Primes Dividing Invariants of CM Picard Curves

Construction of CM Picard curves

In this article we generalize the CM method for elliptic and hyperelliptic curves to Picard curves. We describe the algorithm in detail and discuss the results of our implementation.

The main conjecture for CM elliptic curves at supersingular primes

At a prime of ordinary reduction, the Iwasawa “main conjecture” for elliptic curves relates a Selmer group to a p-adic L-function. In the supersingular case, the statement of the main conjecture is more complicated as neither the Selmer group nor the p-adic L-function is well-behaved. Recently Kobayashi discovered an equivalent formulation of the main conjecture at supersingular primes that is ...

Picard Schemes of Curves 0b92

Edwards curves and CM curves

Edwards curves are a particular form of elliptic curves that admit a fast, unified and complete addition law. Relations between Edwards curves and Montgomery curves have already been described. Our work takes the view of parameterizing elliptic curves given by their j-invariant, a problematic that arises from using curves with complex multiplication, for instance. We add to the catalogue the li...

Orders of CM elliptic curves modulo p with at most two primes

Nowadays the generation of cryptosystems requires two main aspects. First the security, and then the size of the keys involved in the construction and comunication process. About the former one needs a difficult mathematical assumption which ensures your system will not be broken unless a well known difficult problem is solved. In this context one of the most famous assumption underlying a wide...