We develop a general framework for producing deterministic primality tests based on commutative group schemes over rings of integers. Our focus is on the cases of algebraic tori and elliptic curves. The proposed general machinery provides several series of tests which include, as special cases, tests discovered by Gross and by Denomme and Savin for Mersenne and Fermat primes, primes of the form 2 2l+1-2 l+1, as well as some new ones.
Bibliographical noteFunding Information:
Kunyavski˘ı was supported in part by the Minerva foundation through the Emmy Noether Research Institute. This work was finished when he was visiting the MPIM (Bonn) in September 2010. The support of these institutions is gratefully appreciated. We thank the referees for careful reading and thoughtful critical remarks.
- Algebraic torus
- Elliptic curve
- Group scheme
- Primality test