ECM and the Elliott–Halberstam conjecture for quadratic fields
Volume 213 / 2024
Acta Arithmetica 213 (2024), 289-324
MSC: Primary 11N36; Secondary 11Y05, 11G05, 11G15
DOI: 10.4064/aa230110-21-2
Published online: 4 June 2024
Abstract
The complexity of the elliptic curve method of factorization (ECM) has been proven under a strong conjectural form of existence of friable numbers in short intervals. In the present work we use friability to tackle a different version of ECM which is much more studied and implemented, especially because it enables the use of ECM-friendly curves. In the case of curves with complex multiplication (CM) we replace heuristic arguments by rigorous results conditional on the Elliott–Halberstam (EH) conjecture. The proven results mirror recent work concerning the count of primes $p$ such that $p-1$ is friable. In the case of non-CM curves, we explore consequences of a hypothetical statement that can be seen as an elliptic curve analogue of EH.
