Sonny Arora

Sonny Arora

Software, Bioinformatics, and Machine Learning

Publications

Constructing Picard Curves with Complex Multiplication using the Chinese Remainder Theorem

Published in Thirteenth Algorithmic Number Theory Symposium (ANTS XIII) Proceedings, 2019

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.

Recommended citation: Constructing Picard Curves with Complex Multiplication, with Kirsten Eisenträger, Thirteenth Algorithmic Number Theory Symposium (ANTS XIII) Proceedings, pages 21-36, 2019. https://msp.org/obs/2019/2/p02.xhtml

The twisting Sato-Tate group of the curve $y^2 = x^8 - 14x^2 + 1$

Published in Mathematische Zeitschrift, 2018

We determine the twisting Sato–Tate group of the genus 3 hyperelliptic curve $y^2 = x^8 - 14x^2 + 1$ and show that all possible subgroups of the twisting Sato–Tate group arise as the Sato–Tate group of an explicit twist of $y^2 = x^8 - 14x^2 + 1$ . Furthermore, we prove the generalized Sato–Tate conjecture for the Jacobians of all $\mathbb{Q}$-twists of the curve $y^2 = x^8 - 14x^2 + 1$

Recommended citation: The twisting Sato-Tate group of the curve $y^2 = x^8 − 14x^22 + 1$, with Victoria Cantoral-Farfán, Aaron Landesman, Davide Lombardo, Jackson S. Morrow. Mathematische Zeitschrift (2018): 1-32. https://link.springer.com/article/10.1007/s00209-018-2049-6