Vincent Macri | Searcharxiv

arXiv2026

Distinguishing elliptic curves modulo $p$ and identifying images of product representations

Given two elliptic curves defined over $\mathbb{Q}$ and a rational prime $p$, we study the product of their residual Galois representations. Using Goursat's lemma, we explicitly enumerate and completely characterize all possible images of such product representations. We also define associated invariants to these image groups, which we call \textit{witness ratios}, and we explain their computational utility and their relationship to the well-known Sturm bound for testing congruences between modular forms.

math.NT↗

arXiv2026

Improvements to Jacobian Arithmetic in Global Function Fields

We present two improvements to arithmetic in the Jacobian of global function fields based on the approach of Hess. The first reduces the number of expensive reduction steps by optimizing for typical inputs rather than worst-case behavior, assuming the function field contains a degree-one place. The second introduces a memory-time trade-off that speeds up computations by caching frequently used intermediate results. Our asymptotic analysis and empirical experiments show that our improved algorithms are significantly faster in practice than previously published methods. To the best of our knowledge, our publicly-available software implementation of Jacobian arithmetic is the first to support unique representatives of divisor classes.

math.NT↗