Frobenius Traces Recover Three Weierstrass Coefficients

Elliptic curves over $\mathbb{Q}$ can be described by reduced minimal Weierstrass equations, but the coefficients of those equations do not obviously belong to the same data as Frobenius traces. This paper studies a narrow inverse problem: how much of the normalized equation can be recovered from the trace sequence attached to the isogeny class. The answer is sharper than the machine-learning framing might suggest. Decision trees first find the pattern, and the paper then proves exact formulas for the first three reduced minimal coefficients.

Core Contribution

The central result is Theorem 1.1. For a reduced minimal Weierstrass equation

with $w_1,w_3\in\{0,1\}$ and $w_2\in\{-1,0,1\}$, the authors prove

and

Here $a_p(E)$ is the Frobenius trace, $N(E)$ is the conductor, and $[m]_n$ denotes the residue of $m$ modulo $n$. The consequence is arithmetic rather than predictive: $w_1$, $w_2$, and $w_3$ are determined by the isogeny class through local trace data and conductor parity.

Technical Approach

The paper uses decision trees as a discovery tool, not as the final argument. On the LMFDB subset $\mathcal{E}_1$ of isogeny classes with conductor below $10^5$, a tree trained on Frobenius traces $\{a_p(E)\}_{p<100}$ predicts $w_1$ with 100% accuracy, but the fitted tree only uses $a_2(E)$. This leads to the formula $w_1=[a_2(E)]_2$, proved by reducing the minimal equation modulo 2 and exhaustively checking the 32 residue classes.

The same pattern repeats for $w_2$. A decision tree predicts it from $a_2(E)$ and $a_3(E)$ with 100% accuracy, and a smaller tree using $[a_2(E)]_2$ and $[a_3(E)]_3$ gives the formula above. The proof reduces the curve modulo 3 and checks 108 residue classes, using Lemma 2.4 to express the trace as $a_p(E)=p+1-\#\widetilde{E}(\mathbb{F}_p)$ even when the reduction is singular.

The third coefficient needs one extra invariant. The tree for $w_3$ uses $[N(E)]_2$, $a_2(E)$, and $w_2$, matching Lemma 2.5, which ties conductor parity to whether the reduction modulo 2 is singular. The resulting closed form is then verified by two modulo-2 case tables, split according to the parity of $a_2(E)$.

Results and Analysis

The empirical results are strongest where they lead to exact formulas. With all traces available below each cutoff, decision trees reach 100% accuracy for $w_1$ and $w_2$ on $\mathcal{E}_1$. Removing the trace at 2 hurts $w_1$ prediction: accuracy rises only from 51.03% for $p<10^2$ to 62.39% for $p<10^4$. Removing $a_2(E)$ or $a_3(E)$ similarly blocks clean recovery of $w_2$; without both, the tree stays near 33.66–33.73%, essentially the three-class baseline.

For $w_3$, traces alone are less decisive. The best tree trained on $\{a_p(E)\}_{p<B}$ reaches 73.71%, 75.36%, and 76.68% as $B$ moves through $10^2$, $10^3$, and $10^4$. Once conductor parity and the proven intermediate coefficients are supplied, the formula gives exact recovery. That gap is informative: the paper is not claiming generic neural or tree models discover the arithmetic rule unaided, but that interpretable models can expose the right local quantities to test.

Appendix B reinforces this point with a deliberately small encoder-decoder transformer trained on about 2.1 million isogeny classes with conductor at most $5\times10^5$. It obtains 100% validation accuracy for predicting $(w_1,w_2)$ from traces, but for $(w_1,w_2,w_3)$ with conductor included it reports 100% on $w_1$, 89.2% on $w_2$, 80.5% on $w_3$, and 72.6% joint accuracy. The transformer improves over the trace-only decision tree for $w_3$, but it remains far from the exact theorem. The evidence for the main claim therefore comes from the formal proof and exhaustive residue checks, with machine learning serving as provenance rather than proof.