Thirteenth Algorithmic Number Theory Symposium ANTS-XIII University of Wisconsin, Madison July 16 – 20, 2018 Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII) July 16 – 20, 2018

Constructing Picard curves with complex multiplication using the Chinese Remainder Theorem

Sonny Arora and Kirsten Eisenträger

Abstract: We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow one to work over smaller fields than the elliptic curve case. For a sextic CM-field K containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.

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