statistical Analysis: The dataset was analyzed to identify patterns in the size of $n_k$ and its $p$-adic valuations $v_p(n_k)$. The authors examined correlations with the conductor $N_k$, the discriminant $\Delta_k$, and congruence conditions on $k$.

3. Machine Learning Experiments: Using the Axolver framework (based on the Int2Int transformer architecture), the authors trained sequence-to-sequence neural networks to predict properties of $r_k$.

   • Inputs: Integers $k$ (encoded as digit sequences) and various arithmetic features (root number $\omega_k$, conductor $N_k$, factorizations of $N_k$ and $\Delta_k$, traces of Frobenius $a_p$, rank, and Tamagawa numbers).
   • Tasks: Predicting $r_k$ directly, predicting the magnitude $|n_k|$, and predicting $p$-adic valuations $v_p(n_k)$ for $p \in \{2, 3, 5, 7\}$.
   • Evaluation: Accuracy was measured via greedy exact-match rates and correlation coefficients between predicted and actual values.

Key Contributions and Results

1. Statistical Observations on $n_k$

• Integrality: With few exceptions ($k \in \{3, 4, 5, 8, 12, 16, 32\}$), $n_k = 1/r_k$ is an integer.
• Magnitude: The order of magnitude of $n_k$ is governed by the conductor $N_k$ and $k$, following the estimate $|n_k| \asymp \frac{N_k}{\log k}$. The term $L(E_k, 2)$, which scales this relationship, is empirically observed to lie in a narrow range $(0.52, 1.75)$ with a mean of 0.83.
• Primes $p \ge 5$: For $p \ge 5$, the $p$-adic valuations generally follow a simple probabilistic law: $P(v_p(n_k) = r) \approx p^{-r}$ for $r \ge 1$. However, the prime $p=5$ shows a significant deviation from this random model, exhibiting a higher probability of divisibility.
• Prime $p=3$: The 3-adic valuation exhibits complex structure. Divisibility by 3 occurs with high frequency (~73%). The valuation is strongly influenced by congruence conditions on $k$ (specifically modulo 27) and the parity of the number of bad primes of $E_k$ congruent to $1 \pmod 3$. On specific subfamilies (e.g., where $k(k^2-16)$ is squarefree), strict parity laws emerge relating $v_3(n_k)$ to the parity of these bad primes.
• Prime $p=2$: The 2-adic valuation is modeled by a deterministic term plus a stochastic error. The authors propose the model:
  $$v_2(n_k) \approx \hat{v}(k) + s(k) + X_k$$
  where $\hat{v}(k) = \omega_{odd}(k) + 2\omega_{odd}(k^2-16) - c(k)$ counts weighted odd prime factors of $k$ and $k^2-16$, $s(k)$ is a correction term for $k$ being 2-adically close to $\pm 8$, and $X_k$ follows a negative-binomial distribution $NB(r=3, p=3/5)$. This leads to Conjecture 3.2: $\hat{v}(k) \le v_2(n_k)$ for all $k \neq 4, 8$.
• Conditional Proof: The authors provide a conditional proof of Conjecture 3.2 for a specific family of $k$ (where $k=4u$, $u \equiv 4 \pmod 8$, etc.) assuming the Bloch-Kato conjecture and specific hypotheses from Dummigan et al. [DGdJK26].

2. Machine Learning Results

• Magnitude Prediction: The neural network failed to predict $r_k$ exactly. However, it successfully learned the magnitude $|n_k|$ (achieving ~100% accuracy in factor-of-two bounds) when provided with the conductor $N_k$ or its factorization. This confirms the statistical link between the conductor and the size of $n_k$.
• Valuation Prediction ($p \ge 5$): Models for $p=5$ and $p=7$ collapsed to predicting the most frequent outcome ($v_p=0$), failing to reveal structure due to the unbalanced distribution of valuations.
• Valuation Prediction ($p=3$): The model learned to distinguish $v_3(n_k)=0$ from $v_3(n_k)=1$ with high accuracy (~99%) on specific subfamilies, successfully capturing the parity laws identified in the statistical analysis.
• Valuation Prediction ($p=2$): The model's learning process revealed a two-stage evolution:
  • First Plateau: The model's predictions closely correlated with the deterministic predictor $\hat{P}(k)$ derived from the statistical model.
  • Second Plateau: As training continued, the model's accuracy increased while its correlation with $\hat{P}(k)$ decreased. Crucially, the model became significantly better at identifying cases where the lower bound is sharp ($v_2(n_k) = \hat{v}(k)$). This suggests the network detected arithmetic information not captured by the current statistical model.

Significance and Claims
The paper does not claim to have proven Boyd's conjectures or derived a closed-form formula for $r_k$. Instead, it claims to have:

1. Characterized the Arithmetic Structure: Identified that $r_k$ is governed by a hierarchy of arithmetic properties: the root number (sign), the conductor (magnitude), and complex local $p$-adic laws (divisibility).
2. Proposed New Conjectures: Formulated specific conjectures regarding the distribution of $p$-adic valuations, particularly the lower bound for $v_2(n_k)$ and the probabilistic laws for $p \ge 5$.
3. Demonstrated AI Utility: Showed that machine learning can recover known statistical patterns (like the conductor-magnitude relationship) and, more importantly, detect subtle arithmetic structures (like the sharpness of the 2-adic bound) that are not immediately obvious from raw data or current theoretical frameworks.
4. Highlighted Gaps: The work highlights that while the "size" of $r_k$ is well-understood via the conductor, the "fine structure" of its valuations (especially at $p=2$ and $p=3$) requires deeper arithmetic explanations, potentially involving Selmer groups and regulator indices as suggested by the Bloch-Kato framework.

The authors conclude that the combination of large-scale computation and machine learning serves as a powerful tool for generating concrete conjectures and identifying phenomena that can guide future theoretical proofs in the context of Mahler measures and $L$-functions.