Computation-Guided Discoveries in Arithmetic

I will discuss two recent results that arose from computation-guided exploration. The first concerns elliptic curves over Q, where machine-learning experiments on LMFDB data revealed seemingly new expressions for reduced minimal Weierstrass coefficients in terms of Frobenius traces, which we then proved. The second concerns the inverse Galois problem for sporadic simple groups: between 1984 and 1989, researchers realized 25 of the 26 sporadic finite simple groups as Galois groups over Q, leaving M23 as the sole remaining case. We resolve this case through an explicit construction involving Belyi maps and computation, in a setting where the classical rigidity method does not directly apply. We are still seeking a conceptual explanation underlying this resolution.